SYMMETRIC KRONECKER PRODUCTS AND SEMICLASSICAL WAVE PACKETS

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1 SYMMETRIC KRONECKER PRODUCTS AND SEMICLASSICAL WAVE PACKETS GEORGE A HAGEDORN AND CAROLINE LASSER Abstract We investigate the iterate Kronecker prouct of a square matrix with itself an prove an invariance property for symmetric subspaces This motivates the efinition of an iterate symmetric Kronecker prouct an the erivation of an explicit formula for its action on vectors We apply our result for escribing a linear change in the matrix parametrization of semiclassical wave packets Introuction The Kronecker prouct of matrices is known to be ubiquitous [VL00], an our aim here is to investigate the n-fol Kronecker prouct of a complex square matrix M C with itself, M n M M, n N, }} n times an to apply our finings to the parametrization of semiclassical wave packets Two-fol symmetric Kronecker proucts In semiefinite programming See for example [AHO98] or [Kle02, Appenix E], the two-fol Kronecker prouct has notably occurre in combination with subspaces of a particular symmetry property One consiers the space X 2 x C 2 : x vecx, X X t C }, that contains those vectors that can be obtaine by the row-wise vectorization of a complex symmetric matrix The imension of the space X 2 is L One can prove that this space is invariant uner Kronecker proucts, in the sense that for all matrices M C, one has M M x X 2, whenever x X 2 Now one uses the stanar basis of C 2 for constructing an orthonormal basis of the subspace X 2 an efines a corresponing sparse L 2 2 matrix P 2 that has the basis vectors as its rows The symmetric Kronecker prouct of M with itself is then the L 2 L 2 matrix S 2 M P 2 M M P 2 Date: March 4, Mathematics Subject Classification 5A69, 5B0, 8Q20 Key wors an phrases Kronecker prouct, symmetry, semiclassical wave packet

2 2 GEORGE A HAGEDORN AND CAROLINE LASSER 2 n-fol symmetric Kronecker proucts How oes one exten this construction to symmetrizing n-fol Kronecker proucts? It is instructive to revisit the secon orer space in two imensions an to write a vector x X 2 as x x 2,0, x,, x,, x 0,2 t This labelling uses the multi-inices k k, k 2 N 2 with k + k 2 2 in the reunant enumeration ν 2 2, 0,,,,, 0, 2 t This escription allows for a straightforwar extension to higher orer n an imension One works with a reunant enumeration ν n of the multi-inices k k,, k N with k + + k n, an efines } X n x C n : x j x j if ν n j ν n j The imension of X n equals the number of mult-inices in N of orer n, that is the binomial coefficient n + L n n An again, we can prove invariance in the sense that for all M C M n n x X n, whenever x X n See Proposition 2 Then, we use the stanar basis of C n to buil an orthonormal basis of X n an assemble the corresponing sparse L n n matrix P n All this motivates the efinition of the n-fol symmetric Kronecker prouct as S n M P n M n P n The matrix S n M is of size L n L n an inherits structural properties as invertibility or unitarity from the matrix M See Lemma 4 Our main result Theorem provies an explicit formula for the action of the matrix S n M in terms of multinomial coeffients an powers of the entries of the original matrix M Labelling the components of a vector y C Ln by multi-inices of orer n, we obtain for all k N with k n that S n M y k k! α k α k k α k α m α mα α + + α! y α+ +α, where m,, m C enote the row vectors of M The summations weighte with multinomial coefficients stem from the n-fol Kronecker prouct M n, whereas the square roots of the factorials originate in the orthonormalization of the row vectors of the matrix P n 3 Application to semiclassical wave packets Our interest in n-fol symmetric Kronecker proucts emerge when stuying linear changes in the parametrization of semiclassical wave packets Semiclassical wave packets have first been propose in [Hag85] as a multivariate non-isotropic generalization of the Hermite functions See also [Hag98] A family of semiclassical wave packets ϕk [A, B] : k N }

3 SYMMETRIC KRONECKER PRODUCTS AND SEMICLASSICAL WAVE PACKETS 3 is parametrize by two invertible matrices A, B GL, C an forms an orthonormal basis of the Hilbert space of square integrable functions More recently, semiclassical wave packets have been also use for the numerical iscretization of semiclassical quantum ynamics by E Faou, V Grainaru, an C Lubich [FGL09, Lub08] The computationally emaning step of this metho is the assembly of the Galerkin matrix for the potential function V : R R accoring to ϕ k [A, B], V ϕ l [A, B] ϕ k [A, B]x V x ϕ l [A, B]x x R In an effort to evelop a new variant [BGH] that performs the quarature for semiclassical wave packets with ifferent parametrizing matrices A, B, the search began for the matrices escribing a linear change in the parametrization Our main application is Corollary in Section 54 It gives an explicit formula in terms of an n-fol symmetric Kronecker prouct 4 Organization of the paper In the next Section,we start with some combinatorics for explicitly relating the lexicographic enumeration of multi-inices of orer n with the reunant enumeration ν n Then we introuce the symmetric subspaces X n in Section 3 an construct an orthonormal basis together with the corresponing matrix P n In Section 4, we efine the n-fol symmetric Kronecker prouct an prove our main result Theorem An introuction to semiclassical wave packets an the escription of linear changes in their parametrization by symmetric Kronecker proucts is given in Section 5 5 Notation On some occasions we shall use the binomial coefficient n n!, for non-negative integers n j j n j! j! We write a multi-inex k N as a row vector k k,, k moulus k k + + k, an the multinomial coefficient k k! k k! k!, for k N We use the We aopt the convention that any multinomial coefficient with any negative argument is efine to be 0 We also use the kth power of a vector, x k x k x k, x C 2 Combinatorics 2 Reverse Lexicographic orering First we enumerate the set multi-inices of orer n in imensions, k N : k n }, n N, in reverse lexicographic orering an collect them as components of a formal vector enote by l n The length of the vector l n is the binomial coefficient n + L n n One can think of this in the following way [JHT]: The multi-inices k of orer n in imensions are in a one-to-one corresponence with the sequences of n ientical balls an ientical sticks The sticks partition the line into bins into which

4 4 GEORGE A HAGEDORN AND CAROLINE LASSER one can insert the n balls The first bin is to the left of all the sticks, an contains k balls; the last bin is to the right of all the sticks, an contains k balls; for 2 j, the j th bin is between sticks j an j, an it contains k j balls Eg, the multi-inex 3, 2, 0, in four imensions correspons to If all these objects were istinguishable, there woul be n +! permutations, but since the balls are all ientical, one must ivie by n!, an since the sticks are all ientical, one must ivie by! 22 A reunant enumeration Next we reunantly enumerate an collect multi-inices of moulus n in a vector ν n of length n We procee recursively an set ν 0 0,, 0 t, ν e,, e t, an ν n+ ν n + e ν n n + e vec ν n + e ν n n + e, n 0, where e,, e C are the stanar basis vectors of C, an vec enotes the row-wise vectorization of a matrix For example, for 2, we have ν 2 2, 0,,,,, 0, 2 t, ν 3 3, 0, 2,, 2,,, 2, 2,,, 2,, 2, 0, 3 t 23 A partition For relating the lexicographic an the reunant enumeration, we efine the mapping σ n :,, L n } P,, n } so that for all i,, L n } an j,, n } the following hols: j σ n i ν n j l n i For example, for 2, we have σ 2 }, σ 2 2 2, 3}, σ 2 3 4} an σ 3 }, σ 3 2 2, 3, 5}, σ 3 3 4, 6, 7}, σ 3 4 8} We observe the following partition property Lemma We have an #σ n i n, i,, L n, l n i i,, L n σ n i, n }, where the union is pairwise isjoint

5 SYMMETRIC KRONECKER PRODUCTS AND SEMICLASSICAL WAVE PACKETS 5 Proof We first prove that we have a partition property For any j,, n } there exists i,, L n } so that ν n j l n i So, we clearly have i,,l n σ n i,, n } Moreover, since j σ n i σ n i is equivalent to l n i l n i, that is, i i, the union is isjoint For proving the claime carinality, we argue by inuction For n 0, we have l 0 0,, 0 ν 0, σ 0 0 }, #σ 0 For the inuctive step, we observe that in the reunant enumeration ν n, the multi-inex k k,, k can be generate from possible entries in ν n, k, k 2,, k,, k,, k, k by aing e,, e, respectively Of course, such an entry only belongs to ν n if all its components are non-negative For each of these inices with all entries non-negative, there is a unique number j, 2,, L n }, such that l n j is the given inex If one of these inices has a negative entry, we efine j an σ n to be the empty set, ie, σ n }, whose carinality is 0 We list the numbers efine this way as i,, i, an note that all the positive values in this list must be istinct Then, #σ n i m m m #σ n i m n k,, k m,, k } n! k! k m! k! k m > 0 0 k m 0 n! k + + k k! k! n k 3 Symmetric subspaces We next analyze the symmetric spaces } X n x C n : x j x j if ν n j ν n j, n N We have X 0 C an X C, whereas X n is a proper subset of C n for n 2

6 6 GEORGE A HAGEDORN AND CAROLINE LASSER For example, for 2, X 2 x C 4 : x 2 x 3 }, X 3 x C 8 : x 2 x 3 x 5, x 4 x 6 x 7 } Any vector x X n has n components, but the components that correspon to the same multi-inex in the reunant enumeration ν n,, ν n n have the same value Hence, at most L n components of x X n are ifferent They may be labelle by the multi-inices l n,, l n L n, an we often refer to them by x lni, i,, L n The symmetric subspace X 2 has also been consiere in [VLV5], an we next relate the alternative secon orer constructions to ours 3 The secon orer space The secon orer subspace } X 2 x C 2 : x j x j if ν 2 j ν 2 j can also be escribe in terms of matrices Since e + e e + e ν 2 vec, e + e e + e we may write X 2 x C 2 : x vecx, X X t C } Alternatively, as in [VLV5, 23], one may permute the stanar basis vectors e,, e 2 C 2 accoring to the 2 2 permutation matrix Π e +0, e +,, e +,, e +0, e +,, e 2 an characterize the symmetric subspace as } X 2 x C 2 : Π x x This way of writing the secon orer subspace correspons to the efinition of symmetric tensor spaces by permutations as given in [Hack2, Chapter 35] 32 Relation between the subspaces Due to the recursive efinition of the reunant multi-inex enumeration, the symmetric subspaces of neighboring orer can be relate to each other as follows Lemma 2 The symmetric subspace X n+ is containe in the -ary Cartesian prouct of the symmetric subspace X n, Proof We ecompose a vector X n+ X n X n, n N x x,, x t X n+ into subvectors with n components each labelle by the multi-inices The n+ components of x can be ν n + e,, ν n n + e,, ν n + e,, ν n n + e,

7 SYMMETRIC KRONECKER PRODUCTS AND SEMICLASSICAL WAVE PACKETS 7 so that the components of the subvector x m, m,,, can be labelle by Hence, x m j ν n + e m,, ν n n + e m x m j if ν nj ν n j, an x m X n for all m,, The two-imensional example X C 2 an X 2 x C 4 : x 2 x 3 } shows that the inclusion of Lemma 2 is in general not an equality 33 An orthonormal basis We now use the stanar basis of C n to construct an orthonormal basis of X n Lemma 3 Let e,, e n be the stanar basis vectors of C n, an efine the vectors p i e j, i,, L n #σn i j σ ni Then, p,, p Ln } forms an orthonormal basis of the space X n Proof For all i 0,, L n an j, j,, n, we have p i j #σn i /2, if j σ n i, 0, otherwise Since j σ n i if an only if ν n j l n i, we have p i j p i j if ν n j ν n j, an thus p i X n We also observe, that for all i, i,, L n, p i, p i e j, e j δ i,i #σn i #σ n i j σ ni j σ ni Hence, the vectors p,, p Ln are orthonormal Moreover, for all x X n, we have p i, x e j, x #σ n i x lni, #σn i an therefore x n j L n i e j, x e j p i, x p i L n j σ ni i j σ ni e j, x e j L n i x lni #σn i p i 34 An orthonormal matrix The orthonormal basis vectors p,, p Ln X n allow us to efine the sparse rectangular L n n matrix p P n p L n

8 8 GEORGE A HAGEDORN AND CAROLINE LASSER that has the L n basis vectors as its rows For example, for 2, we have e P 2 2 e 2 + e , P 3 e 4 e 2 e 2 + e 3 + e 5 2 e 4 + e 6 + e 7 e We summarize some properties of the matrix P n an of its ajoint P n an calculate explicit formulas for their actions on vectors Proposition The L n n matrix P n an its ajoint P n satisfy P n P n I Ln L n, rangep n X n Moreover, for all x X n, P n x i n x lni, i,, L n, l n i an for all y C Ln, P n y lni / n l n i y i, i,, L n In particular, P n P n x x, whenever x X n Proof The two properties P n P n I Ln L n an rangep n X n equivalently say, that the row vectors of P n buil an orthonormal basis of X n For any y C Ln, the vector P n y is a linear combination of the vectors p,, p Ln an therefore in X n Labelling its components by l n,, l n L n, we obtain P n y lni L n i p i lni y i For x X n an i,, L n, we obtain P n x i n j #σn i y i, i,, L n p i νnj x νnj #σ ni #σn i x l ni #σ n i x lni, since #σ n i components of p i o not vanish In particular, P n P n x lni #σn i P nx i x lni

9 SYMMETRIC KRONECKER PRODUCTS AND SEMICLASSICAL WAVE PACKETS 9 4 Symmetric Kronecker proucts 4 Iterate Kronecker proucts We next investigate the action of an n-fol Kronecker prouct on the symmetric subspace X n, n N Proposition 2 Let M C, an enote by m,, m C the row vectors of the matrix M Then, M n x X n, whenever x X n Moreover, for all k N with k n, an all x X n, M n x k where α,, α N α k α k k α k α m α m α x α+ +α, Proof For n, we have M n M an X n C, an our formula reuces to usual matrix-vector multiplication written as Mx ek j m e k j x ej, k,, For the inuctive step, we consier x x,, x X n+ ecompose as in Lemma 2 with x j X n We compute m M n m M n M n+ x m M n m M n x x m M n x + + m M n x m M n x + + m M n x By the inuctive hypothesis, we have for all j,,, that m j M n x + + m j M n x X n The components of these vectors can be labelle by k N with k n, an we have m j M n x + + m j M n x k k k m jm α α α mα x α + +α + α k α k + m j m α mα x α + +α

10 0 GEORGE A HAGEDORN AND CAROLINE LASSER We now observe that kj m αj+e j x α + +α + + m αj+e j x α + +α α j k j α j α j e k j α j k j+ kj α j e kj + α j m αj j x α + +α j e + +α + + α j e k j m αj j x α+ +α α k kj α j e Therefore, for all j,,, m j M n x + + m j M n x k k kj + α α j α k α j k j+ m αj j k α x α + +α j e + +α m α mα x α + +α, Altogether, we have proven for all k N with k n +, an all x X n+ M n+ x k k m α m α x α+ +α k α α α k α k 42 Symmetric Kronecker proucts Having proven that n-fol Kronecker proucts leave the nth symmetric subspace invariant, we efine the n-fol symmetric Kronecker prouct as follows: Definition For M C an n N, we efine the L n L n matrix S n M P n M n P n an call it the n-fol symmetric Kronecker prouct of the matrix M The n-fol symmetric Kronecker prouct has useful structural properties Lemma 4 The n-fol symmetric Kronecker prouct S n M of a matrix M C satisfies S n M S n M If M GL, C, then S n M GLL n, C with S n M S n M In particular, if M U, then S n M UL Proof Since M M M M an M n M n, we have S n M P n M n P n S n M If M is invertible, then M M is invertible with M M M M By Proposition 2, M n p j X n, j,, L n Proposition yiels P n P n x x for all x X n an P n p j e j, j,, L n,

11 SYMMETRIC KRONECKER PRODUCTS AND SEMICLASSICAL WAVE PACKETS where e,, e Ln C Ln are the stanar basis vectors Therefore, S n M S n M e j P n M n P n P n M n p j P n p j e j, that is, S n M S n M I Ln L n 43 The main result The explicit formula of Proposition 2 for the n-fol Kronecker prouct also allows a etaile escription of the n-fol symmetric Kronecker prouct Theorem Let M C, an enote by m,, m C the row vectors of the matrix M Then, the n-fol symmetric Kronecker prouct satisfies S n M y k k! α k α k k α k α for all y C Ln an k N with k n m α mα α + + α! y α+ +α Proof By Proposition, we have Pn y X n an / n Pn y lni y i, for i,, L n l n i By Proposition 2, the n-fol Kronecker prouct leaves X n invariant, an we have for all k N with k n, M n Pn y k k m α k m α α α P n y α+ +α α k α k k k m α m α α + + α! y α+ +α α α n! α k α k By Proposition, we have so that Pn M n Pn y k n! k! α k k! α k P n x i α α k k α k n x lni, for i,, L n, l n i k α k α k α m α mα α + + α! n! y α+ +α m α mα α + + α! y α+ +α

12 2 GEORGE A HAGEDORN AND CAROLINE LASSER 5 Application to semiclassical wavepackets 5 Parametrizing Gaussians We consier two complex invertible matrices A, B GL, C that satisfy the conitions 2 A t B B t A 0, A B + B A 2 I These two conitions imply that B A is a complex symmetric matrix such that its real part satisfies ReB A A A is a Hermitian an positive efinite matrix, see [Hag80] Let > 0 an efine the multivariate complex-value Gaussian function ϕ 0 [A, B] : R C, ϕ 0 [A, B]x π /4 eta /2 exp x, B A x 2 Then, ϕ 0 [A, B] is a square-integrable function, an the constant factor eta /2 ensures normalization accoring to R ϕ 0 [A, B]x 2 x Changing the parametrization by a unitary matrix changes the Gaussian function only by constant multiplicative factor of moulus one: Lemma 5 Let A, B GL, C satisfy the conitions 2 Then, for all unitary matrices U U, the matrices A A U an B B U also satisfy the conitions 2 Moreover, ϕ 0 [A, B ] etu /2 ϕ 0 [A, B] Proof We observe that A t B B t A U t A t B BAU 0 A B + B A U A B + BAU 2 I an B A B U U A BA Therefore, ϕ 0 [A, B ] π /4 eta U /2 exp etu /2 ϕ 0 [A, B] x, B A x 2 52 Semiclassical wave packets Following the construction of [Hag98], we consier A, B GL, C satisfying the conitions 2 an introuce the vector of raising operators R[A, B] B x ia i x 2 that consists of components, R[A, B] R [A, B] R [A, B]

13 SYMMETRIC KRONECKER PRODUCTS AND SEMICLASSICAL WAVE PACKETS 3 The raising operator acts on Schwartz functions ψ : R C as R[A, B]ψ x 2 B x ψx ia i x ψx, x R Powers of the raising operator generate the semiclassical wave packets accoring to ϕ k [A, B] k! R[A, B] k ϕ 0 [A, B], k N, where the kth power of the raising operator R[A, B] k R [A, B] k R [A, B] k oes not epen on the orering, since the components commute with one another ue to the compatibility conitions 2 The set ϕk [A, B] : k N } forms an orthonormal basis of the space of square-integrable functions L 2 R, C 53 Hermite polynomials By its construction, the kth semiclassical wave packet is a multivariate polynomial of egree k times the Gaussian function, that is, ϕ k [A, B] 2 k k! p k[a] ϕ 0 [A, B] The polynomials p k [A] are etermine by the matrix A GL, C an satisfy the three-term recurrence relation pk+ej [A] 2 A x p j k [A] 2A A k j p k ej [A] j Whenever the parameter matrix A has all entries real, A GL, R, then the polynomials factorize accoring to p k [A]x H kj A x j, x R, j where H n is the nth Hermite polynomial, n N, efine by the univariate threeterm recurrence relation H n+ y 2 y H n y 2 n H n y, y R The real parameter case, however, is rather expectional when using semiclassical wave packets for their key application in molecular quantum ynamics There, the parameter matrices At an Bt, t R, are time-epenent an etermine by a system of orinary ifferential equations For the particularly simple, but instructive example of harmonic oscillator motion, one can even write the solution explicitly as At cost A0 + i sint B0, Bt i sint A0 + cost B0 Hence, the matrix At cannot be expecte to have only real entries, an the crucial matrix factor At At in the three term recurrence relation generates multivariate polynomials beyon a tensor prouct representation

14 4 GEORGE A HAGEDORN AND CAROLINE LASSER 54 Changing the parametrization If A, B GL, C satisfy the compatibility conitions 2, then A AA is a real symmetric, positive efinite matrix, an the singular value ecomposition of A, A V ΣW with Σ iagσ,, σ positive efinite, is given by an orthogonal matrix V O an a unitary matrix W U This provies two natural ways for transforming A A U with A GL, R an U U One may work with the polar ecomposition of A, or alternatively with A A V Σ V an U W V, A V Σ an U W Both choices provie a unitary transformation to the real case, an we ask how to relate ifferent families of wave packets that correspon to unitarily linke parametrizations For an explicit escription, we collect the semiclassical wave packets of orer n in one formal vector ϕ ln[a, B] ϕ n [A, B], ϕ lnl n[a, B] whose components are labelle by the multi-inices l n,, l n L n Then, we use the n-fol symmetric Kronecker prouct in the following way: Corollary Let A, B GL, C satisfy the conitions 2, an consier the matrices A A U, B B U with U U Then, ϕ n [A, B ] etu /2 S n U ϕ n [A, B], n N Proof We observe that the raising operators transform accoring to which means for the components that R[A, B ] R[A U, B U] U R[A, B], R j [A, B ] m u mj R m [A, B], j,, Since all components of the raising operators commute which each other, we can use the multinomial theorem an obtain for all n N that R j [A, B ] n n u α j α R[A, B]α, α n where u,, u C enote the column vectors of the matrix U This implies for any k N with k n, R[A, B ] k k k u α u α R[A, B] α+ +α, α α α k α k

15 SYMMETRIC KRONECKER PRODUCTS AND SEMICLASSICAL WAVE PACKETS 5 where the summations run over α,, α N Together with Lemma 5, we therefore obtain ϕ k [A, B ] R[A, B ] k ϕ 0 [A, B ] k! etu k! etu k! α k α k α k α k By Theorem, we then obtain k α k α k α k α u α u α R[A, B] α+ +α ϕ 0 [A, B] u α u α ϕ n [A, B ] etu /2 S n U ϕ n [A, B] α + + α! ϕ α+ +α [A, B] Acknowlegements This research was partially supporte by the US National Science Founation Grant DMS an the German Research Founation DFG, Collaborative Research Center SFB/TRR 09 References [AHO98] F Alizaeh, J Haeberly, M Overton, Primal-ual interior-point methos for semiefinite programming: convergence rates, stability an numerical results, SIAM J Optim 8, no 3, [BGH] R Bourquin, V Grainaru, an GA Hageorn, In Preparation [FGL09] E Faou, V Grainaru, an C Lubich, Computing semiclassical quantum ynamics with Hageorn wavepackets SIAM J Sci Comp 3, [Hack2] W Hackbusch, Tensor Spaces an Numerical Tensor Calculus, Springer, 202 [Hag80] GA Hageorn, Semiclassical quantum mechanics I: The 0 limit for coherent states, Commun Math Phys 7, [Hag85] G A Hageorn, Semiclassical quantum mechanics IV: The large orer asymptotics an more general states in more than one imension, Ann Inst Henri Poincaré Sect A 42, [Hag98] GA Hageorn, Raising an lowering operators for semi-classical wave packets, Ann Physics 269, [Lub08] C Lubich, From quantum to classical molecular ynamics: reuce moels an numerical analysis European Math Soc, 2008 [Kle02] E e Klerk, Aspects of Semiefinite Programming Kluwer, 2002 [JHT] JH Toloza, Private Communication [VL00] C Van Loan, The ubiquitous Kronecker prouct, J Comput Appl Math 23, [VLV5] C Van Loan, J Vokt, Approximating matrices with multiple symmetries, SIAM J Matrix Anal Appl 36, no 3, George A Hageorn Department of Mathematics an Center for Statistical Mechanics, Mathematical Physics, an Theoretical Chemistry, Virginia Polytechnic Institute an State University, Blacksburg, Virginia , USA aress: hageorn@mathvteu Caroline Lasser Zentrum Mathematik M3, Technische Universität München, D München, Germany aress: classer@matume