Informing the structure of complex Hadamard matrix spaces using a flow
IInforming the structure of complex Hadamardmatrix spaces using a flow
Francis C. Motta ∗ and Patrick D. Shipman † Abstract
The defect of a complex Hadamard matrix H is an upper bound forthe dimension of a continuous Hadamard orbit stemming from H . Weprovide a new interpretation of the defect as the dimension of the cen-ter subspace of a gradient flow and apply the Center Manifold Theoremof dynamical systems theory to study local structure in spaces of com-plex Hadamard matrices. Through examples, we provide several applica-tions of our methodology including the construction of affine families ofHadamard matrices. The principal objects of interest in this paper are dephased complex Hadamardmatrices : matrices H ∈ M d × d ( S ) such that HH ∗ = dI d , and[ H ] i, = [ H ] ,i = 1 for i = 1 , . . . , d. Here I d is the d × d identity matrix, ∗ denotes conjugate transpose, and S ⊂ C is the complex unit circle. We will adhere to the common practice of declaringtwo Hadamards H and K equivalent if there exist unitary diagonal matrices D and D and permutation matrices P and P such that H = D P KP D . Complex Hadamards are natural generalizations of their real analogue: squarematrices with entries in {− , } with mutually orthogonal rows and columns,discovered to have the largest determinant among all real matrices whose en-tries have absolute values bounded by one [1]. While both real and complexHadamards are mathematically interesting objects in their own right, theyalso have a wide range of applications including uses in coding theory [2],the design of statistical experiments [3], numerous constructions in theoreticalphysics [4, 5, 6] and quantum information theory [7]. ∗ Department of Mathematics, Duke University, Box 90320, Durham, NC [email protected] † Department of Mathematics, Colorado State University, 1874 Campus Delivery, FortCollins, CO 80523-1874. [email protected] a r X i v : . [ m a t h - ph ] O c t he focus of much of the current mathematical research is aimed at completeclassification of equivalence classes of complex Hadamards, at least for smalldimensions [8, 9, 10]. In dimensions d ≤
5, complex Hadamards have beencompletely described [11, 12], while the classification of 6 × H , the defect d ( H ) ∈ N can be computed [17]. This quantity (definedin Section 2) bounds the dimension of affine orbits stemming from a dephasedHadamard [18]. Notably, if d ( H ) = 0, then there exists a neighborhood of H which does not contain any other dephased Hadamards, and H is said to be iso-lated . The main contributions of this paper include 1) introducing an equivalentdefinition of the defect using classical dynamical systems theory, 2) demonstrat-ing through examples how this new framework can be used to investigate thelocal structure of complex Hadamards and, in particular, 3) constructing severalnew affine families stemming from known Hadamards.This paper is organized as follows: in Section 2 we motivate and define ourmethodology. Section 3 serves to justify and explicate the technique by applyingit to the space of 4 × Fix an integer d ≥ H d ( θ ) . = · · · e i θ e i θ · · · e i θ d − e i θ d e i θ d +1 · · · e i θ d − ... ... ... . . . ...1 e i θ ( d − d − e i θ ( d − d − · · · e i θ ( d − , depending on θ . = (cid:2) θ , θ , . . . , θ ( d − (cid:3) with θ i ∈ [0 , π ). If H d ( θ ) is a dephasedHadamard, then H d ( θ ) H d ( θ ) ∗ = dI d . Naturally this imposes conditions on theallowed phases in the entries of H d ( θ ). In particular, θ must be chosen to satisfy d ( d −
1) equations stemming from the requirement that the off-diagonal entriesof H d ( θ ) H d ( θ ) ∗ must be identically 0:[ H d ( θ ) H d ( θ ) ∗ ] i,j = (cid:40) , if i (cid:54) = jd, if i = j . (1)2sing the equations in (1) we define a scalar potential V d : R ( d − → R , whichcan be thought of as measuring the extent of the failure of a matrix to beHadamard, by V d ( θ ) . = d (cid:88) i (cid:54) = j (cid:12)(cid:12)(cid:12) [ H d ( θ ) H d ( θ ) ∗ ] i,j (cid:12)(cid:12)(cid:12) . (2)Observe, V d ( θ ) vanishes exactly when H d ( θ ) is a complex Hadamard matrix. Bycomputing the negative gradient of V d , we define a gradient system of ordinarydifferential equations, Φ d ( θ ) . = −∇V d , (3)whose stationary points are the dephased complex Hadamard matrices. Equa-tion (3) can be thought of as defining a flow on the ( d − -torus, T ( d − , sinceonly the core – the ( d − × ( d −
1) lower-right submatrix – is allowed to vary(i.e. we insist that the matrices H d ( θ ) be dephased). Note that this systemdoes not take into account permutation equivalences, and so the set to whichΦ d ( θ ) converges is not the space of inequivalent Hadamards, but rather thespace of inequivalent Hadamards together with all copies of this space derivedfrom permutations of the cores of its members.Let be a fixed point of the nonlinear system d x dt = A x + F ( x ) , x ∈ R N , (4)where A is an N × N matrix and F is C r in a neighborhood of . Then theeigenvalues of A can be used to determine the local dynamics of the systemnear the origin. In particular, there exist stable ( E s ), unstable ( E u ), and center( E c ) subspaces spanned by the generalized eigenvectors corresponding to theeigenvalues of A with negative, positive, and zero real-parts respectively. Thegeneralized eigenvectors spanning the stable and unstable subspaces are tangentat the origin to the stable and unstable manifolds ( W s and W u ): invariant setswith consistent asymptotic behavior with respect to the origin. Similarly, thecenter subspace is tangent to every center manifold, W c : invariant sets on which F ( x ) (the non-linear part of the vector field) governs the dynamics as given bythe Center Manifold Theorem [19, 20]. Local Center Manifold Theorem.
Assume that A has c eigenvalues withreal part equal to zero and N − c eigenvalues with negative real part. Then thesystem defined by Equation (4) can be written in diagonal form d x dt = C x + G ( x , y ) d y dt = S y + H ( x , y ) , where x ∈ R c , y ∈ R N − c , C is a square matrix whose eigenvalues all have 0real part, S is a square matrix whose eigenvalues have negative real part, and G ( ) = H ( ) = DG ( ) = DH ( ) = . Furthermore, for some δ > there xists h ∈ C r ( B δ ( )) that defines the local center manifold W c ( ) = { [ x , y ] ∈ R N | y = h ( x ) for | x | < δ } and satisfies Dh ( x )[ C x + G ( x , h ( x ))] = Sh ( x ) + H ( x , h ( x )) for | x | < δ . The flow on the center manifold is governed by the system d x dt = C x + G ( x , h ( x )) for all x ∈ R c with | x | < δ . Note that the linearized system given by the Center Manifold Theorem has noeigenvalues with positive real part, and thus no unstable subspace or manifoldat . This assumption is not required but is appropriate for us since Hadamardmatrices are global minima of the potential (2).The Center Manifold Theorem provides a means to bound the local dimen-sion of the space of degenerate fixed points, F , containing since the centermanifold at must contain F . Thus, by applying the theorem to the gradientsystem (3), we can estimate the local dimension of the space of dephased com-plex Hadamards at H by computing the dimension of the center manifold at H . This construction is reminiscent of the definition of the defect of a d × d Hadamard matrix H as the dimension of the solution space of the real linearsystem R i, = 0 , for 1 ≤ i ≤ dR ,j = 0 , for 2 ≤ j ≤ d (5) d (cid:88) k =1 [ H ] i,k [ H ∗ ] j,k ([ R ] i,k − [ R ] j,k ) = 0 , for 1 ≤ i < j ≤ d, where R ∈ M d × d ( R ) is a matrix of variables. The linear system (5) is derivedby considering a matrix H ◦ EXP(i R ), (cid:0) [EXP(i R )] i,j = e i[ R ] i,j (cid:1) , and computingthe Jacobian of the non-linear system R i, = 0 , for 1 ≤ i ≤ dR ,j = 0 , for 2 ≤ j ≤ d (6) d (cid:88) k =1 [ H ] i,k [ H ∗ ] j,k e i([ R ] i,k − [ R ] j,k ) = 0 , for 1 ≤ i < j ≤ d. The d + d − H ◦ EXP(i R ) is a dephasedHadamard.The dimension of the center manifold of system (3) at a Hadamard H is ac-tually equal to the defect d ( H ) since we are linearizing the dephased Hadamardconditions in both cases. In particular, recall that if the defect of a Hadamard4atrix is 0, then it must be isolated. Likewise, if there does not exist a centersubspace of Φ d at H d ( θ ), then the stable subspace is ( d − -dimensional andall points sufficiently close to H d ( θ ) must flow to it.Since a center manifold is not necessarily (and not usually) comprised en-tirely of fixed points, the Center Manifold Theorem could improve upon an over-estimate of the local dimension of the space of complex Hadamards suggestedby the the dimension of the center subspace and the defect. If Φ d ( θ ) (cid:54) = 0, then H d ( θ ) is not a complex Hadamard matrix and therefore any flow on the centermanifold, as slow as it may be, will shrink the bound given by the defect.A center manifold reduction is typically accomplished by performing a changeof coordinates on the system into eigen-coordinates so that the center manifoldcan be written as a graph over the center subspace. This becomes impracticalfor a high-dimensional systems since the process first requires diagonalization of(large) symbolic matrices. We have adopted an alternative method in which thecenter manifold, W c , is written as an embedding over the c -dimensional centersubspace, E c = span( v , . . . , v c ) [21]. One begins by writing W c = X ( t , . . . , t c ) = t v + . . . + t c v c + w ( t , . . . , t c ) , and expands w ( t , . . . , t c ) ∈ ( E c ) ⊥ in a Taylor expansion by repeated differen-tiation of the vector field. By observing that the center manifold is an invariantset (i.e. the flow at a point on the W c is tangent to W c there) one derives f ( X ( t , . . . , t c )) = α ( t , . . . , t c ) ∂ X ∂t + . . . + α c ( t , . . . , t c ) ∂ X ∂t c , (7)for some real-valued functions α i : R c → R , which can be shown to be the time-rates-of-change of the embedding parameters, t , . . . , t c . More precisely, α i = dt i /dt for i = 1 , . . . , c . As one expands w ( t , . . . , t c ), they also approximate each α i ( t , . . . , t c ) and thus the flow on the center manifold. Notice, if flow is nowherepresent on the center manifold, then X ( t , . . . , t c ) represents a local embeddingof dephased Hadamards and so this framework yields a series approximation ofthe manifold of interest.The principal computational limitation is the memory requirements of stor-ing high-order tensors, which are needed if one wishes to expand the centermanifold to high orders and which grow exponentially with order. This is ex-aggerated for large matrices, since the base of the exponential growth dependson the order of the Hadamard. Although identification of flow on the centermanifold may not require expansion to high orders – as shown by example inSection 3 – it should be noted that the absence of flow at any finite order cannotguarantee that flow is not present at some higher order.5 Explanatory Example
It is known that every 4 × F (1)4 ( a ) . = e i a − − i e i a − − − i e i a − e i a . Moreover, for all but one choice of parameter a ∈ [0 , π ], the defect of F (1)4 ( a ) co-incides with the dimension of this topological circle of inequivalent Hadamards.However, for a = π/
2, when the Hadamard is real, the defect jumps from 1 to 3.Thus, this is an example where we know that the defect overestimates the localfreedom of inequivalent Hadamards. This section serves as a illustrative exam-ple of the methodology developed in Section 2. In particular we use the centermanifold reduction to show what is already known: The space of inequivalentHadamards near F (1)4 ( π/
2) is one-dimensional, despite what the defect theremight suggest.For the remainder of this section we will denote the one-parameter family ofinequivalent 4 × F ( a ) . = F (1)4 ( a ). We will interchangeably referto elements of this space as either the matrix F ( a ), for a particular a ∈ [0 , π/ θ ( a ) . = (cid:104) a − π , π, a + π , π, , π, a + π , π, a − π (cid:105) . Derivation of the gradient system Φ ( θ ) (Equation 3) begins with the ma-trices H ( θ ) . = e i θ e i θ e i θ e i θ e i θ e i θ e i θ e i θ e i θ . With the aid of a computer algebra system, we compute explicit formulae forthe eigenvalues of the Jacobian matrix, D Φ | θ ( a ) , as functions of the parameter a ∈ [0 , π ]: D Φ | θ ( a ) = 4 − − sin a − − − − sin a a − −
11 1 − − a − − sin a − − a − − sin a a − a − sin a − a − − − sin a − a − − − − a − sin a − − − − sin a − . The characteristic polynomial of D Φ | θ ( a ) is p ( λ ; a ) = − λ [ λ + 8][ λ + (32 − a ) λ + 128(1 − sin a )]6 λ + (32 + 8 sin a ) λ + 128(1 + sin a )][ λ + 36 λ + (352 −
64 sin a ) λ + 512(1 − sin a )] . Let λ ( a ) = 0, λ ( a ) = −
8, and λ ( a ), λ ( a ) and λ ( a ) equal the three realroots of the cubic factor c ( λ ; a ) . = λ + 36 λ + (352 −
64 sin a ) λ + 512(1 − sin a ) ,λ ( a ) and λ ( a ) the roots of the quadratic q ( λ ; a ) . = λ + (32 − a ) λ + 128(1 − sin a ) , and λ ( a ) and λ ( a ) the roots of the quadratic q ( λ ; a ) . = λ + (32 + 8 sin a ) λ + 128(1 + sin a ) . The constant terms of c ( λ ; a ) and q ( λ ; a ) vanish precisely at a = π/
2, increasingthe multiplicity of the root λ = 0, of p ( λ ; a ), to three there. A plot of theeigenvalues of D Φ | θ ( a ) is given in Figure 1.Figure 1: Plot of the eigenvalues of the linearization of Φ at θ ( a ) for a ∈ [0 , π ]. λ ( a ) (blue), λ ( a ) (green), and λ ( a ) (red) simultaneously vanish at a = π/ a ∈ [0 , π ].As stated, for every parameter value (other than a = π/
2) there are exactly8 negative eigenvalues and 1 eigenvalue equal to 0. The latter corresponds to theone-dimensional manifold of fixed points parametrized by a , as its eigenvectoris the tangent vector to the manifold F (1)4 embedded in T , v . = [1 , , , , , , , , . At a = π/ λ and λ also vanish, giving rise to a three-dimensional center7anifold spanned by v and the eigenvectors v . = [0 , , , , , , , ,
0] and v . = [0 , , , , , , , , . We anticipate nonlinear flow at all nearby points off of the lines spanned by each v , v and v since the space of inequivalent Hadamards is one-dimensional at F ( π/ E c =span( v , v , v ), X ( t , t , t ) . = t v + t v + t v + w ( t , t , t ) , we derive cubic approximations of the time-rates-of-change of the embeddingparameters:˙ t = − t t − t t + 49 t t + 49 t t + 49 t t + 49 t t ˙ t = − t t − t t + 49 t t + 49 t t + 49 t t + 49 t t (8)˙ t = − t t − t t + 49 t t + 49 t t + 49 t t + 49 t t . Thus, the motion of the point X ( t , t , t ) on W c is governed by (8).Because every nonzero cubic-term in (8) is a mixed monomial, setting anytwo embedding parameters to zero will result in no flow. For example, setting t = t = 0 amounts to choosing a point X ( t , , ∈ F (1)4 on the manifoldof fixed points (since moving in the direction of v amounts to varying theparameter a ). Any solution trajectory of (8) converges to a point on one of theaxes, where two of the embedding parameters vanish.The important point is that there is flow for any choice of embedded point X ( t , t , t ) not directly over the lines spanned by v , v or v (i.e. on an axisin t − t − t space). Therefore, the local dimension of the space of inequivalentHadamards at F ( π/
2) cannot be greater than one.Recall that Φ does not converge to the space of inequivalent Hadamards, butrather to the superset containing all row and column permutations of the coreof dephased 4 × P r ( i, j ) and P c ( i, j ) be the 4 × i and j and columns i and j respectively. Thereare exactly five unique row and column permutations of F ( a ) which, for somechoice of parameter a , are again equal to the matrix F ( π/ F ( π/
2) = F (3 π/ P c (2 , P r (2 , F (3 π/ P r (2 , F ( π/ P c (2 , P r (2 , F ( π/ P c (3 , P r (3 , F ( π/ P c (2 , = [ θ , . . . , θ ] of T . For example, P r (2 , H ( θ ) P c (3 ,
4) = e i θ e i θ e i θ e i θ e i θ e i θ e i θ e i θ e i θ , corresponds to the permutation σ . = (14)(26)(35)(7)(89)[ θ , θ , θ , θ , θ , θ , θ , θ , θ ] (cid:55)→ [ θ , θ , θ , θ , θ , θ , θ , θ , θ ] . Let σ . = (12)(3)(48)(57)(69); the permutation of the coordinates resulting fromthe action of P r (3 ,
4) and P c (2 , v ,which we recall is tangent to F ( a ) at a = π/
2, in the following ways: v = [1 , , , , , , , , σ (cid:55)−→ [0 , , , , , , , ,
0] = v , and v = [1 , , , , , , , , σ (cid:55)−→ [0 , , , , , , , ,
0] = v , while v remains fixed under P r (2 ,
4) and P c (2 , F ( π/
2) caused bythe permutations of its core. We conclude that the three-dimensional centermanifold emerges at the real Hadamard to account for three copies of the spaceof dephased, permutation-equivalent Hadamards intersecting here.The local geometric consequence of this result can be seen quite convincinglyby directly visualizing the flow of core phases taken from a neighborhood of F ( π/ F ( π/ F ( π/ ( θ )in R near F ( π/
2) is provided. (i) (ii) (iii) (iv) -5-10-15-20-25-30
Figure 2: Snapshots of 500 initial phases – drawn from R uniformly at randomfrom a neighborhood of the core phases corresponding to F ( π/
2) – as they evolveunder the flow defined by Φ ( θ ), at times (i) 5, (ii) 20, (iii) 70, and (iv) 500.Each point cloud has been projected onto its top three principal components,and each point θ is colored by log of the magnitude of the vector field Φ ( θ ).9 Applications
The smallest order for which classification of complex Hadamards is incom-plete is order six. Numerical searches, analysis of known families, and a generalmethod of construction due to Sz¨oll˝osi, which depends on four free parameters,all support the conjecture that Sz¨oll˝osi’s generic four-parameter family and theisolated matrix S (0)6 capture all inequivalent 6 × S (0)6 the defectis always found to be four. We further support the conjecture by expanding theflow on the center manifold of Φ at selected Hadamards without ever encoun-tering non-zero coefficients in the Taylor expansion of the time-rates-of-changeof the embedding parameters. The Affine Fourier Family
Stemming from the Fourier matrix F = w w w w w w w w w w w w w w w w w w w w w , (where w = e π i / ) are two, two-parameter affine orbits of Hadamards F (2)6 ( a, b ) := F ◦ EXP (i R ( a, b )) F (2)6 ( a, b ) T := F ◦ EXP (cid:0) i R ( a, b ) T (cid:1) , where R ( a, b ) = • • • • • •• a b • a b • • • • • •• a b • a b • • • • • •• a b • a b . Note that H ◦ K denotes the Hadamard (entrywise) product of the matrices H and K , and the numeric value 0 has been replaced with • to improve readability.The defect of F is four, and numerical evidence supports the conjecture thatthere is a four-dimensional non-affine family stemming from it [27]. In kind thecenter subspace of Φ at F (i.e. the kernel of the symmetric matrix D Φ | F ) isfour dimensional and is spanned by the vectors v = [1 , , , , , , , , , , , , , , , , , , , , , , , , v = [0 , , , , , , , , , , , , , , , , , , , , , , , , v = [1 , , , , , , , , , , , , , , , , , , , , , , , , v = [0 , , , , , , , , , , , , , , , , , , , , , , , , . Recalling the coordinates of T in which Φ is expressed, it is clear that v and v span F (2)6 ( a, b ) and v and v span F (2)6 ( a, b ) T .10e compute the center manifold X ( t , t , t , t ) = w ( t , t , t , t ) + (cid:88) i =1 t i v i , as an embedding and, as before, expand the functions α i ( t ) = ˙ t i in powerseries. In support of the conjecture that there exists a four-dimensional manifoldof complex Hadamards passing through F , all partial derivatives of each α i through fifth order are found to vanish. This means that if the four-dimensionalcenter manifold stemming from F does not consist entirely of fixed points, theflow near F must be very slow. This may be interpreted in the following way:points on the local center manifold are very close to being Hadamard, althoughthey may not be. If the local manifold is four-dimensional, our expansion of X ( t , t , t , t ) gives a series approximation of the space of dephased Hadamardsnear F . Again we cannot rule out the possibility that a higher-order expansionmight reveal flow on the center manifold. The affine family D (1)6 Another maximal affine order-6 family, this one foundby Dit¸˘a [8], stems from the symmetric matrix D = − − i − i i1 i − − i − i1 − i i − − i1 − i − i i − − i − i i − . One representative of the five, permutation-equivalent families stemming from D is D ( c ) := D (1)6 ( c ) = D ◦ EXP (i R ( c )), where R ( c ) = • • • • • •• • • • • •• • • c c •• • − c • • − c • • − c • • − c • • • c c • , Computing D Φ | D ( c ) gives the linearized flow explicitly in terms of c . Thecharacteristic polynomial of D Φ | D ( c ) is found to be p ( λ ; c ) = − λ ( λ + 24) f f f , where f ( λ ; c ) = 32 λ (cid:0) λ + 64 λ + 1024 (cid:1) cos(2 c ) + 512 λ cos(4 c )+ λ + 128 λ + 6096 λ + 131840 λ + 1241600 λ + 3686400 f ( λ ; c ) = 256 λ cos(2 c ) + λ + 88 λ + 2608 λ + 29312 λ + 92160 f ( λ ; c ) = 64 (cid:0) λ + 76 λ + 1408 λ − λ − (cid:1) cos(2 c )+ λ + 132 λ + 6448 λ + 142272 λ + 1386496 λ + 5140480 λ + 5308416 . λ in p ( λ ; c ) guarantees that the center subspace at D ( c ) is at leastfour dimensional for all values of c . A plot of the eigenvalues of D Φ | D ( c ) –given in Figure 3 – suggests that the stable subspace is 21 dimensional for allchoices of c . In fact, careful consideration of the factors that depend on c revealsthat the center subspace is exactly four dimensional for all c since λ = 0 is nevera root of f , f , or f . This follows from the observation that λ is not a divisorof f or f for any c since the constant terms in these factors do not depend on c .In f , the constant (in λ ) term is − c ) + 5308416 which cannot bemade to vanish with c ∈ R . This proves that d (cid:16) D (1)6 ( c ) (cid:17) = 4 for every memberof this affine family.Figure 3: Plot of the 25 eigenvalues of D Φ | D ( c ) for c ∈ [ − π/ , π/ f ( λ ; c ) (green) havemultiplicity two, and all other eigenvalues (gray) are simple.Permutations of the core of D ( c ) give four other families which are per-mutation equivalent. These permutations act on the coordinates of the vectortangent to the curve D ( c ) at D (0), v = [0 , , , , , , , , , , , − , , , − , , − , , , − , , , , , , to give v = [0 , , , , , , , , , , , , , , , − , , − , , , − , , − , , v = [0 , − , , − , , , , , , , , , , , , , , , , , , − , , − , v = [0 , , , , , − , , , − , , − , , , − , , , , , , , , , , , v = [0 , , − , , − , , , − , , − , , , , , , , , , , , , , , , . Since − v = v + v + v + v , we take { v , v , v , v } as a natural choiceof basis for the center subspace at D . If flow exists on the center manifold,our choice of basis ensures that it will present itself as nonzero coefficients of12he mixed monomials in the Taylor expansions of ˙ t , . . . , ˙ t . Unsurprisingly, wedid not encounter any nonzero coefficients in these expansions, reinforcing theconjecture that the space of dephased Hadamards near D is four dimensional. Beauchamp and Nicoara’s B (0)9 Often one is able to obtain explicit eigen-vectors and eigenvalues for the linearizations of Φ i ( θ ) and explicit coefficientsin the Taylor expansions of the center manifold and the time-rates-of-change ofthe embedding parameters. If numerical computations are undertaken, numeri-cal approximations of the time-rates-of-change of the embedding parameters aredetermined. If these approximate coefficients are bounded away from 0, one candeduce that there exists non-linear flow on parts of the center manifold, evenwithout exact knowledge of that flow.It is known that the defect of the matrix B (0)9 = − (cid:15) (cid:15) − (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) − (cid:15) (cid:15) (cid:15) − (cid:15) (cid:15) (cid:15) (cid:15) − (cid:15) (cid:15) (cid:15) (cid:15) − (cid:15) (cid:15) − − (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) − (cid:15) (cid:15) − (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) − − (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) − (cid:15) − (cid:15) − (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) − , where (cid:15) = e π i / , is d (cid:16) B (0)9 (cid:17) = 2 [22]. Since no family was known that con-tained B (0)9 , it was a longstanding open problem to determine if this matrixwas actually isolated, despite having a positive defect. Recently, it was shownthat, in fact, the defect arises from a non-affine 2-dimensional family [23] con-taining it. Prior to learning of these results, we computed – to 200 decimalplaces of accuracy – numerical approximations of the basis vectors, v and v , which span the center subspace. We then expanded the center manifold X ( t , t ) . = t v + t v + w ( t , t ) as a numerical embedding over the centersubspace. Naturally, the numerical derivatives of the embedding parametersremain zero, as they must at all orders.Visualizing the flow of Φ ( θ ) in a neighborhood of B (0)9 , which appears tobe drawn to a 2-dimensional plane (See Figure 4 and ancillary files), providesanother type of evidence of the conclusions which were proven definitively in[23]. Thus, this serves as an example where the numerical series expansionand numerical integration of the flow was only able to provide evidence for theexistence of a center manifold of fixed points. Golay sequence affine family G (1)10 & non-Dit¸˘a-type D (3)10 In this sec-tion we derive two 10 ×
10 affine families stemming from a member of the one13 i) (ii)(iii) (iv) (v) -2-4-6-8-10-12
Figure 4: Snapshots of 500 initial phases – drawn from R uniformly at randomfrom a neighborhood of the core phases corresponding to B (0)9 – as they evolveunder the flow defined by Φ ( θ ), at times (i) 5, (ii) 20, (iii) 70, and (iv,v) 500.Each point cloud has been projected onto its top three principal components,and each point θ is colored by log of the magnitude of the vector field Φ ( θ ).parameter family G (1)10 ( a ) = e i a i e i a e i a − i e i a − e i a − i e i a − e i a − e i a − − i e i a − e i a e i a i − i e i a − e i a i e i a − i − i e i a i e i a − i i − − e i a e i a − e i a i − − i − i − i e i a − i e i a − e i a e i a e i a − − e i a − e i a e i a i e i a e i a − e i a − i e i a − i − e i a e i a − e i a − − i i e i a − e i a i 1 − e i a − i e i a − i i i − e i a − i − i i − e i a − i − − i − − , presented here in dephased form, discovered by Lampio et al . [24]. The defectof G (0) := G (1)10 (0) and the dimension of the center subspace of Φ there is8, so it may be possible to introduce additional affine parameters. As we haveseen, the center subspace at a Hadamard, H , must contain all vectors tangentto dephased families stemming from H , affine or otherwise. For example, thevector v ∈ R with 1’s in the coordinates corresponding to the core entriesof G (1)10 ( a ) whose phases depend on a (and 0’s elsewhere) will be in the centersubspace because it is tangent to the line of Hadamards, G (1)10 ( a ). Thus, one cansearch for affine families by considering linear combinations of vectors in a basisfor the center subspace and introducing appropriately-scaled parameters in thephases corresponding to nonzero coordinates.14ontained in the kernel of D Φ | G (0) is the vector V := • • • • • • • • • •• − • − − • • • • − • • • − − • • • • •• • • − − • • • • •• • • • • • • •• • • • • • • • • •• • • • • •• • • • • •• − • − − • • • • − • • • • • • • • , shown here as the lower-right 9 × V to establish its relationshipwith the core of G (0). By computing G (0) ◦ EXP ( a i V ) (where a is a freeparameter) we uncover a one-parameter affine family; explicitly M (1)10 ( a ) := e − i a i e − i a − i e − i a − − i − − e − i a − − i − e − i a e − i a i − i 1 − − i − i e − i a i e − i a − i i − − e i a − − − i e i a − i − i1 − i − − − − e i a − − i − i e i a − e i a i e i a − − − i e i a i − e i a e i a − e i a − i1 − i e − i a i i e − i a − e − i a − i − i i − e − i a − i − e i a − i − e i a i − . Similarly, the null vectors of D Φ | G (0) , expressed in the cores of U := • • • • • • • • • •• • • • • • • • • •• • • • • • • • • • • •• • • • • • • • • •• • • • • • • •• • • • • • • • • •• • • • • • • • • • • •• • • • • • • • and W := • • • • • • • • • •• • • •• • • • • • • •• • • • • • • • • •• • • • • • • • • •• • • • • • • • • •• • • •• • • • • • • •• • • • • • • •• • • • • • • • , together reveal the two-parameter family G (0) ◦ EXP ( a i U + b i W ) M (2)10 ( a, b ) := e i b e i b − i − e i b e i b − i e i b − e i b − − − i e i b − e i a e i a i e i b − i e i a e i a − e i a i e i a − i − i e i a i − i i e i a − − − − − i − i − i1 − i − e i a − − e i a − e i b − e i b − i − i e i b − e i b i e i b e i b − − − i e i b i e i a − e i a i e i b e i a − e i a e i a − i e i a − i i e i b i e i a − − i e i b − i e i a i − − i − e i b − i e i a − e i b i e i a − .M (1)10 and M (2)10 were verified to be Hadamard by symbolic computation.Interestingly, these families and G (1)10 are independent in the sense that they do15ot, together, form a three-parameter affine family. That being said, a centermanifold reduction did not rule out the possibility that they belong to someunifying family that has yet to be discovered.We applied the same approach to the non-Dit¸˘a-type matrix D = D (3)10 (0 , ,
0) := − − i − i − i − i i i i i1 − i − − i − i − i i i1 − i i − − i i − i i − i i1 − i i − i − − i i − i1 − i − i i i − − i − i1 i − i − i i i − − i − i i1 i − i i − i i − i − − i1 i i − i i − i − i i − − i1 i i i − i − i i − i − i − , whose membership in a three-parameter affine family D (3)10 was already estab-lished by Sz¨oll˝osi [26]. Notice that the defect of D is 16, and so there maybe more parameters which can be introduced to the family D (3)10 ( a, b, c ). To-wards identifying these additional parameters, we searched for and found abasis, { V , . . . , V } , for the kernel of D Φ | D whose elements each describe aone-parameter affine family stemming from D . In Table 1 we list the nonzerocoordinates of each V , . . . , V – each of which happens to be 1 or -1 – byidentifying the coordinates having either value.Table 1: Nonzero coordinates of the vectors in the basis { V , . . . , V } for D Φ | D . A nonzero coordinate has value 1 or -1, indicated by the subcolumnto which it belongs. vector coordinate V V
10 12 16 18 64 66 70 72 2 8 20 26 56 62 74 80 V
28 29 35 36 46 47 53 54 4 6 13 15 67 69 76 78 V V
37 40 47 54 55 58 65 72 5 7 15 17 32 34 78 80 V V
19 27 29 34 38 43 64 72 3 8 13 14 58 59 75 80 V
37 39 43 45 46 48 52 54 5 6 23 24 59 60 77 78 V V
47 48 49 50 74 75 76 77 15 18 24 27 33 36 42 45 V V
12 14 30 32 48 50 75 77 20 22 24 27 38 40 42 45 V
47 50 56 59 65 68 74 77 15 16 17 18 42 43 44 45 V
25 26 34 35 47 50 74 77 15 18 42 45 57 58 66 67 V
19 23 28 32 64 68 73 77 3 4 8 9 39 40 44 45 V
19 23 49 53 58 62 73 77 3 9 33 34 39 45 69 70For each k = 1 , . . . ,
16, the collection of matrices D ◦ EXP (i σ k V k ) – where σ i is a parameter – is an affine family. This does not say that there is a 16-dimensional affine family stemming from D ; in fact, simultaneous inclusion of16ll σ i destroys the Hadamard property for most choices of parameters. How-ever, many combinations of inclusion of parameters do give two- and three-dimensional families. For example, D (3)10 ( a, b, c ) = D ◦ EXP (i( a V + b V + c V )) . Another example of a three-parameter family found among these vectors is D ◦ EXP (i( a V + b V + c V )) . Note that no combination of four (or more) of these vectors give an affine family,although this does not guarantee one does not exist.The existence of this particular basis for the kernel of D Φ | D does pro-vide one possibility for the genesis of a 16-dimensional center subspace: cer-tainly the defect must be at least 16 to account for the intersection of 16, possi-bly permutation-equivalent, one-parameter affine (sub-)families passing through D . In this paper we have given a new interpretation of the defect of a Hadamardmatrix as the dimension of a center subspace of a gradient flow whose fixedpoints are exactly the dephased Hadamard matrices. We have applied thistechnique to a simple example, using dynamical systems theory to explain whythe defect of the real 4 × F (1)4 . We have used tools from dynamical systems theory toprove that the d ( D ( c )) = 4 for all values of c and have presented a new type ofevidence in support of existing conjectures concerning 6 × ×
10 affine families.It is a virtue of the formalism built in Section 2 that we need not have ex-plicit values for the Taylor coefficients of the functions α i ( t , . . . , t c ) to concludethat flow on the center manifold exists. It is a shortcoming of center manifoldreduction, in general, that it cannot prove flow does not exist on some part ofa center manifold. This limitation is a consequence of the fact that, a priori ,one has no knowledge of the smallest order in the Taylor expansion where onemight first encounter nonzero contribution to the time-rates-of-change of theembedding parameters. We are not certain that our application is bound withthis deficiency and are hopeful that a deeper understanding of the derivativesof the vector field Φ d may be exploited to further expand the use of dynamicalsystems theory to attack questions about complex Hadamards. In particular, ifone could show that all coefficients in the expansion of the time-rates-of-changeof the embedding parameters vanish in the center manifold reduction of Φ d ata Hadamard H , then one would prove the existence of a positive-dimensionalfamily of complex Hadamards stemming from H .Finally, vectors in a basis for the center subspace of Φ n at an order- n matrix17 may be tangent to affine families stemming from H . This fact can be exploitedto find new affine families, in the manner of Section 4. References [1] J. Hadamard,
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