GGrassmann angles between real or complexsubspaces
André L. G. Mandolesi ∗ August 13, 2020
Abstract
The Grassmann angle unifies and extends various concepts of angle be-tween subspaces, being defined for equal or different dimensions, in real orcomplex inner product spaces. It is an angle in Grassmann algebra, givesthe Fubini-Study distance in Grassmannians, and its cosine (squared, inthe complex case) describes volume contraction in orthogonal projections.The angle is asymmetric when dimensions are different, giving a quasi-pseudometric on the total Grassmannian, and measuring Hausdorff dis-tances between certain sets of simple multivectors. Odd features, presentin similar angles but overlooked, are explained, like the angle with an or-thogonal complement, and how the angle between complex subspaces isnot the same as for the underlying real spaces. We use them to get anobstruction on complex structures for pairs of real subspaces.
Keywords: angle between subspaces, Grassmann angle, exterior algebra,Grassmann algebra.
MSC:
Many distinct concepts of angle between subspaces appear in the literature[5, 7, 10, 12, 20, 28, 50, 52], with applications in geometry, linear algebra,and other areas as diverse as functional analysis [33], statistics [24] anddata mining [27]. For recent works on the subject, see [3, 11, 18, 23].The reason for such diversity is that, while in R a single angle de-scribes the relative position of two subspaces, in higher dimensions this isno longer true. With enough ‘wiggle room’ for more complex positionings,a list of principal angles [28] is needed for a full description. Still, it isoften convenient to synthesize in a single number whichever characteristicof the relation between subspaces is most important in a given applica-tion. This has led to many different concepts, each suitable for certainpurposes: minimal angle, Friedrichs angle, gap, etc. ∗ Instituto de Matemática e Estatística, Universidade Federal da Bahia, Av. Adhemar deBarros s/n, 40170-110, Salvador - BA, Brazil. E-mail: [email protected] a r X i v : . [ m a t h . M G ] A ug he literature on the subject can be quite confusing to the uninitiated.Many authors call their favorite concept the angle between subspaces, asif the term had a clear and unique meaning. Angle definitions are oftengiven without mention of their peculiarities, limitations, or alternatives.Equivalent concepts are given different names, or presented in ways whichseem unrelated. All this can lead even experienced researchers to error:for example, a certain angle is interpreted in [8, p.69] as a dihedral one,which is only true in very special cases [37].There is a kind of angle that keeps reappearing [11, 12, 15, 18, 23, 26,47, 50] under various names and different definitions, in terms of princi-pal angles, projections of volumes, determinants, Grassmann or Cliffordalgebras. Often it is defined, or results proven, only for real subspaces ofsame dimension. And sometimes it lies hidden behind formalisms whichdo not identify it [1, 42].This is the first of a set of articles which organize and extend thetheory on this angle, showing when the various approaches are equivalent,and unifying them in terms of a new Grassmann angle. Its propertiesare developed in a more systematic and detailed way, and results spreadthroughout the literature are brought together, with a formalism whichmakes them clearer and the proofs, in many cases, simpler.But this is not just a compilation of old results using new tools. Weintroduce a small but important modification in the angle definition, whichsimplifies its use with subspaces of distinct dimensions, allowing us to getnew and more general results. This change makes the angle asymmetricwhen dimensions are distinct, but in such unique feature lies much of itspower, as it reflects the dimensional asymmetry between the subspaces,and helps ‘keep track’ of which subspace is larger.Our work includes both real and complex spaces. With few exceptions[11, 50, 52], most research on the subject deals only with the real case.This is unfortunate, for the geometry of complex spaces is important inareas like quantum information and computation [4, 43, 44]. We showmost results remain valid in the complex case, but there are differences,mainly in how the angle relates to volume projections, which some authorssee as its defining property. This leads to complex Pythagorean theorems[39], and may have important consequences for quantum theory [40].This article is focused on fundamental properties of Grassmann angles.In [37] we relate them to products of Grassmann and Clifford algebras.These are used in [38] to get new formulas for computing the angles, andidentities for angles with certain families of subspaces, which are relatedto generalized Pythagorean theorems [39].Section 2 reviews concepts and results which will be needed. We alsointroduce a weaker, asymmetric, concept of orthogonality between sub-spaces, which will relate to our asymmetric angle.Section 3 presents the Grassmann angle and its basic properties. Werelate it to the Grassmann algebra and volume projections, and discussits asymmetry and other counterintuitive features.Section 4 studies its metric properties. We obtain a triangle inequalityvalid even for distinct dimensions, and relate the angle to the Fubini-Studydistance in Grassmannians. The total Grassmannian of all subspaces isgiven a new geometric structure, of quasi-pseudometric space, and the ngle is related to Hausdorff distances between certain sets.Section 5 shows the angle with the orthogonal complement of a sub-space has special properties, and is related to a product of sines studiedby some authors. Its relation to the angle with the original subspace isanalyzed in detail, being more complicated than an usual complement. Itgives an obstruction on complex structures satisfying a certain condition.Section 6 closes with a few remarks. Appendix A lists some resultsfrom [37, 38], and appendix B reviews similar angles. Throughout this article X is a n -dimensional vector space over R (realcase) or C (complex case), with an inner product (cid:104)· , ·(cid:105) (Hermitian productin the complex case). Unless otherwise indicated, whenever we refer tosubspaces or other linear algebra concepts it will be with respect to thesame field as X . In the complex case, for any subspace V ⊂ X we denoteits underlying real vector space by V R .A line L ⊂ X is a 1-dimensional subspace. Given v ∈ X , the sets R v = { cv : c ∈ R } and, in the complex case, C v = { cv : c ∈ C } are,respectively, the real line and the complex line of v , if v (cid:54) = 0 . When weuse R v in the complex case, it is to be understood as a real line in X R .Given subspaces V, W ⊂ X , we denote by Proj W and Proj VW the or-thogonal projections X → W and V → W , respectively. It will be convenient to extend the concept of angle between vectors toinclude the null vector.
Definition.
In the real case, the angle θ v,w ∈ [0 , π ] between nonzerovectors v, w ∈ X is defined by cos θ v,w = (cid:104) v, w (cid:105)(cid:107) v (cid:107)(cid:107) w (cid:107) . We also define θ , = 0 , θ ,v = 0 and θ v, = π .This definition of angles with the vector is unusual, and breaks thesymmetry between the vectors, as θ ,v (cid:54) = θ v, . As we will see, this isactually helpful, and the asymmetry is due to dim R (cid:54) = dim R v .For complex vectors there are several angle concepts [49], which de-scribe different relations between the vectors and the real or complexsubspaces they determine. Definition.
In the complex case, the (Euclidean) angle θ v,w ∈ [0 , π ] between nonzero vectors v, w ∈ X is defined by cos θ v,w = Re (cid:104) v, w (cid:105)(cid:107) v (cid:107)(cid:107) w (cid:107) . We also define θ , = 0 , θ ,v = 0 and θ v, = π .As Re (cid:104) v, w (cid:105) gives a real inner product in X R , this is the same as theangle between v and w considered as real vectors. Definition.
In the complex case, the
Hermitian angle γ v,w ∈ [0 , π ] be-tween nonzero vectors v, w ∈ X is defined by cos γ v,w = |(cid:104) v, w (cid:105)|(cid:107) v (cid:107)(cid:107) w (cid:107) . (1)We also define γ , = 0 , γ ,v = 0 and γ v, = π . Example 2.1. In C , for v = (1 , and w = ( i − , we have θ v,w = 120 ◦ and γ v,w = 45 ◦ .The Hermitian angle γ v,w is actually the angle v makes with the realplane C w (fig. 1), as the next proposition shows. Proposition 2.2.
Let v, w ∈ X and P = Proj C w . Then γ v,w = θ v,Pv .Proof. Assume (cid:104) v, w (cid:105) (cid:54) = 0 . As
P v = (cid:104) w,v (cid:105)(cid:107) w (cid:107) w (cid:54) = 0 and (cid:104) v, P v (cid:105) = (cid:107) P v (cid:107) > , cos θ v,Pv = Re (cid:104) v, P v (cid:105)(cid:107) v (cid:107)(cid:107) P v (cid:107) = |(cid:104) v, P v (cid:105)|(cid:107) v (cid:107)(cid:107) P v (cid:107) = |(cid:104) v, w (cid:105)|(cid:107) v (cid:107)(cid:107) w (cid:107) = cos γ v,w . Corollary 2.3.
Let v, w ∈ X .i) If P = Proj R w then (cid:107) P v (cid:107) = (cid:107) v (cid:107) · | cos θ v,w | .ii) If P = Proj C w then (cid:107) P v (cid:107) = (cid:107) v (cid:107) · cos γ v,w , in the complex case. Corollary 2.4.
Let v, w ∈ X be nonzero. Any nonzero v (cid:48) ∈ C v makeswith the real plane C w the same angle γ v,w . In other words, the real planes C v and C w are isoclinic [49], i.e. allvectors in one make the same angle with the other. In higher dimensions the relative position of two subspaces can be morecomplicated than in R , and many angle concepts have been used todescribe it. Here we present the minimal, maximal and principal angles,and specify some simple cases in which we drop the qualifiers and justtalk about the angle between subspaces. Definition.
The minimal angle θ min V,W ∈ [0 , π ] between nonzero subspaces V, W ⊂ X is θ min V,W = min { θ v,w : v ∈ V, w ∈ W, v (cid:54) = 0 , w (cid:54) = 0 } . learly, θ min V,W = θ min W,V . In the complex case, γ v,w can be used insteadof θ v,w in this definition, with the same result (by proposition 2.2).This angle is also called Dixmier angle , having been introduced in [7].A review of its properties can be found in [6]. Even if it is useful inmany situations, the information it provides is quite limited at times. Forexample, θ min V,W = 0 if, and only if, V ∩ W (cid:54) = { } , in which case it tells usnothing else about the relative position of V and W . Definition.
Given subspaces
V, W ⊂ X , the directed maximal angle θ max V,W ∈ [0 , π ] from V to W is θ max V,W = max v ∈ V min w ∈ W θ v,w . The maximal angle ˆ θ max V,W ∈ [0 , π ] between V and W is ˆ θ max V,W = max (cid:8) θ max V,W , θ max
W,V (cid:9) . As we describe below, a recursive search for minimal angles yields a listof principal angles. If dim V ≤ dim W then θ max V,W is their largest principalangle θ m , otherwise θ max V,W = π . Hence ˆ θ max V,W = θ m if dim V = dim W , and ˆ θ max V,W = π otherwise.The maximal angle seems to appear first in [35], and its sine givesthe gap or aperture between the subspaces, used in operator perturbationtheory [29]. Other notable angle is the Friedrichs angle [6, 10], which isthe first nonzero principal angle (or 0 if they are all null), and whose sinegives the minimal gap [33].
More detailed information about the relative position of subspaces requiresa list of principal angles [1, 11, 12, 52]. These were introduced by Jordan[28], being also called
Jordan or canonical angles . Definition.
Let
V, W ⊂ X be nonzero subspaces, p = dim V , q = dim W ,and m = min { p, q } . The principal angles ≤ θ ≤ . . . ≤ θ m ≤ π of V and W can be defined recursively as follows. For each i = 1 , . . . , m , let V i = { v ∈ V : (cid:104) v, e j (cid:105) = 0 ∀ j < i } ,W i = { w ∈ W : (cid:104) w, f j (cid:105) = 0 ∀ j < i } ,θ i be the minimal angle between V i and W i ,e i ∈ V i and f i ∈ W i be unit vectors such that θ e i ,f i = θ i . The e i ’s and f i ’s are principal vectors , and others are chosen for i > m to form corresponding orthonormal principal bases ( e , . . . , e p ) of V and ( f , . . . , f q ) of W .Principal vectors and angles can also be obtained via a singular valuedecomposition [11, 14]: for P = Proj VW , the e i ’s and f i ’s are orthonormaleigenvectors of P ∗ P and P P ∗ , respectively, and the square roots of the m largest eigenvalues of either give the cos θ i ’s.Note that θ = θ min V,W , and the number of null principal angles equals dim V ∩ W . The θ i ’s are uniquely defined, but the e i ’s and f i ’s are not(e.g., − e i and − f i are alternative principal vectors). R . The minimality condition implies (cid:104) e i , f j (cid:105) = 0 if i (cid:54) = j . And, in thecomplex case, γ e i ,f i = θ i for any ≤ i ≤ m . In fact, the characterizationvia singular value decomposition gives: Proposition 2.5.
Orthonormal bases ( e , . . . , e p ) of V and ( f , . . . , f q ) of W , and angles ≤ θ ≤ . . . ≤ θ m ≤ π , with m = min { p, q } , constitutecorresponding principal bases and angles of V and W if, and only if, (cid:104) e i , f j (cid:105) = δ ij cos θ i . (2) Corollary 2.6. If P = Proj W then P e i = (cid:40) f i · cos θ i if ≤ i ≤ m, if i > m. Corollary 2.7.
Proj VW is represented, in the principal bases, by a q × p diagonal matrix, with the cos θ i ’s in the diagonal. This construction has a nice geometric interpretation (fig. 2). The unitsphere of V projects to an ellipsoid in W . In the real case, for ≤ i ≤ m ,the e i ’s project onto its semi-axes, of lengths cos θ i , and the f i ’s pointalong them. The complex case is similar, but for each ≤ i ≤ m thereare two semi-axes of equal lengths, corresponding to projections of e i and i e i (in particular, this means each principal angle will be twice repeatedin the underlying real spaces). Example 2.8. In R , e = (1 , , , / √ , e = (0 , , , / √ , f =(1 , , , and f = (0 , , , are principal vectors for V = span( e , e ) and W = span( f , f ) , with principal angles θ = θ = 45 ◦ . Example 2.9. In C , e = (1 , , , / √ , e = (0 , , i , √ / , f =(1 + i , − i , , / and f = (0 , , i , are principal vectors for V =span C ( e , e ) and W = span C ( f , f ) , with principal angles θ = 45 ◦ and θ = 60 ◦ . In the underlying R , these subspaces have as principal vectors e = (1 , , , , , , , / √ , f = (1 , , , − , , , , / , ˜ e = i e = (0 , , , , , , , / √ , ˜ f = i f = ( − , , , , , , , / ,e = (0 , , , , , , √ , / , f = (0 , , , , , , , , ˜ e = i e = (0 , , , , − , , , √ / , ˜ f = i f = (0 , , , , − , , , , ith principal angles θ = ˜ θ = 45 ◦ and θ = ˜ θ = 60 ◦ .The importance of principal angles is due to the fact that, in general,all of them are necessary to fully describe the relative position of twosubspaces. The following result is proven in [54] for the real case, but theproof also works for the complex one. Proposition 2.10.
Given two pairs ( V, W ) and ( V (cid:48) , W (cid:48) ) of subspaces of X , with dim V (cid:48) = dim V and dim W (cid:48) = dim W , there is an orthogonaltransformation (unitary, in the complex case) taking V to V (cid:48) and W to W (cid:48) if, and only if, both pairs have the same principal angles. Dealing with a list of principal angles can be cumbersome, and inapplications one often uses minimal, maximal, Friedrichs, or whicheverangle better describes the properties of interest. But even if an angleconcept captures some important relation between the subspaces, it canmiss on other information. Unfortunately, angles between subspaces areoften presented without an explanation of their shortcomings, and thiscan lead to misunderstandings, specially since inequivalent concepts arein many cases simply called the angle .Anyway, the above result tells us any good concept of angle betweensubspaces must be a function of principal angles. Several concepts ofdistance between subspaces, or metrics in Grassmann manifolds, are alsogiven in terms of them [9, 19, 45, 54].
Some cases are simple enough that the relation between the subspaces canbe expressed in easier terms.
Definition.
We define the angle θ V,W between subspaces
V, W ⊂ X onlyin the following cases:i) θ { } , { } = 0 , θ { } ,V = 0 and, if V (cid:54) = { } , θ V, { } = π .ii) If their principal angles are (0 , . . . , , θ m ) then θ V,W = θ m .iii) If their principal angles are (0 , . . . , , π , . . . , π ) then θ V,W = π .Case (i) is for convenience. In (ii) there might be no 0’s, and θ m canbe 0. In (iii) there might be no 0’s, but there must be at least one π .In particular, θ V,W is defined if V or W is a line, or if V ∩ W hascodimension 1 in V or W , coinciding with the usual angle between lineand subspace, or the dihedral angle between hyperplanes. Proposition 2.11.
Given nonzero subspaces
V, W ⊂ X for which θ V,W is defined:i) θ V,W = θ W,V .ii) θ V,W = 0 ⇔ V ⊂ W or W ⊂ V .iii) θ V,W = π ⇔ there are nonzero v ∈ V , w ∈ W , such that (cid:104) v, w (cid:105) = 0 . Proposition 2.12.
For any v, w ∈ X :i) θ R v, R w = min { θ v,w , π − θ v,w } .ii) θ C v, C w = γ v,w , in the complex case. Corollary 2.13.
For any v, w ∈ X , if V = span( v ) and W = span( w ) then |(cid:104) v, w (cid:105)| = (cid:107) v (cid:107)(cid:107) w (cid:107) cos θ V,W . .3 Partial orthogonality We define a weaker, asymmetric, concept of orthogonality between sub-spaces, which will relate to orthogonality in the Grassmann algebra andto our asymmetric angle.
Definition.
Given subspaces
V, W ⊂ X , if there is a nonzero v ∈ V suchthat v ⊥ W then V is partially orthogonal to W , and we write V ⊥ / W (otherwise V (cid:54)⊥ / W ).This relation is not symmetric (e.g. for a plane V and a line W notperpendicular to it, V ⊥ / W but W (cid:54)⊥ / V ), and even partial orthogonalityboth ways does not imply V ⊥ W (e.g. perpendicular planes in R ).Some authors [1, 3] say V is completely inclined to W if V (cid:54)⊥ / W , and thesubspaces are totally inclined if V (cid:54)⊥ / W and W (cid:54)⊥ / V .The following results are immediate. Proposition 2.14.
For any subspaces
V, W, V (cid:48) , W (cid:48) ⊂ X :i) { } (cid:54)⊥ / W .ii) V ⊥ / { } ⇔ V (cid:54) = { } .iii) V ⊥ / W ⇔ V ∩ W ⊥ (cid:54) = { } .iv) V ⊥ / W ⇔ dim P ( V ) < dim V , where P = Proj W .v) V ⊥ / W ⇔ dim V > dim W or a principal angle is π .vi) V ⊥ / W and W ⊥ / V ⇔ a principal angle is π .vii) If dim V = dim W then V ⊥ / W ⇔ W ⊥ / V .viii) V ⊥ W ⇔ V = { } , W = { } or all principal angles are π .ix) V ⊥ W ⇒ V ⊥ / W if V (cid:54) = { } .x) If θ V,W is defined then V ⊥ / W ⇔ θ V,W = π .xi) V ⊥ W ⇒ θ V,W = π if V (cid:54) = { } . If dim V = 1 the converse holds.xii) V (cid:48) ⊂ V, V (cid:48) ⊥ / W ⇒ V ⊥ / W .xiii) W (cid:48) ⊂ W, V ⊥ / W ⇒ V ⊥ / W (cid:48) . Let V and W be nonzero subspaces, with principal bases ( e , . . . , e p ) and ( f , . . . , f q ) , respectively, and principal angles θ ≤ . . . ≤ θ m , where m = min { p, q } , and let P = Proj W . Then: Proposition 2.15. If V (cid:54)⊥ W then P ( V ) = span( f , . . . , f r ) , where r isthe number of principal angles which are not π , and the principal anglesof V and P ( V ) are θ , . . . , θ r . Corollary 2.16. If V (cid:54)⊥ / W then P ( V ) = span( f , . . . , f p ) , and the prin-cipal angles of V and P ( V ) are the same as those of V and W . This motivates the following decomposition.
Definition.
Given nonzero subspaces
V, W ⊂ X , with dim V = p ≤ q = dim W , and a principal basis ( f , . . . , f q ) of W with respect to V , a projective-orthogonal decomposition of W with respect to V is W = W P ⊕ W ⊥ , (3)where W P = span( f , . . . , f p ) and W ⊥ = span( f p +1 , . . . , f q ) are, respec-tively, projective and orthogonal subspaces of W with respect to V . orollary 2.17. If V (cid:54)⊥ / W then W P = P ( V ) and W ⊥ = W ∩ V ⊥ . If V ⊥ / W then P ( V ) (cid:40) W P and W ⊥ (cid:40) W ∩ V ⊥ , and in this case thedecomposition depends on the choice of principal basis. The
Grassmann or exterior algebra [41, 53, 55] of a vector space V is agraded algebra Λ V = (cid:76) mp =0 Λ p V , where m = dim V , equipped with an exterior product ∧ : Λ p V × Λ q V → Λ p + q V which is bilinear, associative,and alternating on elements of V .Elements of Λ V are multivectors , and an element of the p th exteriorpower Λ p V is a multivector of grade p , or p -vector , which, for ≤ p ≤ m ,is a linear combination of ( p -)blades v ∧ . . . ∧ v p , where v , . . . , v p ∈ V .Blades are also called decomposable or simple p -vectors. Also, Λ p V = { } for p > m , and Λ V = { scalars } ( R or C , depending on the case). Weconsider scalars as -blades.The alternativity of ∧ has important consequences, such as:i) If ν ∈ Λ p V and ω ∈ Λ q V then ν ∧ ω = ( − pq ω ∧ ν .ii) v ∧ . . . ∧ v p = 0 if, and only if, these vectors are linearly dependent.iii) Given a basis β of V , the set of all exterior products of p distinctelements of β is a basis of Λ p V .iv) dim Λ p V = (cid:0) mp (cid:1) .v) ( v , . . . , v m ) is a basis of V ⇔ Λ m V = span( v ∧ . . . ∧ v m ) .With these results, the next lemma, which we will use later, becomes alinear algebra exercise. Note that ν and ω need not be blades or p -vectors. Lemma 2.18.
Let
V, W ⊂ X be disjoint subspaces, ν ∈ Λ V and ω ∈ Λ W . Then ν ∧ ω = 0 ⇔ ν = 0 or ω = 0 . A nonzero p -blade ν = v ∧ . . . ∧ v p ∈ Λ p X is said to represent the p -dimensional subspace V = span( v , . . . , v p ) , which can also be charac-terized by Λ p V = span( ν ) or as the annihilator of ν , V = Ann( ν ) = { x ∈ X : x ∧ ν = 0 } . We also say that any ν ∈ Λ X represents V = { } .The inner product in Λ p X , for p > , is defined by extending linearly(sesquilinearly, in the complex case) the following formula for any blades ν = v ∧ . . . ∧ v p and ω = w ∧ . . . ∧ w p : (cid:104) ν, ω (cid:105) = det (cid:0) (cid:104) v i , w j (cid:105) (cid:1) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) v , w (cid:105) · · · (cid:104) v , w p (cid:105) ... . . . ... (cid:104) v p , w (cid:105) · · · (cid:104) v p , w p (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . For ν, ω ∈ Λ X we define (cid:104) ν, ω (cid:105) = ¯ ν · ω . The inner product is extended to Λ X by defining the Λ p X ’s as mutually orthogonal.The norm (cid:107) ν (cid:107) = (cid:112) (cid:104) ν, ν (cid:105) of a blade ν = v ∧ . . . ∧ v p gives, in the realcase, the p -dimensional volume of the parallelotope spanned by v , . . . , v p . i.e. with intersection { } . n fact, ν is usually represented by such (oriented) parallelotope, and two p -blades are equal if, and only if, their vectors span parallelotopes of same p -volume and orientation, in the same p -dimensional subspace.The next result relates orthogonal projections in X and Λ X . Proposition 2.19.
Given a subspace W ⊂ X , let P = Proj XW and P =Proj Λ X Λ W . Then P ν = P v ∧ . . . ∧ P v k for any ν = v ∧ . . . ∧ v k ∈ Λ X .Proof. Decomposing each v i as v i = P v i + v ⊥ i , with v ⊥ i ∈ W ⊥ , we get ν = P v ∧ . . . ∧ P v k + ν ⊥ , where ν ⊥ is a sum of blades, each with at leastone v ⊥ i , so that ν ⊥ ∈ (Λ W ) ⊥ .Given an orthogonal projection P : V → W , we use the same letterfor the orthogonal projection P : Λ V → Λ W , so that P ( v ∧ . . . ∧ v k ) = P v ∧ . . . ∧ P v k . In a slight abuse of terminology, we refer to P ν as theorthogonal projection of ν ∈ Λ X on W . Proposition 2.20.
Let
V, W ⊂ X be subspaces, with V represented by ν ,and P = Proj W . If V ⊥ / W then P ν = 0 , otherwise
P ν represents P ( V ) .Proof. We can assume ν = e ∧ . . . ∧ e p for a principal basis ( e , . . . , e p ) of V w.r.t. W . If V ⊥ / W then P e p = 0 , otherwise p ≤ dim W and, bycorollary 2.6, P ν = k · f ∧ . . . ∧ f p for some k (cid:54) = 0 . The result followsfrom corollary 2.16. Proposition 2.21.
Let
V, W ⊂ X be subspaces, with p = dim V (cid:54) = 0 .Then V ⊥ / W ⇔ Λ p V ⊥ Λ p W .Proof. Assume W (cid:54) = { } and let ν = e ∧ . . . ∧ e p for a principal basis ( e , . . . , e p ) of V w.r.t. W , and P = Proj W . By corollary 2.6 and (2), (cid:107) P ν (cid:107) = (cid:104) ν, P ν (cid:105) = (cid:104) e , P e (cid:105) · . . . · (cid:104) e p , P e p (cid:105) . As Λ p V = span( ν ) , byproposition 2.19 Λ p V ⊥ Λ p W ⇔ P ν = 0 ⇔ P e p = 0 ⇔ V ⊥ / W . Corollary 2.22.
Let
V, W ⊂ X be subspaces, with V (cid:54) = { } , P = Proj W ,and ν be a blade representing V . Then V ⊥ / W ⇔ P ν = 0 . Corollary 2.23.
Let ν and ω be nonzero blades of same grade, represent-ing subspaces V, W ⊂ X . Then (cid:104) ν, ω (cid:105) = 0 ⇔ V ⊥ / W . The following angle is designed to reflect many important properties ofthe relation between two subspaces. It is based on similar ones found inthe literature (see appendix B for a review), unifying and extending them,but also introducing a small but important modification.
Definition.
Let
V, W ⊂ X be nonzero subspaces, with principal an-gles θ , . . . , θ m , where m = min { dim V, dim W } . The Grassmann angle Θ V,W ∈ [0 , π ] of V with W is Θ V,W = (cid:40) arccos(cos θ · . . . · cos θ m ) if dim V ≤ dim W, π if dim V > dim W. We also define Θ { } , { } = 0 , Θ { } ,V = 0 and Θ V, { } = π . esides being defined for subspaces of any dimensions, in the real andcomplex cases, Θ V,W differs from similar angles in that we set it as π when dim V > dim W . We refer to this modification as asymmetrization ,as it causes the unusual asymmetry Θ V,W (cid:54) = Θ
W,V .Such change may seem artificial and hardly worth the trouble, but itleads to simpler and more general results, which hold regardless of whichsubspace is larger (see section 3.3.2 for a discussion). It is so relevant,in fact, that we defined the Grassmann angle as above just to make itexplicit, even though there are simpler definitions (e.g. proposition 3.4).Anyway, as principal angles are symmetric with respect to interchangeof V and W , so is Θ V,W when dimensions are equal.
Proposition 3.1. If dim V = dim W then Θ V,W = Θ
W,V . Though it may not be obvious that our definition gives a good angleconcept, Θ V,W does have many usual angle properties.
Proposition 3.2.
For any subspaces
V, W ⊂ X , and any v, w ∈ X :i) Θ V,W = 0 ⇔ V ⊂ W .ii) Θ V,W = π ⇔ V ⊥ / W .iii) If θ V,W is defined and dim V ≤ dim W then Θ V,W = θ V,W .iv) Θ R v, R w = min { θ v,w , π − θ v,w } .v) Θ C v, C w = γ v,w , in the complex case.vi) |(cid:104) v, w (cid:105)| = (cid:107) v (cid:107)(cid:107) w (cid:107) cos Θ V,W , if V = span( v ) and W = span( w ) . Proposition 3.3.
Let
U, V, W ⊂ X be subspaces, and P = Proj W . Then:i) Θ V,W = Θ
V,P ( V ) .ii) If U ⊥ V + W then Θ V,W = Θ
V,W ⊕ U .iii) If L ⊂ X is a line and v ∈ L then (cid:107) P v (cid:107) = (cid:107) v (cid:107) · cos Θ L,W .iv) If V = (cid:76) i V i , and all V i ’s are spanned by vectors of the same prin-cipal basis of V w.r.t. W , then cos Θ V,W = (cid:81) i cos Θ V i ,W .v) If V (cid:48) and W (cid:48) are the orthogonal complements of V ∩ W in V and W , respectively, then Θ V,W = Θ V (cid:48) ,W (cid:48) .vi) Θ T ( V ) ,T ( W ) = Θ V,W for any orthogonal transformation T : X → X (unitary, in the complex case).Proof. ( i ) If V (cid:54)⊥ / W it follows from corollary 2.16. Otherwise Θ V,W = π ,and so is Θ V,P ( V ) since dim P ( V ) < dim V . ( ii ) Proj W ⊕ U ( V ) = P ( V ) .Grassmann angles can be computed via projection matrices, as follows.Propositions 3.15 and A.4 give more general formulas. Proposition 3.4. If P is a matrix representing Proj VW in orthonormalbases of V and W then cos Θ V,W = det( ¯ P T P ) . If dim V = dim W then cos Θ V,W = | det P | .Proof. It is enough to consider principal bases of V and W , for which theresult follows from corollary 2.7.They also satisfy a spherical Pythagorean theorem (fig. 3). cos Θ V,U = cos Θ
V,P ( V ) · cos Θ P ( V ) ,U . Theorem 3.5.
Given subspaces
V, W ⊂ X and U ⊂ W , let P = Proj W .Then cos Θ V,U = cos Θ
V,P ( V ) · cos Θ P ( V ) ,U . Proof.
We can assume V (cid:54)⊥ / W , so dim P ( V ) = dim V . Let P , P and P be matrices representing Proj VU , Proj P ( V ) U and Proj VP ( V ) , respectively, inorthonormal bases. Then P = P P and, as P is square, det( ¯ P T P ) = | det P | det( ¯ P T P ) . The result follows from proposition 3.4. Corollary 3.6. Θ V,W ≤ Θ V,W (cid:48) for any subspace W (cid:48) ⊂ W , with equalityif, and only if, V ⊥ / W or P ( V ) ⊂ W (cid:48) , where P = Proj W . Corollary 3.7. Θ V,W = min W (cid:48) ⊂ W Θ V,W (cid:48) = min W (cid:48) ⊂ W, dim W (cid:48) = p Θ V,W (cid:48) , for p = dim V . Corollary 3.8.
Let p = dim V . If V (cid:54)⊥ / W then P ( V ) is the p -dimensionalsubspace of W with which V makes the smallest Grassmann angle. Proposition 3.9. Θ V,W ≥ Θ V (cid:48) ,W for any subspace V (cid:48) ⊂ V , with equalityif, and only if, V (cid:48) ⊥ / W or V (cid:48)⊥ ∩ V ⊂ W .Proof. If V ⊥ / W the inequality is trivial, as Θ V,W = π , and equality isequivalent to V (cid:48) ⊥ / W , which also happens if V (cid:48)⊥ ∩ V ⊂ W , as in this casethe nonzero v ∈ V which is orthogonal to W must be in V (cid:48) .If V (cid:54)⊥ / W then V (cid:48) (cid:54)⊥ / W , so dim P ( V ) = dim V and dim P ( V (cid:48) ) =dim V (cid:48) , where P = Proj W . Given orthonormal bases of V (cid:48) and P ( V (cid:48) ) ,complete them to orthonormal bases of V and P ( V ) . If P , P and P arematrices representing Proj VP ( V ) , Proj V (cid:48) P ( V (cid:48) ) and Proj V (cid:48)⊥ ∩ VP ( V (cid:48) ) ⊥ ∩ P ( V ) , respec-tively, in these bases, then P is block triangular of the form P = (cid:0) P B P (cid:1) ,for some matrix B . Thus cos Θ V,P ( V ) = | det P | = | det P | · | det P | ≤ | det P | = cos Θ V (cid:48) ,P ( V (cid:48) ) , with equality if, and only if, V (cid:48)⊥ ∩ V ⊂ P ( V (cid:48) ) ⊥ ∩ P ( V ) , which happensif, and only if, V (cid:48)⊥ ∩ V ⊂ W . Corollary 3.10. Θ V,W = max V (cid:48) ⊂ V Θ V (cid:48) ,W = max V (cid:48) ⊂ V min W (cid:48) ⊂ W Θ V (cid:48) ,W (cid:48) . .1 An angle in the Grassmann algebra In section 3.3.3 we show Θ V,W is not really an angle (in the usual sense)in X . Here we show it is in fact an angle in the Grassmann algebra Λ X . Lemma 3.11.
Given principal bases ( e , . . . , e p ) and ( f , . . . , f q ) of V and W , respectively, with p ≤ q , let ν = e ∧ . . . ∧ e p and ω = f ∧ . . . ∧ f p .Then Θ V,W = θ ν,ω and, in the complex case, also Θ V,W = γ ν,ω .Proof. By (2), (cid:104) ν, ω (cid:105) = (cid:81) pi =1 (cid:104) e i , f i (cid:105) = (cid:81) pi =1 cos θ i = cos Θ V,W , where θ , . . . , θ p are the principal angles, and (cid:107) ν (cid:107) = (cid:107) ω (cid:107) = 1 . Theorem 3.12.
Given any subspaces
V, W ⊂ X , let p = dim V andconsider Λ p V, Λ p W ⊂ Λ p X . Then Θ V,W = θ Λ p V, Λ p W .Proof. If V = { } both angles are 0. If p > q = dim W then Λ p W = { } and by definition θ Λ p V, { } = π . If < p ≤ q , let ( e , . . . , e p ) and ( f , . . . , f q ) be principal bases of V and W . Then { e ∧ . . . ∧ e p } and { f i ∧ . . . ∧ f i p : 1 ≤ i < . . . < i p ≤ q } are principal bases of Λ p V and Λ p W , and the only principal angle is θ e ∧ ... ∧ e p ,f ∧ ... ∧ f p .So Θ V,W is the angle, in Λ p X , between the line Λ p V and the subspace Λ p W . The importance of this theorem is that, in turning an angle betweensubspaces into an angle with a line, it creates a link between Grassmannangles and elliptic geometry (which, ultimately, is behind theorem 3.5).With corollary 2.13, we get: Corollary 3.13.
Let
V, W ⊂ X be subspaces of same dimension, repre-sented by blades ν, ω ∈ Λ p X . Theni) Θ V,W = min { θ ν,ω , π − θ ν,ω } , in the real case;ii) Θ V,W = γ ν,ω , in the complex case.iii) |(cid:104) ν, ω (cid:105)| = (cid:107) ν (cid:107)(cid:107) ω (cid:107) cos Θ V,W . Proposition A.1 ii generalizes ( iii ) for the case of distinct grades. Corollary 3.14.
Let
V, W ⊂ X be subspaces, with V represented by ablade ν , and P = Proj W . Then:i) Θ V,W = θ ν,Pν .ii) (cid:107) P ν (cid:107) = (cid:107) ν (cid:107) cos Θ V,W . In the real case, (cid:107) ν (cid:107) and (cid:107) P ν (cid:107) are the p -dimensional volumes of aparallelotope and its orthogonal projection on W (fig. 4), so cos Θ V,W measures how volumes contract when projecting from V to W , as onemight have guessed from proposition 3.4. Some authors [12, 18] takethis as the defining characteristic of the angle between subspaces, but insection 3.2 we show this changes in the complex case.We now get another formula for Θ V,W . Note that the matrix B repre-sents Proj VW , so this is a direct generalization of proposition 3.4. Formulasfor arbitrary bases are given in proposition A.4. (cid:107) P ν (cid:107) = (cid:107) ν (cid:107) · cos Θ V,W . Proposition 3.15. If ( v , . . . , v p ) is any basis of V , and ( w , . . . , w q ) isan orthonormal basis of W , then cos Θ V,W = det( ¯ B T B )det D , (4) where B = (cid:0) (cid:104) w i , v j (cid:105) (cid:1) and D = (cid:0) (cid:104) v i , v j (cid:105) (cid:1) .Proof. Let ν = v ∧ . . . ∧ v p and P = Proj W . Then (cid:107) ν (cid:107) = det D and (cid:107) P ν (cid:107) = det (cid:0) (cid:104) P v i , P v j (cid:105) (cid:1) = det (cid:16) (cid:88) k (cid:104) v i , w k (cid:105)(cid:104) w k , v j (cid:105) (cid:17) = det( ¯ B T B ) . Even if Θ V,W is not an (ordinary) angle in X itself, corollary 3.14 ii gave cos Θ V,W a geometric interpretation in X , at least in the real case, as thecontraction factor for p -volumes in V ( p = dim V ). We can reach thesame conclusion noting that each cos θ i is the factor by which lengths ina principal axis R e i ⊂ V contract when projected to W .In the complex case, the same holds for p -volumes in span R ( e , . . . , e p ) or span R (i e , . . . , i e p ) , but for p -volumes in V we must take cos Θ V,W ,as each cos θ i describes the contraction of 2 axes, R e i and R (i e i ) .So we have the following result, where in the complex case the Lebesguemeasures are taken in the underlying real spaces V R and W R (with twicethe complex dimension). Theorem 3.16.
Let
V, W ⊂ X be subspaces, p = dim V , P = Proj W , S ⊂ V be any Lebesgue measurable set, and | · | k be the k -dimensionalLebesgue measure. Then:i) | P ( S ) | p = | S | p · cos Θ V,W in the real case;ii) | P ( S ) | p = | S | p · cos Θ V,W in the complex case.Proof.
Consider first the real case, and assume dim V ≤ dim W (otherwisethe result is trivial, as |·| k = 0 on W for k > dim R W ). As P is linear, theratio of | P ( S ) | p to | S | p is independent of S . Take S to be the unit cubespanned by the principal vectors e , . . . , e p of V . By corollary 2.6, P ( S ) is the orthogonal parallelotope spanned by f cos θ , . . . , f p cos θ p , so that | P ( S ) | p = cos Θ V,W . he complex case is similar, but with S being the unit cube spannedby e , i e , . . . , e p , i e p , so each cos θ i is multiplied twice.As noted, some authors [12, 18] take ( i ) as characterizing the anglebetween real subspaces. ( ii ) shows such characterization must be adjustedin the complex case (see section 3.3.5 for a discussion).A comparison with corollary 3.14 ii suggests that to interpret (cid:107) ν (cid:107) in thecomplex case we should consider the square root of some volume of twicethe complex dimension. Indeed, one can check that the squared norm ofa complex p -blade ν = v ∧ . . . ∧ v p gives the p -dimensional volume ofthe parallelotope spanned by v , i v , . . . , v p , i v p .This theorem establishes a connection between Grassmann angles andprojection factors, defined as follows [39]. Definition.
For subspaces
V, W ⊂ X , let P = Proj W and k = dim R V .The projection factor of V on W is π V,W = | P ( S ) | k | S | k , where S is anyLebesgue measurable subset of V with | S | k (cid:54) = 0 . Corollary 3.17. π V,W = (cid:40) cos Θ V,W in the real case ;cos Θ V,W in the complex case.
In section 3.3.5 this will clarify the relation between Θ V,W and Θ V R ,W R . Besides its nice properties, Θ V,W also has some strange characteristics.Some of them affect similar angles as well, but are not usually discussed,and this can lead to errors. Still, if properly used these features can beuseful. For example, in section 5.2 we use them to get an obstruction oncomplex structures satisfying a certain condition.
By proposition 2.10, in general Θ V,W does not describe completely therelative position of V and W . But it was never supposed to. Its purposeis just to capture, in a single number, important properties of such relativeposition.Anyone who works with angles between subspaces is used to this, butas it goes against most people’s intuition regarding angles, a warningmay help avoid misunderstandings: even for pairs of subspaces with equaldimensions, having equal Grassmann angles is no guarantee that there isan orthogonal transformation (unitary, in the complex case) taking onepair to the other. The Grassmann angle is asymmetric by definition: if a line V makes a ◦ angle with a plane W then Θ V,W = 20 ◦ , but we chose to let Θ W,V = 90 ◦ .This choice is unusual, as appendix B shows: authors who considersubspaces of different dimensions take the angle between the smaller oneand its projection on the other, or some equivalent construction. Θ V,W = Θ
W,V ⊕ W ⊥ , if dim V ≤ dim W . But there are good reasons for it. For example, if p = dim V > dim W we have | P ( S ) | p = 0 in theorem 3.16 i , and Λ p W = { } in theorem 3.12.In both cases, consistency requires that we set Θ V,W = π .Also, this asymmetry, far from being a problem, is quite beneficial.It reflects the asymmetry between subspaces of different dimensions, andleads to simpler proofs and more general results. For example, it gives usthe equivalence Θ V,W = π ⇔ V ⊥ / W ⇔ dim V > dim Proj W V , used insection 4 to extend a triangle inequality for subspaces of different dimen-sions, and give the total Grassmannian a new geometric structure. It isalso behind many results of section 5, and the formulas for the contractionand exterior product of blades in proposition A.1.Anyway, the projective-orthogonal decomposition allows us to par-tially interchange V and W in the Grassmann angle (fig. 5), if necessary. Proposition 3.18. If dim V ≤ dim W then Θ V,W = Θ
W,V ⊕ W ⊥ , with W ⊥ as in (3) .Proof. If V and W have principal bases ( e , . . . , e p ) and ( f , . . . , f q ) , re-spectively, with principal angles θ , . . . , θ p , then W and W ⊥ ⊕ V haveprincipal bases ( f p +1 , . . . , f q , f , . . . , f p ) and ( f p +1 , . . . , f q , e , . . . , e p ) , re-spectively, with principal angles , . . . , , θ , . . . , θ p .Symmetrized versions of Θ V,W can also be useful at times.
Definition.
The min- and max-symmetrized Grassmann angles are ˇΘ V,W = min { Θ V,W , Θ W,V } , ˆΘ V,W = max { Θ V,W , Θ W,V } . The min-symmetrized angle may seem more natural than Θ V,W , as itagrees, for example, with how one usually talks about the angle betweena plane and a line (in the above example, ˇΘ V,W = ˇΘ
W,V = 20 ◦ ). In fact,many authors adopt it implicitly. But this angle loses information aboutprincipal vectors in the larger space not corresponding to any principalangle, and this leads to worse properties. A simple example (2 lines andthe plane containing them) shows ˇΘ V,W does not even satisfy a triangle nequality if dimensions are different. Still, ˇΘ V,W is related to the dot and(Hestenes) inner products of geometric algebra (proposition A.2).The max-symmetrized one may seem like an even worse choice, as for aline and a plane, or whenever dimensions are different, ˆΘ V,W = π . But itdoes have its uses, as in corollary 4.9 and propositions 4.15, 4.19 and A.2. X Another counterintuitive feature of the Grassmann angle is the followingconsequence of proposition 3.9.
Corollary 3.19.
Given nonzero subspaces
V, W ⊂ X , there is a line L ⊂ V with θ L,W = Θ
V,W if, and only if, V ⊥ / W or there is at most onenonzero principal angle between V and W . So, in general, Θ V,W does not correspond to any angle between a linein V and its projection. This shows that, as noted before, it is not reallyan (ordinary) angle in X . In fact, it follows from the definition that Θ V,W is, usually, strictly greater than all principal angles.
Proposition 3.20. If θ m is the largest principal angle of V and W then Θ V,W ≥ θ m , with equality if, and only if, θ m = π or all other principalangles are . Example 3.21.
Let V and W be as in example 2.8. All vectors in V make a ◦ angle with W , but Θ V,W = 60 ◦ . The next example reveals another strange feature: the Grassmann anglewith the orthogonal complement of a subspace is not the usual complementof an angle.
Example 3.22.
For V and W as in example 2.8 again, we have W ⊥ =span( f , f ) for f = (0 , , , and f = (0 , , , , and both principalangles for V and W ⊥ are also ◦ . Thus Θ V,W ⊥ = 60 ◦ (cid:54) = 90 ◦ − Θ V,W .In particular, sin Θ
V,W (cid:54) = cos Θ
V,W ⊥ , i.e. the sine of Θ V,W does notcorrespond to a projection on W ⊥ . In section 5 we discuss this in detail.For now, note that Grassmann angles are angles in the Grassmann algebra,and in general Λ( W ⊥ ) (cid:54) = (Λ W ) ⊥ . From a metric perspective, complex spaces are the same as their underly-ing real ones, so one might expect to have Θ V,W = Θ V R ,W R in the complexcase. But this is not valid, as the following example shows. Example 3.23.
For the complex subspaces V and W of example 2.9, Θ V,W = arccos( √ · ) ∼ = 69 . ◦ , but for their underlying real vector spaces Θ V R ,W R = arccos( √ · √ · · ) ∼ = 82 . ◦ .Since V R and W R always have the same principal angles as V and W ,but twice repeated, Θ V R ,W R and Θ V,W are related as follows. roposition 3.24. In the complex case, cos Θ V R ,W R = cos Θ V,W for anysubspaces
V, W ⊂ X . Corollary 3.25.
In the complex case, Θ V R ,W R ≥ Θ V,W for any subspaces
V, W ⊂ X , with equality if, and only if, V ⊂ W or V ⊥ / W . This too can be understood in the Grassmann algebra, since the alge-bras over R and C are different, with Λ( X R ) and (Λ X ) R not even havingthe same dimension.Also, the Lebesgue measures used to define projection factors refer, inthe complex case, to the underlying real spaces, so that π V,W = π V R ,W R .Thus corollary 3.17 shows Θ V,W and Θ V R ,W R are different ways to encodeinformation about the same projection factor.One might say the angle between complex subspaces ought to be de-fined as equal to Θ V R ,W R , but this has many inconveniences. Workingwith underlying real spaces doubles the dimension and does not explorethe symmetries of the complex structure, leading to the redundancy ofeach principal angle appearing twice. Also, this would complicate mat-ters, conflicting with other definitions [11, 50], and leading to differentformulas than in the real case. Proposition 3.2 vi , for example, wouldbecome |(cid:104) v, w (cid:105)| = (cid:107) v (cid:107)(cid:107) w (cid:107) (cid:112) cos Θ V R ,W R in the complex case. In this section we prove a triangle inequality for Grassmann angles, andgive conditions for equality. The angles are related to the Fubini-Studydistance in Grassmannians, to Hausdorff distances between total Grass-mannians or sets of unit blades, and endow the total Grassmannian witha quasi-pseudometric.
For subspaces U , V and W of same dimension, theorem 3.12 translates thespherical triangle inequality of elliptic geometry (or, in the complex case,a triangle inequality for Hermitian angles) into Θ U,W ≤ Θ U,V + Θ
V,W .The asymmetrization allows us to extend this for distinct dimensions.If Θ U,V = π or Θ V,W = π the inequality is trivial, and if not then U , P V ( U ) and P P V ( U ) have equal dimensions, where P = Proj W and P V = Proj V . By corollary 3.6 and proposition 3.9, Θ U,W ≤ Θ U,PP V ( U ) ≤ Θ U,P V ( U ) + Θ P V ( U ) ,PP V ( U ) = Θ U,V + Θ P V ( U ) ,W ≤ Θ U,V + Θ
V,W .We give below a more detailed proof, to help us get conditions forequality later on.
Theorem 4.1. Θ U,W ≤ Θ U,V + Θ
V,W for any subspaces
U, V, W ⊂ X .Proof. Let P = Proj W and P V = Proj V . The result is trivial unless Θ U,W (cid:54) = 0 , Θ U,V (cid:54) = π and Θ V,W (cid:54) = π , which implies Θ P V ( U ) ,W (cid:54) = π .And if P V ( U ) ⊂ W then, by corollary 3.6, Θ U,W ≤ Θ U,P V ( U ) = Θ U,V ,so we can also assume Θ P V ( U ) ,W (cid:54) = 0 . With these conditions, we have p = dim U = dim P V ( U ) ≤ dim V ≤ dim W . Θ U,W ≤ Θ U,V + Θ
V,W . Let µ and ν = P V µ/ (cid:107) P V µ (cid:107) be unit blades representing U and P V ( U ) ,respectively, and ω ν , ω µ ∈ Λ p W , ω ⊥ ν , ω ⊥ µ ∈ (Λ p W ) ⊥ be given by ω ν = P ν (cid:107)
P ν (cid:107) , ω µ = (cid:40) Pµ (cid:107) Pµ (cid:107) if Θ U,W (cid:54) = π ,ω ν if Θ U,W = π ,ω ⊥ ν = ν − P ν (cid:107) ν − P ν (cid:107) , ω ⊥ µ = µ − P µ (cid:107) µ − P µ (cid:107) . With corollary 3.14 ii we obtain µ = ω µ · cos Θ U,W + ω ⊥ µ · sin Θ U,W ,ν = ω ν · cos Θ P V ( U ) ,W + ω ⊥ ν · sin Θ P V ( U ) ,W . (5)As (cid:104) µ, ν (cid:105) > , corollary 3.13 iii gives cos Θ U,P V ( U ) = (cid:104) ω µ , ω ν (cid:105) cos Θ U,W cos Θ P V ( U ) ,W + (cid:104) ω ⊥ µ , ω ⊥ ν (cid:105) sin Θ U,W sin Θ P V ( U ) ,W ≤ cos(Θ U,W − Θ P V ( U ) ,W ) , (6)and therefore Θ U,V = Θ
U,P V ( U ) ≥ Θ U,W − Θ P V ( U ) ,W ≥ Θ U,W − Θ V,W , byproposition 3.9.Figure 6 illustrates the inequality for lines U and V and a plane W .Taking U, V ⊂ W we see that Θ U,W ≥ | Θ U,V − Θ V,W | is not always valid.The asymmetry of the Grassmann angle prevents us from getting this, asusual, from the triangle inequality. Instead, we have: Corollary 4.2.
For any subspaces
U, V, W ⊂ X , Θ U,W ≥ max { Θ U,V − Θ W,V , Θ V,W − Θ V,U } . Corollary 4.3.
For any subspaces
U, V, W ⊂ X of same dimension, Θ U,W ≥ | Θ U,V − Θ V,W | . To get conditions for equality in theorem 4.1, we need a lemma.
Lemma 4.4.
Let µ, ν, ω ∈ Λ p X be nonzero blades representing distinctsubspaces U, V, W ⊂ X , respectively, and A = U ∩ V ∩ W .If ν = aµ + bω for some nonzero a, b ∈ C then dim A = p − andthere are u ∈ U , v ∈ V , w ∈ W and a unit blade ξ ∈ Λ p − A such that v = au + bw , µ = u ∧ ξ , ν = v ∧ ξ and ω = w ∧ ξ .Moreover, u , v and w can be chosen to be in any given complement of A in X . If they are in A ⊥ then (cid:104) u, w (cid:105) = (cid:104) µ, ω (cid:105) . roof. For any x ∈ U ∩ W we have x ∧ ν = x ∧ ( aµ + bω ) = 0 , so x ∈ V .As µ and ω are also linear combinations of the other blades, repeatingthis argument we get U ∩ V = U ∩ W = V ∩ W = A . Let r = dim A < p ,and X (cid:48) be any complement of A in X . Then U (cid:48) = U ∩ X (cid:48) , V (cid:48) = V ∩ X (cid:48) and W (cid:48) = W ∩ X (cid:48) are disjoint ( p − r ) -dimensional complements of A in U, V and W , respectively.Given an unit ξ ∈ Λ r A , we have µ = µ (cid:48) ∧ ξ , ν = ν (cid:48) ∧ ξ and ω = ω (cid:48) ∧ ξ for some blades µ (cid:48) ∈ Λ p − r U (cid:48) , ν (cid:48) ∈ Λ p − r V (cid:48) and ω (cid:48) ∈ Λ p − r W (cid:48) . Then ν = aµ + bω means ( ν (cid:48) − aµ (cid:48) − bω (cid:48) ) ∧ ξ = 0 , and since ν (cid:48) − aµ (cid:48) − bω (cid:48) ∈ Λ X (cid:48) lemma 2.18 gives ν (cid:48) = aµ (cid:48) + bω (cid:48) .For any nonzero vectors u (cid:48) ∈ U (cid:48) and w (cid:48) ∈ W (cid:48) we have ν (cid:48) ∧ u (cid:48) ∧ w (cid:48) =( aµ (cid:48) + bω (cid:48) ) ∧ u (cid:48) ∧ w (cid:48) = 0 . Since U (cid:48) and V (cid:48) are disjoint, ν (cid:48) ∧ u (cid:48) (cid:54) = 0 ,thus w (cid:48) ∈ V (cid:48) ⊕ span( u (cid:48) ) . As w (cid:48) (cid:54)∈ V (cid:48) and u (cid:48) was arbitrary, this implies dim U (cid:48) = 1 , so r = p − . Thus µ (cid:48) , ν (cid:48) and ω (cid:48) are vectors u ∈ U (cid:48) , v ∈ V (cid:48) and w ∈ W (cid:48) , respectively, with v = au + bw .In case X (cid:48) = A ⊥ , we have (cid:104) µ, ω (cid:105) = (cid:104) u ∧ ξ, w ∧ ξ (cid:105) = (cid:104) u, w (cid:105) · (cid:107) ξ (cid:107) . Proposition 4.5.
Given subspaces
U, V, W ⊂ X , Θ U,W = Θ
U,V + Θ
V,W (7) if, and only if, one of the following conditions is satisfied:i) U ⊂ V and either U ⊥ / W or U ⊥ ∩ V ⊂ W ;ii) V ⊂ W and either U ⊥ / W or P ( U ) ⊂ V , where P = Proj W ;iii) There are nonzero u, w ∈ X with (cid:104) u, w (cid:105) ≥ , v = au + bw with a, b > , and subspaces A, B, C ⊂ X orthogonal to span( u, w ) andto each other, such that U = span( u ) ⊕ A,V = span( v ) ⊕ A ⊕ B,W = span( w ) ⊕ A ⊕ B ⊕ C. Moreover, in this last case θ u,v = Θ U,V , θ u,w = Θ U,W and θ v,w = Θ V,W .Proof. ( i ) and ( ii ) correspond, by proposition 3.9 and corollary 3.6, towhen Θ U,V = 0 or Θ V,W = 0 , and the other two angles are equal.Suppose (7) holds, but ( i ) and ( ii ) do not. Then Θ U,W , Θ U,V , Θ V,W (cid:54) = 0 and Θ U,V , Θ V,W (cid:54) = π , which also implies Θ P V ( U ) ,W (cid:54) = π . Since Θ U,W ≤ Θ U,P V ( U ) + Θ P V ( U ) ,W ≤ Θ U,V + Θ
V,W , we get Θ U,W = Θ
U,P V ( U ) + Θ P V ( U ) ,W , (8)so that Θ U,W > Θ P V ( U ) ,W , since Θ U,P V ( U ) = Θ U,V > . We also get Θ P V ( U ) ,W = Θ V,W , which, by proposition 3.9 and since P V ( U ) (cid:54)⊥ / W ,gives P V ( U ) ⊥ ∩ V ⊂ W . As V (cid:54)⊂ W , this implies Θ P V ( U ) ,W (cid:54) = 0 .Let µ, ν, ω µ , ω ν , ω ⊥ µ , ω ⊥ ν be the unit p -blades in the proof of theorem 4.1.As (8) implies equality in (6), Θ U,W (cid:54) = 0 , and Θ P V ( U ) ,W (cid:54) = 0 or π , we get ω µ = ω ν (by definition, if Θ U,W = π ) and ω ⊥ µ = ω ⊥ ν . So (5) becomes ν = ω µ · cos Θ P V ( U ) ,W + µ − P µ (cid:107) µ − P µ (cid:107) · sin Θ P V ( U ) ,W . s P µ = ω µ · cos Θ U,W and (cid:107) µ − P µ (cid:107) = sin Θ
U,W , we obtain ν = ω µ · sin(Θ U,W − Θ P V ( U ) ,W )sin Θ U,W + µ · sin Θ P V ( U ) ,W sin Θ U,W , so that ν = aµ + b ω µ with a, b > .Let A = U ∩ P V ( U ) ∩ K , where K = Ann( ω µ ) ⊂ W . As U (cid:54)⊂ V , U (cid:54)⊂ W and P V ( U ) (cid:54)⊂ W , the subspaces U , P V ( U ) and K are distinct. Lemma 4.4gives nonzero vectors u ∈ U ∩ A ⊥ , v ∈ P V ( U ) ∩ A ⊥ and w ∈ K ∩ A ⊥ suchthat v = au + bw , (cid:104) u, w (cid:105) = (cid:104) µ, ω µ (cid:105) ≥ , and U = span( u ) ⊕ A,P V ( U ) = span( v ) ⊕ A,K = span( w ) ⊕ A. Let B = P V ( U ) ⊥ ∩ V ⊂ W . Then V = P V ( U ) ⊕ B , with B orthogonalto A , v , u , w (as w ∈ span( u, v ) ) and K .Let C = ( K ⊕ B ) ⊥ ∩ W . Then W = K ⊕ B ⊕ C , with C orthogonalto B , K , w and A . If Θ U,W (cid:54) = π then K = Ann( P µ ) = P ( U ) , and if Θ U,W = π then u ⊥ W , as A ⊂ W . In either case, C is orthogonal to u .So ( iii ) is satisfied. Under its conditions, it is immediate, in the realcase, that Θ U,W = θ u,w , Θ U,V = θ u,v , Θ V,W = θ v,w and θ u,w = θ u,v + θ v,w .In the complex case, we must use (cid:104) u, w (cid:105) ≥ to get Θ U,W = γ u,w = θ u,w ,and also a, b > for the other angles and (7).When dimensions are equal the conditions become simpler. Corollary 4.6.
Given subspaces
U, V, W ⊂ X of same dimension p , Θ U,W = Θ
U,V + Θ
V,W if, and only if, V = U or W , or dim( U ∩ V ∩ W ) = p − and there are nonzero u ∈ U , v ∈ V and w ∈ W in an isotropic real plane orthogonal to U ∩ V ∩ W , with R v in the smaller pair of anglesformed by R u and R w . Note that, in the complex case, such real plane must be orthogonalwith respect to the Hermitian product, not the underlying real product.
Example 4.7. If α, β and γ are the dihedral angles between the faces ofa nondegenerate trihedral angle (fig. 7) then min { α, ◦ − α } < β + γ .This can also be obtained from the fact that the sum of the angles of anondegenerate spherical triangle is strictly greater than ◦ . In a complex projective space, the Fubini-Study distance [13] (geodesicdistance for the Fubini-Study metric) between complex lines equals theangle between them, as given in proposition 2.12 ii . In a real projectivespace, with the round metric (i.e. as a quotient of the unit sphere), thedistance between real lines is also their angle, now as in proposition 2.12 i . A real subspace R ⊂ X is isotropic if (cid:104) u, w (cid:105) ∈ R ∀ u, w ∈ R . Other terms are totally real [13] or antiholomorphic [48], although some authors use these differently. min { α, ◦ − α } < β + γ . We will refer to the round metric as a Fubini-Study metric as well, and inboth cases we call the angle between lines their Fubini-Study distance dist FS (cid:0) span( v ) , span( w ) (cid:1) = arccos |(cid:104) v, w (cid:105)|(cid:107) v (cid:107)(cid:107) w (cid:107) . Given a vector space V and ≤ k ≤ dim V , the Grassmannian G k ( V ) is the set of k -dimensional subspaces of V , which, given the appropriatedifferentiable structure, is a compact manifold [16, 34]. The full or totalGrassmannian of all subspaces of V is G ( V ) = ∪ pk =0 G k ( V ) .For each ≤ p ≤ n = dim X , the Plücker embedding of G p ( X ) into theprojective space P (Λ p X ) maps each p -dimensional subspace V ⊂ X to thecorresponding line Λ p V . This also gives an embedding G ( X ) (cid:44) → P (Λ X ) .We identify G p ( X ) and G ( X ) with their images, so that dist FS gives thema metric. Since distinct Λ p X ’s are mutually orthogonal, points in different G p ( X ) ’s are at a fixed distance of π , and in this metric the geometry of G ( X ) is reduced to a disjoint union of the G p ( X ) ’s.Theorem 3.12 means the Grassmann angle between subspaces of samedimension equals the Fubini-Study distance between them. A similarresult, for the real case, appears in [9]. Theorem 4.8. If V, W ∈ G p ( X ) then Θ V,W = dist FS ( V, W ) . Corollary 4.9.
Given any subspaces
V, W ∈ G ( X ) , dist FS ( V, W ) = ˆΘ
V,W = (cid:40) Θ V,W if dim V = dim W, π if dim V (cid:54) = dim W. Proposition 4.10.
Given V ∈ G p ( X ) and W ∈ G ( X ) , consider thesubsets G p ( W ) , G ( W ) ⊂ P (Λ X ) . Then Θ V,W = dist FS ( V, G p ( W )) = dist FS ( V, G ( W )) . Proof. Θ V,W = min W (cid:48) ∈ G p ( W ) Θ V,W (cid:48) = min W (cid:48) ∈ G p ( W ) dist FS ( V, W (cid:48) ) , by corol-lary 3.7, and by the one above we may take the minimum over G ( W ) .Geodesics for the Fubini-Study metric are, in RP n , quotients of greatcircles of the sphere S n , and in CP n they are great circles in the complexprojective line CP ∼ = S determined by any two points. The equality U,V + Θ
V,W = Θ
U,W means that, in the Plücker embedding, V lies in aminimal geodesic between U and W in P (Λ p X ) . With corollary 4.6, thisimplies the following result. Proposition 4.11.
Given distinct points
U, W ∈ G p ( X ) , a minimalgeodesic in P (Λ p X ) connecting them intercepts G p ( X ) at some other pointif, and only if, dim( U ∩ W ) = p − . When this happens:i) Any geodesic through U and W lies entirely in G p ( X ) , and is givenby V ( t ) = ( U ∩ W ) ⊕ span( u · cos t + w · sin t ) , with t ∈ [0 , π ) , for somenonzero u ∈ U and w ∈ W such that u, w ⊥ U ∩ W and (cid:104) u, w (cid:105) ∈ R .ii) In the complex case, the complex projective line determined by U and W lies in G p ( X ) , and its elements can be described by V ( t, ϕ ) =( U ∩ W ) ⊕ span( u · cos t + w · e iϕ sin t ) , with t ∈ [0 , π ) and ϕ ∈ [0 , π ) ,for some nonzero u ∈ U and w ∈ W such that u, w ⊥ U ∩ W . Example 4.12.
To see why u and w must be in an isotropic plane, let U, W, V ⊂ X = C be spanned by u = (1 , , w = (1 , √ / and v = u + w ,respectively. Then γ u,w = 60 ◦ and γ u,v = γ v,w = 30 ◦ , so V lies in thegeodesic segment between U and W in P ( X ) = CP .Taking u = ( i, instead, we get γ u,w = 60 ◦ and γ u,v = γ v,w ∼ = 38 ◦ ,so U , V and W are not in the same geodesic of P ( X ) anymore. However, θ u,w = 90 ◦ and θ u,v = θ v,w = 45 ◦ , so span R ( v ) does lie in the geodesicsegment between span R ( u ) and span R ( w ) in P ( X R ) = RP . Definition.
A subset C of a Riemannian manifold M is weakly convex if any two points of C are connected by a minimal geodesic segment of M which is contained in C . Corollary 4.13. If < p < dim X − then G p ( X ) is not weakly convexin P (Λ p X ) . A similar result is given in [26], but erroneously using a sphere insteadof projective space, and without the restriction p < n − , which is neededas the image of G n − ( X ) is the whole P (Λ n − X ) . The Grassmann angle is defined in the whole G ( X ) , but it only gives ametric in each G p ( X ) , as for subspaces of different dimensions it lackssymmetry and the identity of indiscernibles (i.e. Θ V,W = 0 (cid:54)⇒ V = W ).Still, it gives G ( X ) a weaker kind of metric [30, 31, 36]. Definition. A quasi-pseudometric on a non-empty set M is a function d : X × X → [0 , ∞ ) such that, for all x, y, z ∈ M ,i) d ( x, x ) = 0 ;ii) d ( x, z ) ≤ d ( x, y ) + d ( y, z ) .It satisfies the T condition if d ( x, y ) = 0 = d ( y, x ) implies x = y .If d is a quasi-pseudometric with the T condition then D ( x, y ) =max { d ( x, y ) , d ( y, x ) } gives a metric. This connects corollary 4.9 with thenext result, which follows from Proposition 3.2 i and theorem 4.1. We use the terminology of [51, 2]. Some authors require the geodesic to be unique. heorem 4.14. The total Grassmannian G ( X ) , with distances given bythe Grassmann angle, is a quasi-pseudometric space with the T condition. Hausdorff distances, which are similarity measures used in computervision and pattern matching [25, 32], provide another example of a quasi-pseudometric.
Definition.
Let S and T be non-empty compact subsets of a metric space ( M, d ) . The directed Hausdorff distance from S to T is h ( S, T ) = max s ∈ S d ( s, T ) = max s ∈ S min t ∈ T d ( s, t ) , and the Hausdorff distance between S and T is H ( S, T ) = max { h ( S, T ) , h ( T, S ) } . So h ( S, T ) is the largest distance from points in S to their closest pointsin T , or, equivalently, the smallest (cid:15) such that a closed (cid:15) -neighborhood of T contains S . All points in either set are within distance H ( S, T ) fromthe other set. H is a metric in the set of non-empty compact subsets of M , but h is only a quasi-pseudometric with the T condition. In partic-ular, it satisfies h ( R, T ) ≤ h ( R, S ) + h ( S, T ) for any non-empty compactsubsets R, S, T ⊂ M . For any x ∈ M , as h ( { x } , T ) = d ( x, T ) this implies d ( x, T ) ≤ d ( x, S ) + h ( S, T ) .We now relate Grassmann angles to Hausdorff distances between cer-tain sets, and use this inequality to get others. Proposition 4.15. h ( G ( V ) , G ( W )) = Θ V,W and H ( G ( V ) , G ( W )) =ˆΘ V,W for any subspaces
V, W ⊂ X , considering G ( V ) , G ( W ) ⊂ P (Λ X ) .Proof. By corollary 3.10 and corollary 4.9, Θ V,W = max V (cid:48) ⊂ V min W (cid:48) ⊂ W Θ V (cid:48) ,W (cid:48) =max V (cid:48) ∈ G ( V ) min W (cid:48) ∈ G ( W ) dist FS ( V (cid:48) , W (cid:48) ) . Corollary 4.16. dist FS ( U, G ( W )) ≤ dist FS ( U, G ( V )) + Θ V,W for anysubspaces
U, V, W ∈ G ( X ) ⊂ P (Λ X ) , We denote by S ( V ) the unit sphere in an inner product space V . In S (Λ V ) , let d S be the distance function, [Λ k V ] = { unit blades in Λ k V } and [Λ V ] = ∪ k [Λ k V ] = { unit blades in Λ V } . These sets are compact,since the orthogonal group O ( V ) (unitary group U ( V ) , in the complexcase) acts transitively on [Λ k V ] . Proposition 4.17.
Let V ∈ G p ( X ) and W ∈ G q ( X ) , with p ≤ q . Forany ν ∈ [Λ p V ] we have d S ( ν, [Λ W ]) = d S ( ν, [Λ p W ]) = Θ V,W .Proof.
Let P = Proj W . By corollary 3.14 i and proposition 2.19, and since d S ( ν, ω ) = π whenever ν and ω have distinct grades, Θ V,W = θ ν,Pν =min ω ∈ [Λ p W ] θ ν,ω = min ω ∈ [Λ p W ] d S ( ν, ω ) = min ω ∈ [Λ W ] d S ( ν, ω ) . Proposition 4.18.
Let V ∈ G p ( X ) and W ∈ G q ( X ) , with p ≤ q . Forany ν ∈ [Λ V ] we have d S ( ν, [Λ W ]) ≤ Θ V,W . This inequality is strict forany ν ∈ [Λ k V ] with k < p , unless V ⊥ / W or V ∩ W (cid:54) = { } . Some authors call h the (one-sided) Hausdoff distance, and H the bidirectional or two-sided Hausdorff distance. roof. Any ν ∈ [Λ V ] represents some subspace V (cid:48) ⊂ V , so d S ( ν, [Λ W ]) =Θ V (cid:48) ,W ≤ Θ V,W , by proposition 3.9.
Proposition 4.19. h ([Λ V ] , [Λ W ]) = Θ V,W and H ([Λ V ] , [Λ W ]) = ˆΘ V,W for any subspaces
V, W ⊂ X , considering [Λ V ] , [Λ W ] ⊂ S (Λ X ) .Proof. By definition, h ([Λ V ] , [Λ W ]) = max ν ∈ [Λ V ] d S ( ν, [Λ W ]) . Corollary 4.20. d S ( µ, [Λ W ]) ≤ d S ( µ, [Λ V ])+Θ V,W for any unit µ ∈ Λ X and any subspaces V, W ⊂ X . Corollary 4.21. min ω ∈ Λ W θ µ,ω ≤ min ν ∈ Λ V θ µ,ν + Θ V,W for any µ ∈ Λ X andany subspaces V, W ⊂ X . We now analyze a topic carrying many surprises: the angle between asubspace and the orthogonal complement of another. This should be amatter of interest for similar angles as well, but it has been overlooked,possibly because nice results depend on the asymmetrization. An impor-tant application of this angle is given in proposition A.1 iii . Definition.
The complementary Grassmann angle Θ ⊥ V,W ∈ [0 , π ] of sub-spaces V, W ⊂ X is the Grassmann angle of V with the orthogonal com-plement of W , i.e. Θ ⊥ V,W = Θ
V,W ⊥ .The new name and notation are due to its special properties. Notethat it is not the usual complement of an angle. For example, Θ ⊥ V,W = π for any two planes in R , due to the asymmetrization. This may seemwrong, but we just have to learn how to interpret Θ ⊥ V,W correctly.The benefits of the asymmetrization should become more evident inthis section, as it is responsible for the simplicity and generality of manyresults, which, without it, would be full of conditionals (e.g. ( iii ) and ( iv )below would not even hold for perpendicular planes in R ). Proposition 5.1.
Let
V, W ⊂ X be any subspaces.i) Θ ⊥ V, { } = Θ ⊥{ } ,V = 0 .ii) If V (cid:54) = { } then Θ ⊥ V,X = Θ ⊥ X,V = π .iii) Θ ⊥ V,W = 0 ⇔ V ⊥ W .iv) Θ ⊥ V,W = π ⇔ V ∩ W (cid:54) = { } . Proposition 5.2.
For any line L ⊂ X and any subspace W ⊂ X , wehave Θ L,W + Θ ⊥ L,W = π .Proof. If L = span( v ) , w = Proj W v and u = Proj W ⊥ v then Θ L,W = θ v,w and Θ ⊥ L,W = θ v,u . And θ v,w + θ v,u = π , as v = w + u and (cid:104) u, w (cid:105) = 0 .So for lines (even complex ones) the complementary Grassmann angleis the usual complement, and cos Θ ⊥ L,W = sin Θ
L,W . But as example 3.22and the one above show, this is not valid in general. When dim
V > , cos Θ ⊥ V,W will be given not by a single sine, but by a product of sines. To how this, we will get principal bases and angles for V and W ⊥ . In [37]we give a simpler proof, which avoids this construction.Let ( e , . . . , e p ) and ( f , . . . , f q ) be principal bases of V and W , re-spectively, with principal angles θ , . . . , θ m , where m = min { p, q } . Weget principal bases (˜ e , . . . , ˜ e p ) of V and ( g , . . . , g n − q ) of W ⊥ , where n = dim X , with principal angles θ ⊥ , . . . , θ ⊥ m (cid:48) , where m (cid:48) = min { p, n − q } ,as follows:1) The ˜ e i ’s are the same as the e i ’s, in reverse order: ˜ e p +1 − i = e i .2) For each q < i ≤ p (if any), let g p +1 − i = e i ∈ W ⊥ , so θ ⊥ p +1 − i = 0 .3) For any i ≤ m with θ i (cid:54) = 0 , let g p +1 − i = P ⊥ e i (cid:107) P ⊥ e i (cid:107) , with P ⊥ = Proj W ⊥ .Then θ ⊥ p +1 − i = θ P ⊥ e i ,e i = π − θ i .4) So far we have p − r principal vectors for W ⊥ , where r = dim V ∩ W .If p − r < n − q , add new g ’s to form an orthonormal basis for W ⊥ .Any e i with θ i = 0 is orthogonal to W ⊥ , so pairing as many of themas possible with the new g ’s we get principal angles θ ⊥ p +1 − i = π .Since g p +1 − i ∈ span( e i , f i ) in step 3, one can check that (cid:104) g i , g j (cid:105) = δ ij and (cid:104) ˜ e i , g j (cid:105) = δ ij cos θ ⊥ i , so the conditions of proposition 2.5 are satisfied. Example 5.3. In R , the following are principal vectors and angles for V = span( e , e , e ) and W = span( f , f ) : e = (1 , , , , , f = (1 , , , , , θ = 0 ,e = (0 , , , , , f = (0 , √ , , , / , θ = 30 ◦ ,e = (0 , , , , , and the Grassmann angle is Θ V,W = 90 ◦ by definition, as dim V > dim W .Applying the procedure described above, we get ˜ e = (0 , , , , , g = (0 , , , , , θ ⊥ = 0 , ˜ e = (0 , , , , , g = (0 , , −√ , , / , θ ⊥ = 60 ◦ , ˜ e = (1 , , , , , g = (0 , , , , , θ ⊥ = 90 ◦ , as principal vectors and angles for V and W ⊥ . Hence the complementaryGrassmann angle is also Θ ⊥ V,W = 90 ◦ . The seemingly strange fact thatboth Θ V,W and Θ ⊥ V,W are ◦ will be explained in section 5.1.We can now compute Θ ⊥ V,W in terms of principal angles of V and W ,and obtain an interpretation for the product of their sines, studied in[1, 42] but never linked to an angle. Theorem 5.4. If V, W ⊂ X are nonzero subspaces, with principal angles θ , . . . , θ m , then cos Θ ⊥ V,W = m (cid:89) i =1 sin θ i . Proof.
By proposition 5.1 iv , Θ V,W ⊥ = π ⇔ θ = 0 . If Θ V,W ⊥ (cid:54) = π then p = dim V ≤ dim W ⊥ and since θ (cid:54) = 0 no pair is formed in step 4above. Then cos Θ V,W ⊥ = (cid:81) pj =1 cos θ ⊥ j = (cid:81) pi =1 cos θ ⊥ p +1 − i , with the θ ⊥ ’sas above, and each cos θ ⊥ p +1 − i is either (step 2, if m = q < i ≤ p ) or sin θ i (step 3). his result is only valid unconditionally due to the asymmetrization.On the other hand, as principal angles do not depend on the order of V and W , it implies Θ ⊥ V,W is symmetric (by proposition 5.1 i , even if asubspace is { } ). Proposition 5.5. Θ ⊥ V,W = Θ ⊥ W,V for any subspaces
V, W ⊂ X . Corollary 5.6. Θ V,W = Θ W ⊥ ,V ⊥ . Note the reversal of V ⊥ and W ⊥ in this last formula. It can also beobtained from the fact that the nonzero principal angles of V ⊥ and W ⊥ are the same as those of V and W [12]. Proposition 5.7.
Let
U, V, W, X (cid:48) ⊂ X be subspaces, and P = Proj W .i) Θ ⊥ V,W = Θ ⊥ V,P ( V ) .ii) If U ⊥ V + W then Θ ⊥ V,W = Θ ⊥ V,W ⊕ U .iii) If V, W ⊂ X (cid:48) then Θ ⊥ V,W is the same whether the complement of W is taken in X (cid:48) or X .Proof. ( i ) If V ⊥ W it follows from proposition 5.1, otherwise from theo-rem 5.4 and proposition 2.15. ( ii ) P ( V ) = Proj W ⊕ U ( V ) . ( iii ) V, W ⊂ X (cid:48) implies W ⊥ = ( W ⊥ ∩ X (cid:48) ) ⊕ ( W ⊥ ∩ X (cid:48)⊥ ) and W ⊥ ∩ X (cid:48)⊥ ⊥ V +( W ⊥ ∩ X (cid:48) ) ,so proposition 3.3 ii gives Θ V,W ⊥ = Θ V,W ⊥ ∩ X (cid:48) .Note that ( i ) is not the same as proposition 3.3 i , which actually gives Θ ⊥ V,W = Θ
V,P ⊥ ( V ) , for P ⊥ = Proj W ⊥ , so that Θ V,P ⊥ ( V ) = Θ V,P ( V ) ⊥ .Likewise, ( ii ) is not the same as proposition 3.3 ii .In the complex case, ( W R ) ⊥ = ( W ⊥ ) R for any subspace W ⊂ X ,even though the first is a R -orthogonal complement and the second is C -orthogonal. So proposition 3.24 gives: Proposition 5.8.
In the complex case, cos Θ ⊥ V R ,W R = cos Θ ⊥ V,W for anysubspaces
V, W ⊂ X . Corollary 5.9.
In the complex case, Θ ⊥ V R ,W R ≥ Θ ⊥ V,W for any subspaces
V, W ⊂ X , with equality if, and only if, V ⊥ W or V ∩ W (cid:54) = { } . Example 5.10.
For the complex subspaces V and W of example 2.9, Θ ⊥ V,W = arccos( √ · √ ) ∼ = 52 . ◦ , and for the underlying real vector spaces Θ ⊥ V R ,W R = arccos( √ · √ · √ · √ ) ∼ = 68 ◦ . Θ V,W and Θ ⊥ V,W
The relation between Θ V,W and Θ ⊥ V,W is more flexible than the usual onebetween an angle and its complement, but they are not totally indepen-dent, and there are restrictions on the values they can assume.
Proposition 5.11.
Let ζ ( V, W ) = cos Θ V,W + cos Θ ⊥ V,W for subspaces
V, W ⊂ X , with V (cid:54) = { } . Then:i) ≤ ζ ( V, W ) ≤ .ii) ζ ( V, W ) = 1 ⇔ dim V = 1 , or V ⊥ W , or V ⊂ W .iii) ζ ( V, W ) = 0 ⇔ V ∩ W (cid:54) = { } and V ∩ W ⊥ (cid:54) = { } . Proof. (i) If W = { } then ζ ( V, W ) = 1 , and otherwise V and W haveprincipal angles θ , . . . , θ m and ζ ( V, W ) ≤ (cid:81) mi =1 cos θ i + (cid:81) mi =1 sin θ i ≤ cos θ + sin θ = 1 . (ii) These inequalities are equalities if, and only if,all θ i ’s are π , or dim V ≤ dim W and either m = 1 or all θ i ’s are 0. (iii) ζ ( V, W ) = 0 ⇔ Θ V,W = Θ ⊥ V,W = π .The condition V (cid:54) = { } is necessary, as ζ ( { } , W ) = 2 for any W .If ν is an unit p -blade representing V , corollary 3.14 ii gives ζ ( V, W ) = (cid:107)
P ν (cid:107) + (cid:107) P ⊥ ν (cid:107) , where P = Proj W and P ⊥ = Proj W ⊥ . If p > wecan have ζ ( V, W ) < because ν written in terms of bases of W and W ⊥ can have components mixing both, which are neither in P ν nor in P ⊥ ν .Proposition A.7 shows which squared cosines to add up to get 1. Proposition 5.12.
Let
V, W ⊂ X be subspaces, with V (cid:54) = { } . Then:i) π ≤ Θ V,W + Θ ⊥ V,W ≤ π ;ii) Θ V,W + Θ ⊥ V,W = π ⇔ dim V = 1 , or V ⊥ W , or V ⊂ W .iii) Θ V,W + Θ ⊥ V,W = π ⇔ V ∩ W (cid:54) = { } and V ∩ W ⊥ (cid:54) = { } ;Proof. (i) If Θ V,W + Θ ⊥ V,W < π then cos Θ ⊥ V,W > sin Θ V,W and ζ ( V, W ) > cos Θ V,W + sin Θ V,W = 1 . (ii) Θ ⊥ V,W = π − Θ V,W ⇔ ζ ( V, W ) = 1 .(iii) Θ V,W + Θ ⊥ V,W = π ⇔ Θ V,W = Θ ⊥ V,W = π ⇔ ζ ( V, W ) = 0 .Note that all cases happen even in R , as planes V and W can formany angle ≤ θ V,W ≤ π , for which Θ V,W = θ V,W and Θ ⊥ V,W = π .Figure 8, representing Θ V,W and Θ ⊥ V,W in Λ p X , may help understandtheir relation. For simplicity, it shows Λ p ( W ⊥ ) as a line spanned by g ∧ . . . ∧ g p , where the g ’s are elements of a principal basis of W ⊥ , butin general we can have dim Λ p ( W ⊥ ) > or even Λ p ( W ⊥ ) = { } .Our results might look more natural with an angle Φ V,W = π − Θ ⊥ V,W .For example, Φ V,W = 0 ⇔ V ∩ W (cid:54) = { } , Φ V,W = π ⇔ V ⊥ W , and sin Φ V,W = (cid:81) mi =1 sin θ i . But the geometric interpretation of Φ V,W wouldnot be as nice as that of Θ ⊥ V,W . ith some information on the minimal and directed maximal angleswe can further restrict the admissible values of Θ V,W and Θ ⊥ V,W . Definition.
Given nonzero subspaces
V, W ⊂ X , the angular range of V w.r.t. W is the length ∆ θ V,W = θ max V,W − θ min V,W ∈ [0 , π ] of the interval (cid:2) θ min V,W , θ max
V,W (cid:3) of possible angles between a nonzero v ∈ V and W . Proposition 5.13.
Let
V, W ⊂ X be nonzero subspaces, with dim V > .Then cos Θ V,W + cos Θ ⊥ V,W ≤ cos ∆ θ V,W , with equality if, and only if, oneof the following conditions is satisfied:i) dim V = 2 .ii) V = U ⊕ span( v ) for some U ⊥ W and v ∈ V (possibly v = 0 ).iii) V = U ⊕ span( v ) for some U ⊂ W and v ∈ V (possibly v = 0 ).iv) V ∩ W (cid:54) = { } and V ⊥ / W .Case (ii) can only happen if Θ V,W = π , and (iii) only if Θ ⊥ V,W = π . Case(iv) happens if, and only if, Θ V,W = Θ ⊥ V,W = ∆ θ V,W = π .Proof. Let θ , . . . , θ m be the principal angles of V and W , so θ min V,W = θ .If m = dim V ≤ dim W then θ max V,W = θ m and cos Θ V,W + cos Θ ⊥ V,W = (cid:81) mi =1 cos θ i + (cid:81) mi =1 sin θ i ≤ cos θ cos θ m + sin θ sin θ m = cos( θ m − θ ) .Equality happens if, and only if: (i) m = 2 , or (ii) θ = . . . = θ m = π , or(iii) θ = . . . = θ m − = 0 , or (iv) θ = 0 and θ m = π .If dim V > dim W = m then Θ V,W = θ max V,W = π , in which case cos Θ V,W + cos Θ ⊥ V,W = (cid:81) mi =1 sin θ i ≤ sin θ = cos( π − θ ) . Equalityhappens if, and only if, (ii) m = 1 or θ = . . . = θ m = π , or (iv) θ = 0 .If dim V = 1 then cos Θ V,W + cos Θ ⊥ V,W = cos θ + sin θ ≥ , and theinequality is strict unless θ = 0 or π .With proposition A.1, we get an upper bound for the angular range interms of blade products. Corollary 5.14.
Let ν, ω ∈ Λ X be unit blades representing V, W ⊂ X ,respectively, with dim V > . Then ∆ θ V,W ≤ arccos( (cid:107) ν (cid:121) ω (cid:107) + (cid:107) ν ∧ ω (cid:107) ) . Proposition 5.15.
Let
V, W ⊂ X be nonzero subspaces, with dim V > .Then Θ V,W + Θ ⊥ V,W ≥ π + ∆ θ V,W , with equality if, and only if, Θ V,W = π or Θ ⊥ V,W = π , and one of the conditions of proposition 5.13 is satisfied.Proof. sin(Θ V,W + Θ ⊥ V,W ) = sin Θ
V,W cos Θ ⊥ V,W + cos Θ
V,W sin Θ ⊥ V,W ≤ cos Θ ⊥ V,W + cos Θ
V,W ≤ cos ∆ θ V,W = sin( π + ∆ θ V,W ) . As, by proposi-tion 5.12, Θ V,W + Θ ⊥ V,W ∈ [ π , π ] , the result follows.We can now take a deeper look at the relation between Θ V,W and Θ ⊥ V,W . In fig. 9, the shaded region (both light and dark parts) represents cos Θ
V,W + cos Θ ⊥ V,W ≤ cos ∆ θ V,W . It shrinks as the angular range grows,and is reduced to the vertex C when ∆ θ V,W = π .If dim V = 1 the angles will be on the line Θ V,W + Θ ⊥ V,W = π , andif dim V = 2 they will be on the boundary curve cos Θ V,W + cos Θ ⊥ V,W =cos ∆ θ V,W . If dim
V > they will be in the interior of the shaded region,or on the segments AC or BC of its boundary. Point A corresponds to a) ∆ θ V,W = 0 (b) ∆ θ V,W = π (c) ∆ θ V,W = π Figure 9:
Restrictions on Θ V,W and Θ ⊥ V,W for different values of ∆ θ V,W . The shadedregion (both parts) corresponds to cos Θ
V,W + cos Θ ⊥ V,W ≤ cos ∆ θ V,W , and its darkersubregion to (cos Θ
V,W ) + (cos Θ ⊥ V,W ) ≤ cos ∆ θ V,W . The dashed line correspondsto Θ V,W + Θ ⊥ V,W = π + ∆ θ V,W .case (ii) of proposition 5.13, and B to (iii). As C corresponds to case (iv),it is not admissible unless ∆ θ V,W = π .These restrictions on Θ V,W and Θ ⊥ V,W are not exhaustive, and thereare certainly others. For example, if V and W have very high dimensionsthere will be many principal angles, so that Θ V,W ∼ = π unless θ i ∼ = 0 foralmost all i , and Θ ⊥ V,W ∼ = π unless θ i ∼ = π for almost all i . So in this casethe admissible region for these angles will retract to a small neighborhoodof the segments AC and BC . It would be interesting to have a moredetailed analysis of how the dimensions of V and W , or the distributionof their principal angles, affect the relation between Θ V,W and Θ ⊥ V,W . We now give an example of how the exotic features of Grassmann anglescan be exploited to give interesting results. Using the above results, andthose of section 3.3.5, we get an obstruction to having two given subspacesin a real space X become complex ones with respect to the same complexstructure in X .A complex structure on an even dimensional real vector space X is anautomorphism J : X → X such that J = − I . It turns X into a complexvector space, with multiplication by i defined by i v = Jv for any v ∈ X .An even dimensional real subspace V ⊂ X becomes a complex subspaceif, and only if, it is invariant under J , i.e. J ( V ) = V .This invariance is a strong condition, and for a given J most realsubspaces do not become complex ones. Still, if dim R V is even there arealways complex structures on X for which V is complex. This is no longertrue if we specify two subspaces V, W ⊂ X to become complex. Definition.
Real subspaces
V, W ⊂ X are simultaneously complexifiable if X admits a complex structure for which both are complex subspaces.As the underlying real spaces of complex subspaces ˜ V and ˜ W havethe same principal angles, but twice repeated, a necessary condition for V nd W to be simultaneously complexifiable is that their principal anglesbe pairwise equal , i.e. θ i − = θ i for i = 1 , , . . . . This is a strong require-ment, which most pairs of even dimensional subspaces do not satisfy.By propositions 3.24 and 5.8, if V = ( ˜ V ) R and W = ( ˜ W ) R then cos Θ V,W = cos Θ ˜ V , ˜ W , cos Θ ⊥ V,W = cos Θ ⊥ ˜ V , ˜ W , and ∆ θ V,W = ∆ θ ˜ V , ˜ W . Soproposition 5.13 gives the following obstruction. Proposition 5.16.
Let
V, W ⊂ X be nonzero subspaces of a real vectorspace X , with dim V ≥ . If (cos Θ V,W ) + (cos Θ ⊥ V,W ) > cos ∆ θ V,W then V and W are not simultaneously complexifiable. This can be understood by noting that the proof of proposition 5.13can be adapted to show that (cos Θ
V,W ) + (cos Θ ⊥ V,W ) ≤ cos ∆ θ V,W if dim V ≥ and the principal angles of V and W are pairwise equal. Sothe above condition is in fact an obstruction on pairwise equality.This result expresses in terms of Θ V,W and Θ ⊥ V,W the difficulty of V and W being simultaneously complexifiable. If dim V ≥ , it is necessary(but not sufficient) that these angles be in the darker shaded region offig. 9, which shrinks faster than the lighter one as ∆ θ V,W increases. Thelarger the angular range of one subspace with respect to the other, theharder it is that they can be simultaneously complexifiable.
Grassmann angles have many other interesting properties and applica-tions, which we develop in other articles (see appendix A). Other resultsfound in the literature, for similar angles, can also be readily translatedinto statements about Grassmann angles. For example, a theorem of [12]tells us the generalized Frenet curvature k p of a curve measures the rateof change of the Grassmann angle for the osculating subspaces spannedby the first p Frenet vectors.Some of our results have equivalents in other formalisms. For example,in appendix B we indicate analogous results of [42] referring to propertiesof matrices. Other results we obtained can certainly be translated intonew results about matrices or in terms of other formalisms.The total Grassmannian G ( X ) has received little attention from ge-ometers, even though it is useful in applications [17, 46]. A reason is thatits most obvious geometric structure, as a disjoint union of Grassmani-anns, reduces its study to that of its components. It might be interestingto study it as a quasi-pseudometric space instead, with distances given by Θ V,W . A Further results
We list below some results proven in [37, 38], which show how usefulGrassmann angles can be.These angles are related to the inner product, contraction (interiorproduct) and exterior product of blades as follows. roposition A.1. Let ν, ω ∈ Λ X be blades representing subspaces V, W ⊂ X , respectively. Then:i) |(cid:104) ν, ω (cid:105)| = (cid:107) ν (cid:107)(cid:107) ω (cid:107) cos Θ V,W , if they have equal grades.ii) (cid:107) ν (cid:121) ω (cid:107) = (cid:107) ν (cid:107)(cid:107) ω (cid:107) cos Θ V,W .iii) (cid:107) ν ∧ ω (cid:107) = (cid:107) ν (cid:107)(cid:107) ω (cid:107) cos Θ ⊥ V,W . ( i ) appears in many works on the subject, and the grade condition canbe removed by replacing Θ V,W with ˆΘ V,W . ( ii ) and ( iii ) only hold withoutany conditions due to the asymmetrization. ( iii ) gives a simple way toprove theorem 5.4, and directly implies proposition 5.5.Products of Clifford’s geometric algebra [8, 22] are also related to thevarious Grassmann angles. Proposition A.2.
Let ν, ω ∈ Λ X be blades representing subspaces V, W ⊂ X , respectively. Then | ν ∗ ω | = (cid:107) ν (cid:107)(cid:107) ω (cid:107) cos ˆΘ V,W , (cid:107) ν • ω (cid:107) = (cid:107) ν (cid:107)(cid:107) ω (cid:107) cos ˇΘ V,W , (cid:107) ν · ω (cid:107) = (cid:107) ν (cid:107)(cid:107) ω (cid:107) cos ˇΘ V,W ( V, W (cid:54) = { } ) , (cid:107) ν (cid:99) ω (cid:107) = (cid:107) ν (cid:107)(cid:107) ω (cid:107) cos Θ V,W , (cid:107) ν (cid:98) ω (cid:107) = (cid:107) ν (cid:107)(cid:107) ω (cid:107) cos Θ W,V , (cid:107) ν ∧ ω (cid:107) = (cid:107) ν (cid:107)(cid:107) ω (cid:107) cos Θ ⊥ V,W . The formula for Clifford’s geometric product is more complicated, as ingeneral a product of blades is not a blade. But it still provides interestinggeometric interpretations (see [37] for notation).
Proposition A.3.
Let ν, ω ∈ Λ p X be nonzero blades representing sub-spaces V, W ⊂ X , respectively. Then ˜ νω = σ ν,ω (cid:107) ν (cid:107)(cid:107) ω (cid:107) (cid:88) I ∈I cos Θ V,W I i I . (9)General formulas for computing Grassmann angles are obtained usingblade products. Given any bases ( v , . . . , v p ) of V and ( w , . . . , w q ) of W ,we form matrices A = (cid:0) (cid:104) w i , w j (cid:105) (cid:1) , B = (cid:0) (cid:104) w i , v j (cid:105) (cid:1) , and D = (cid:0) (cid:104) v i , v j (cid:105) (cid:1) . Proposition A.4.
With A , B and D as above, cos Θ V,W = det( ¯ B T A − B )det D . (10) If p = q this reduces to cos Θ V,W = | det B | det A · det D . (11)For the complementary Grassmann angle we have the following.
Proposition A.5.
With A , B and D as above, cos Θ ⊥ V,W = det( A − BD − ¯ B T )det A . If P is a matrix representing Proj VW in orthonormal bases of V and W , cos Θ ⊥ V,W = det( q × q − P ¯ P T ) . he following results give, with theorem 3.16, known generalizationsof the Pythagorean theorem for real spaces, and also new ones for complexspaces [39], which are simpler and may have important implications forquantum theory [40]. Proposition A.6.
Given an orthogonal partition X = W ⊕ · · · ⊕ W k ,for any line L ⊂ X we have (cid:80) ki =1 cos Θ L,W i = 1 . The complex case, combined with theorem 4.8, has a nice interpreta-tion in terms of quantum probabilities [38].
Proposition A.7.
For any p -dimensional subspace V ⊂ X we have (cid:80) I cos Θ V,W I = 1 , where the sum runs over all p -dimensional coordi-nate subspaces W I of an orthogonal basis of X . With (9), this explains geometrically why (cid:107) νω (cid:107) = (cid:107) ν (cid:107)(cid:107) ω (cid:107) for thegeometric product of blades, while other products are submultiplicativefor blades (as propositions A.1 and A.2 show). Proposition A.8.
Let V ⊂ X be a p -dimensional subspace and ≤ q ≤ n = dim X . If p ≤ q then (cid:80) I cos Θ V,W I = (cid:0) n − pn − q (cid:1) , while if p > q then (cid:80) I cos Θ W I ,V = (cid:0) pq (cid:1) , where the sums run over all q -dimensionalcoordinate subspaces W I of an orthogonal basis of X . B Related angles
We present a brief review of angles and results similar to ours found inthe literature. Most works deal only with real spaces, usually with somerestriction on the dimensions. The results are, mostly, particular casesof ours, with extra conditions. We include works [1, 42] which do notreally define an angle, but whose results are closely related to ours, evenif expressed in different formalisms.Venticos [50] defines the angle between complex subspaces of samedimension as that between their blades, as in corollary 3.13, and gives the product cosine formula (the cosine of the angle is the product of cosines ofprincipal angles). If dim
V < dim W , he defines the angle as that between V and Proj W V , if these have equal dimensions, otherwise as π .Afriat [1] uses, for real subspaces, symbols like cos( V, W ) and sin( V, W ) for products of cosines or sines of principal angles. But ( V, W ) is notmeant as an angle (it is not even defined by itself), and the symbols donot satisfy the usual trigonometric relations. He gives a result analogousto proposition A.1 iii , but assuming V ∩ W = { } and expressed in termsof volumes of parallelotopes and sin( V, W ) .Gluck [12] defines the angle for real subspaces of same dimension interms of volume contraction, as in theorem 3.16 i , and obtains the productcosine formula, (11) and particular cases of theorems 3.12 and 4.1.Górski [15] defines the angle for real subspaces V and W of samedimension as in corollary 3.13 iii , and when dim V < dim W he uses a con-struction with blades which, ultimately, corresponds to fig. 5. He obtainsthe product cosine formula and a particular case of proposition A.1 ii .Miao and Ben-Israel [42] use, for real subspaces, cos - and sin -symbolslike those of Afriat, not interpreting them in terms of angles either. They btain results analogous to corollary 3.13 iii , in terms of determinants,matrix volumes (products of singular values) and the cos -symbol, to aparticular case of proposition A.1 iii , in terms of matrix volumes and the sin -symbol, to theorem 3.12, in terms of compound matrices and the cos -symbol, and to proposition A.7 for the cos -symbol.Jiang [26] defines a p -dimensional angle between real subspaces V and W , with dim V ≤ dim W , in a way similar to corollary 3.14 ii , and obtainsthe product cosine formula. Theorems 3.12 and 4.1, and corollaries 3.13 iii and 4.6, are given for the case of equal dimensions, as well as theorem 3.5and proposition 5.11, with some dimensional conditions.Gunawan et al. [18] use volume contraction of parallelepipeds to definethe angle between real subspaces V and W , with dim V ≤ dim W . Theyprove (4) for this case, correcting a mistake of [47], which uses a similarformula to define the angle, without assuming orthonormality. A formulathat works without such condition is also given, but it is more complicatedthan (10).Galántai and Hegedűs [11] define a product angle between complexsubspaces in terms of the product cosine formula.Hitzer [23] uses the same formula to define a total angle between realsubspaces of same dimension. He gives a result like corollary 3.13 iii , anda formula similar to (9), but in terms of products mixing cosines and sinesof principal angles. Later he changes the angle definition to exclude any θ i = π , and also applies it to subspaces of different dimensions.We also mention below some angles which do not correspond directlyto the Grassmann angle, but are related to it.Degen [5] defines, for real subspaces of dimension p , a projection an-gle ψ , using volume projection, and an aperture angle ϕ , by comparingvolumes of parallelepipeds, in such a way that cos ψ (resp. sin ϕ ) is a ge-ometric mean of cosines (resp. sines) of principal angles, and ψ (resp. ϕ )is between the aritmetic mean of the θ i ’s and the largest (resp. smallest)one. But p -powers in the definitions make these angles harder to use, andthey do not even coincide with the dihedral angle for planes in R .Hawidi [20, 21] defines an asymmetric angle operator by (cid:94) ( V, W ) =Proj WV Proj VW . It carries all information about principal angles (its eigen-values are their squared cosines), but it is not as easy to use as a scalarangle. It relates to our angle by cos Θ V,W = (cid:112) det (cid:94) ( V, W ) .Using Clifford’s geometric algebra, in [37] we define, for real subspacesof same dimension p , an angle bivector θ which also codifies all dataabout their relative position. In particular, the norms of the componentsof grades and p in e θ give, respectively, the cosines of Θ V,W and Θ ⊥ V,W .Properties of θ are under study to assess its usefulness. Acknowledgment
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