Generalisations of the determinant to interdimensional transformations: a review
aa r X i v : . [ m a t h . G M ] A p r Generalisations of the determinant to inter-dimensional transformations: a review
Abhimanyu Pallavi Sudhir
Abstract.
Significant research has been carried out in the past half-century on defining generalised determinants for transformations be-tween (typically real) vector spaces of different dimensions. We reviewthree different generalisations of the determinant to non-square matri-ces, that we term for convenience the determinant-like function [1], thevector determinant [4] and the g-determinant [5]. We introduce andmotivate these generalisations, note certain formal similarities betweenthem and discuss their known properties.
Mathematics Subject Classification (2010).
Keywords. linear algebra, clifford algebra, determinant, matrix, exterioralgebra.
1. Introduction
Several generalisations of the determinant to domains consisting of non-square matrices are known, including the determinant-like function [1], thevector determinant [4], and the g-determinant [5]. Each of these generalisa-tions is defined based on different defining properties of the square determi-nant. detl( ~c , . . . ~c n ) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)^ j ~c j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1.1)Notably, the determinant-like function (Eq. 1.1) has perhaps the clear-est geometric description (it is the volume of the image (in R m ) of the unit n -cube under a linear transformation) – however, it is also the most limited.The function is an unsigned determinant, and is only defined for tall matrices(i.e. linear transformations to a higher-dimensional space). Both limitationsarise directly from the geometric interpretation – the function must be un-signed, as determining the handedness of a general n -volume in R m depends Abhimanyu Pallavi Sudhiron the introduction of additional vectors; and an extension of the geomet-ric interpretation to wide matrices (transformations to a lower-dimensionalspace) would require the function to be uniformly zero.Algebraically, the determinant-like function can be calculated as (wewill derive this expression in Sec. 2):detl A = " X i <...i n det A i ,...i n / (1.2)Where A i ,...i n are the “maximal square submatrices” of A , or the n by n submatrices of A . In fact, these submatrices also appear in the expressionsfor the vector determinant and the g-determinant.vdet A = X i <...i n det A i ,...i n ~e τ ( i ,...i n ) (1.3)The vector determinant Eq. (1.3) is – for some bijection τ to (cid:2) , (cid:0) mn (cid:1)(cid:3) from the set of ascending n -tuples of integers i , . . . i n (1 ≤ i j ≤ m ) – avector-valued function vdet : R m × n → R ( mn ) for m ≥ n .One might note that Eq. (1.2) is simply the magnitude of the vector inEq. (1.3) [2] – indeed, in this sense the vector determinant might be thoughtof as a way to “mitigate” the limitations of th determinant-like function. Thisrelation will be discussed in further detail in Sec. 3.1.(Note that in the original reference [4], the left-multiplication conventionwas adopted, i.e. an m × n matrix represented a transformation from R m → R n , and thus the vector determinant was originally claimed to apply to “wide”rather than “tall” matrices – however, we adopt the opposite convention forthe sake of uniformity.)The g-determinant [5] (a term that first appeared in [6]), defines thedeterminant of a tall matrix recursively through Laplace’s expansion, withthe determinant of a m by 1 matrix as the base case:gdet( a , . . . a m ) T = a − a + . . . + ( − m +1 a m (1.4)With a recursion relation given by the Laplace expansion:gdet A = m X i =1 ( − i +1 a i gdet A { } C { i } C (1.5)Where A TS refers to the submatrix of A comprised of the rows in S and the columns in T , and S C is the complement. It is then easy to show(and will be shown in Sec. 4) inductively that the general expression for theg-determinant of an m by n matrix ( m ≥ n ) is given by:gdet A = X i <...i n ( − P j i j + j det A i ,...i n (1.6)eneralised determinants: a review 3 ~a ∈ R ~a ~e ∈ R ~e Figure 1.
A transformation A : R → R
2. The determinant-like function
On the subject of determinants of non-square matrices, it is clear that aninterpretation in terms of volume requires the determinant of a “wide” matrix(one with more columns than rows) to be zero, as it is a transformation froma higher-dimensional space into a lower-dimensional one. However, one maystill assign a non-trivial volume-based interpretation to the determinant ofa tall matrix – despite being a linear map into a higher-dimensional space,such a transformation is not surjective (its rank is still at most the dimensionof the domain).It is reasonable to define a determinant-like function for a tall matrix A : R n → R m as the n -volume of the image in m -dimensional space of the unit n -form (see Fig. 2 for an example). Formally, one may generalise the followingexterior algebraic property of the standard determinant (for a matrix of n vectors in R n ): | det( ~c , . . . ~c n ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)^ i ~c i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2.1)To the determinant-like function detl that satisfies the following naturalgeneralisation: Definition 2.1.
The determinant-like function of a matrix comprised of columns ~c , . . . ~c n with each c j ∈ R m ( m ≥ n ) is defined as the necessarily unsigned n -volume of the n -form spanned by these columns, i.e.detl( ~c , . . . ~c n ) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)^ j ~c j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) We will treat Definition 2.1 as the definition of the determinant-likefunction for tall matrices, and proceed to find a computable, algebraic ex-pression for the function. Abhimanyu Pallavi Sudhir
With Definition 2.1, one may easily calculate the determinant-like functionof elementary examples. Here, we produce a general usable expression forany tall matrix. In doing so, we correct the erroneous factor of p | m − n | !included in the original reference [1]. Theorem 2.1.
The unsigned determinant-like function of a tall m by n matrix A is given by the following expression in terms of the (cid:0) mn (cid:1) n by n squaresubmatrices of A (here, A S where S ⊆ { , . . . m } , is the submatrix of A whose rows are those indexed in S ): detl A = X i <...i n det A i ,...i n ! / (Note that by convention, ≤ i j ≤ m for ≤ j ≤ n .)Proof. We attempt to simplify the wedge product of the columns of A byexpressing them in the standard basis of R m and applying the distributivelaw (note that e i ,...i n := V j ~e i j ): ^ j X i a ij ~e i = X i ,...i n ^ j a i j j ~e i j = X i ,...i n Y j a i j j e i ,...i n Grouping the e i ,...i n terms with their index permutations σ ∈ S n andapplying the antisymmetric nature of the wedge product, one can rewrite thisas a sum over (cid:0) mn (cid:1) ascending combinations of n indices. . . . = X i <...i n X σ Y j a i σ ( j ) j e i σ (1) ,...i σ ( n ) = X i <...i n e i ,...i n X σ sgn( σ ) Y j a i σ ( j ) j = X i <...i n e i ,...i n det A i ,...i n The result then follows from taking the norm of this n -form, which isdefinitionally | X | = |h X, X i| . (cid:3) We note two special cases of Theorem 2.1 immediately: the m by 1 case is thePythagoras theorem in m dimensions, as it measures the 1-volume (length)of a vector in terms of its projections onto each of the m dimensions. Alsonotably, the ( n + 1) × n determinant is the magnitude of the n -vector crossproduct, as it measures the n -volume contained between them (for examplethe 3 by 2 case is simply the elementary-school cross product).eneralised determinants: a review 5It is worth commenting on the question of the sign of the determinant-like function – in general in m dimensions, one may naturally give a signto an m -form by defining a left-handed/right-handed convention. To definea sign on an n -form, one needs a notion of orientation, however, in m di-mensions, this depends on which remaining m − n vectors you use to definethis orientation (e.g. in 3 dimensions, one needs a third vector to “look at”two vectors and determine which vector is clockwise to which). Fixing suchvectors would not make sense for our purposes (e.g. consider a case where the n -form is parallel to one of our fixed vectors – the determinant of the m -formcomprising of the n -form and the fixed vectors is zero, neither positive nornegative, and this does not describe a sign on our n -volume at all).We consider the validity of the corresponding generalisations of somestandard properties of the square determinant to the determinant-like func-tion – unlike the square determinant, the manner in which the determinant-like function treats its columns is fundamentally different from the way thatit treats its rows. : • An m by n matrix with any linearly independent columns has a zerodeterminant-like function (as the n -form “closes in”), but it needs morethan m − n rows to be linearly dependent in order for the determinant-like function to be zero. • The determinant-like function is linear in each of its columns – however,multiplying a row by a scalar k has no such predictable effect. • Adding a linear multiple of a column to another column leaves thedeterminant-like function invariant, but doing so with the columns hasno such predictable effect. • The function is invariant under switching both columns and rows (asthe component square determinants are just permuted). • In general for matrices A : R n → R m and B : R o → R n ( m ≥ n ≥ o ), the determinant-like function is not multiplicative, i.e. detl AB =detl A detl B , as detl A describes the volume transformation of an n -form and not an o -form (indeed, o -forms would not be similarly scaledunder A ).The generalisation of Cramer’s rule to the determinant-like functionwas the subject of [3]. Specifically, in a solvable rank- n system of m linearequations in n variables ( m ≥ n , as usual), represented by Ax = b , thecomponents of x satisfy: | x j | = detl( ~c , . . . ~c j ,~b, ~c j +1 , . . . ~c n )detl A (2.2)This can be proven in the following straightforward way: for the systemto have a solution, b must be expressible as a linear combination of thelinearly independent columns of A – therefore, any n by n submatrix of A has determinant zero if and only if the corresponding square submatrix ofthe matrix “ A with its j th column replaced with b ” has determinant zero,and the ratio of any non-zero determinants of corresponding submatrices is Abhimanyu Pallavi Sudhir ~a ~b~c + – Figure 2.
An example orientation convention on the xz -planeexactly the solution to the system (as the system can be solved by eliminatinglinearly dependent equations). The result follows.One might wonder if considering the values of the solutions themselves(as opposed to the absolute values) in Eq. (2.2) would allow us to define a signon the determinant-like function. Such a definition is possible, but not natural– it would be equivalent to defining an orientation on every n -dimensionalsubspace of R m , each subspace’s definition independent of one another (seeFig. 2.2). Notably the resulting “signed determinant-like function” would notbe continuous under any chosen convention.
3. The vector determinant
The original historic motivation for the vector determinant Eq. (3.1) is notcompletely clear – it may be described in a sense as an accidental discovery.When introduced in [4], it was claimed that the vector determinant wasdefinitionally the generalisation that satisfied the following properties: • Linearity in each column vector • Irreflexivity, i.e. equals zero whenever two columns are equal • The normalisation vdet( e , e . . . e n ) = (1 , , . . .
0) or similarvdet A = X i <...i n det A i ,...i n ~e τ ( i ,...i n ) (3.1)However, no proof of uniqueness was presented, and in fact, a distinctgeneralisation satisfying these properties was presented later in [8], and isrecorded below in Eq. (3.2). X i <...i n det A i ,...i n (3.2)eneralised determinants: a review 7The key theorem relating to the vector determinant that makes it note-worthy has to do with determinants of products of the form det( A T B ) where X and Y have the same dimensions. We give the statement (and proof) ofthis theorem for real-valued matrices below. Theorem 3.1.
For two m by n matrices A and B , the determinant of theirsquare n by n inner product A T B is given by the dot product of their vectordeterminants: det( A T B ) = (vdet A ) T vdet B Proof.
Consider the j th column of A T B , ( A T B ) j . One may write this as asum of m vectors as follows, where i runs from 1 to m :( A T B ) j = X i B ji A i The determinant of A T B is then the determinant of a matrix whosecolumns are sums of simpler columns – therefore we may inductively apply themultilinearity of the determinant in these sums to write (where 1 ≤ i j ≤ m ):det A T B = X i ,...i n det( B i A i , . . . B ni n A i n )= X i ,...i n B i . . . B ni n det A i ,...i n As in Theorem 2.1, we group permutations of the same combination i , . . . i n together – under such a permutation σ ∈ S n , the determinant of thesubmatrix of A remains unchanged up to multiplication by the sign of thepermutation sgn( σ ):det A T B = X i <...i n X σ B i σ (1) . . . B ni σ ( n ) det A i σ (1) ,...i σ ( n ) = X i <...i n det A i ,...i n X σ sgn( σ ) B i σ (1) . . . B ni σ ( n ) = X i <...i n det A i ,...i n det B i ,...i n Which is precisely the dot product of vdet A and vdet B . (cid:3) Theorem 3.1 provides a crucial insight into the inner product structure onthe space of vector determinants in R ( mn ). In particular, it resolves the fol-lowing two facts as special cases: (1) letting A and B be square, det( AB ) =det( A ) det( B ), (2) letting A and B be equal, detl A = | vdet A | (the latterprovides a geometric interpretation to the magnitude of the vector determi-nant). Abhimanyu Pallavi SudhirA way to think about the vector determinant is that it is a “vector repre-sentation” to V j A j , similar to how a cross product is a vector representationto the wedge product in three dimensions. In a sense, the vector determi-nant may be viewed as a “signed” version of the determinant-like function,but with a direction rather than a sign. For example, one may restate resultEq. (2.2) as follows – if the equation Ax = b is solvable, its solutions x j canbe written as the ratio between proportional vector determinants: x j vdet A = vdet( A , . . . A j , b, A j +1 , . . . A n ) (3.3)If the system is not solvable, the vector determinants are not propor-tional, and Eq. (3.3) has no solution for any j .
4. The g-determinant
We provide an inductive proof of Eq. (1.6) from the defining base caseEq. (1.4) and Laplace recursion Eq. (1.5).
Theorem 4.1.
The following generalised determinant is the unique functionon m by n satisfying Eq. (1.4) for m by 1 matrices and Laplace’s expansionalong the first column: gdet A = X i <...i n ( − P j i j + j det A i ,...i n Proof.
The base case is clear. It suffices (by diagonal induction from the n = 1line) to show that the statement for ( m − × ( n −
1) matrices implies thestatement for m × n matrices. Expanding the ( m − × ( n −
1) g-determinantsin the Laplace expansion,gdet A = m X i =1 ( − i +1 a i X i <...i n = i ( − n P j =2 ( i j + j ) det A ,...ni ,...i n = m X i =1 X i <...i n = i ( − n P j =1 ( i j + j ) a i det A ,...ni ,...i n We group all permutations of a given set i , i , ...i n together, allowingus to write each term with ascending i j – the determinant terms in each suchcase are multiplied by ( − k +1 in bringing the rows of the submatrix to theirnatural order:gdet A = X i <...i n ( − P j i j + j n X k =1 ( − k +1 a i k det A { k } C { i j = k } = X i <...i n ( − P j i j + j det A i ,...i n Uniqueness follows from the well-foundedness of the recursion. (cid:3) eneralised determinants: a review 9
It’s worth noting that in order to preserve parallels with the other generaliseddeterminants, we have used the opposite convention to [5] with regards to therows and columns of the matrix – our g-determinant is the g-determinant ofthe transpose as per the original convention.Although the g-determinant does not have a conventional geometricinterpretation in terms of scaling of volumes under a non-square transforma-tion, the m by 2 g-determinant does have a rather unique geometric applica-tion in relation to the areas of polygons – explicitly, the area of an m -vertexpolygon in R whose vertices are given by the column vectors A i is given by[ A , ...A m ] = 12 gdet( A + A , A + A , . . . A m + A ) T (4.1)We state the above result without proof, as it follows in a straight-forward fashion from the shoelace formula [6]. Notably, it is easy to showthat in the continuous limit with m → ∞ , Eq. (4.1) reduces to the standardexpression for the area contained within a curve:[ C ] = 12 Z C x dy − y dx (4.2)The proof does not differ much from a standard continuous generalisa-tion of the shoelace formula [9], and is left as an exercise to the reader.
5. Conclusion
We have discussed the properties of three significant tall-matrix determinants,as well as some insight into their geometric interpretations. Our treatment isby no means comprehensive – notably, we did not discuss the generalisationin [8], with the exception of a brief statement of its definition in Eq. (3.2),as there is relatively less literature on this generalisation, and its knownproperties are mostly elementary.The g-determinant is perhaps the most well-studied of the generaliseddeterminants, and we have only covered a selection of its properties, includ-ing of its geometric properties. An interested reader might wish to consultreferences [5] [6] [7] for a deeper look at the research in this area.Although we have covered the vector determinant as a vector in R ( mn )in agreement with the literature, it may be more revealing to consider itsimply as an n -vector in R m . All its discussed properties of the generalisationare preserved under this interpretation – additionally, one obtains a rathernatural way to think about the signs of each component determinant as thesigns of the projections of the determinant multivector onto basis n -forms.The projection (and its sign) onto an arbitrary n -dimensional subspace canbe obtained via an inner product with a unit n -form on that subspace –this “directional determinant” can be said to be analogous to directionalderivatives in analysis.0 Abhimanyu Pallavi SudhirIt is not clear if the rather similar roles of the determinants of the squaresubmatrices in the various generalised determinants may offer any furtherinsight, or if there is any fundamental relation between the g-determinant,the vector determinant and the determinant in [8]. References [1] A. Pallavi Sudhir,
Defining the determinant-like function for m by n matricesusing the exterior algebra , Advances in Applied Clifford Algebras. (2013),787-792.[2] A. Pallavi Sudhir, On the determinant-like function and the vector determinant ,Advances in Applied Clifford Algebras. (2014). 805-807.[3] A. Pallavi Sudhir,
On the properties of the determinant-like function , Interna-tional Conference on Mathematical Sciences. Chennai (2014).[4] H. Pyle,
Non-square determinants and multilinear Vectors , Mathematics Asso-ciation of America. (1962), 65-69.[5] M. Radic,
A definition of determinant of rectangular matrix , Glas. Mat. (1966), 17-22.[6] M. Radic,
About the determinant of a 2 by n matrix and its geometric interpre-tation , Contributions to Algebra and Geometry. (2005), 321-349.[7] R. Susanj & M. Radic, Geometric meaning of one generalisation of the deter-minant of square matrix , Glas. Mat. III. Ser. (1994), 217-233.[8] V. N. Joshi,