Schur-Like Forms for Matrix Lie Groups, Lie Algebras and Jordan Algebras

Transcription

1 Schur-Like Forms for Matrix Lie Groups, Lie Algebras and Jordan Algebras Gregory Ammar Department of Mathematical Sciences Northern Illinois University DeKalb, IL 65 USA Christian Mehl Fakultät für Mathematik Technische Universität Chemnitz D-97 Chemnitz, Germany Volker Mehrmann Fakultät für Mathematik Technische Universität Chemnitz D-97 Chemnitz, Germany Dedicated to Ludwig Elsner on the occasion of his sixtieth birthday Abstract We describe canonical forms for elements of a classical Lie group of matrices under similarity transformations in the group Matrices in the associated Lie algebra and Jordan algebra of matrices inherit related forms under these similarity transformations In general, one cannot achieve diagonal or Schur form, but the form that can be achieved displays the eigenvalues of the matrix We also discuss matrices in intersections of these classes and their Schur-like forms Such multistructured matrices arise in applications from quantum physics and quantum chemistry Introduction Many problems that arise in applications have structures that give rise to eigenvalue problems for matrices that are members of a classical Lie group, its Lie algebra, or an associated Jordan algebra of matrices Any nonsingular matrix K C m,m defines a nondegenerate sesquilinear form <, > on C m by < x, y > x H Ky for x, y C m, Research supported by Deutsche Forschungsgemeinschaft Me79/7- Singuläre Steuerungsprobleme

2 where x H denotes the conjugate transpose of the column vector x We restrict ourselves to the case that < x, y > if and only if < y, x > This condition implies that K is either Hermitian or skew-hermitian If K is Hermitian, we can perform a change of basis on C m so that the sesquilinear form is represented by the matrix Ip Σ p,q, I q where p + q m, p, and q If K is skew-hermitian, we analogously obtain that after a change of basis the sesquilinear form is represented by the matrix iσ p,q, where p + q m, p, and q If K is real and skew-symmetric, then the nonsingularity of K implies that m is even, and after a change of basis, the <, > form is represented by the matrix J In I n where m n The classical Lie groups we consider here are the matrices that are unitary with respect to J or Σ p,q 5 We will discuss only classes of complex matrices here, but analogous results also exist for the real case Definition ) The Lie group O p,q of Σ p,q -unitary matrices is defined by O p,q {G C p+q,p+q : G H Σ p,q G Σ p,q } An important special case is the unitary group O n O,n ) The Lie group Sp n of symplectic matrices is defined by Sp n {G C n,n : G H JG J} Developing structure-preserving numerical methods for solving eigenvalue problems for matrices in these groups remains an active area of recent research 7, 9,, 3, motivated by applications arising in signal processing and optimal control for discrete-time or continuous-time linear systems, see 3 and the references therein Of equal importance are the Lie algebras A p,q and H n corresponding to the Lie groups O p,q and Sp n Definition ) The Lie algebra of Σ p,q -skew Hermitian matrices is defined by A p,q {A C p+q,p+q : Σ p,q A + A H Σ p,q } { } F G G H : F F H H C p,p, H H H C q,q, G C p,q A special case is the Lie algebra of skew Hermitian matrices A n A,n,

3 ) The Lie algebra of J-Hermitian matrices (also called Hamiltonian matrices or infinitesimally symplectic matrices) is defined by H n {A { C n,n : JA + A H J } } F G H F H : F, G, H C n,n, G G H, H H H These Lie algebras also have great importance in practical applications, see 4, 5, 6 for applications of Σ p,q -skew symmetric matrices and 8, 3, for applications of Hamiltonian matrices The third class of matrices we consider plays a role similar to that of the Lie algebras These are the Jordan algebras 3, 6 associated with the two Lie groups Definition 3 ) C p,q {C { C p+q,p+q : Σ p,q C C H Σ p,q } } F G G H : F F H H C p,p, H H H C q,q, G C p,q, is the Jordan algebra of Σ p,q -Hermitian matrices A special case is the Jordan algebra of Hermitian matrices C n C,n ) SH n {C { C n,n : JC C H J } } F G H F H : F, G, H C n,n, G G H, H H H, is the Jordan algebra of J-skew Hermitian matrices or skew Hamiltonian matrices For applications of these classes see, for example, 5, 7, 8 In this paper we discuss structure-preserving similarity transformations to condensed forms from which the eigenvalues of the matrices can be read off in a simple way For general matrices these are the Jordan canonical form (under similarity transformations with nonsingular matrices), see eg 3, and the Schur form (under similarity with unitary matrices), see eg 4 While both the Jordan form and Schur form display all the eigenvalues, the transformation to Jordan form gives the eigenvectors and principle vectors, and the transformation to Schur form displays one eigenvector and a nested set of invariant subspaces However, the numerical computation of the Schur form is a well-conditioned problem, while the reduction to Jordan canonical form is in general an ill-conditioned problem, see eg 4 See 9 for classifications of the structured Jordan forms for the classes of matrices defined above The Jordan structure of a matrix can be computed, with considerably more effort than computing the Schur form, by computing the Wyer characteristics, which are invariants under unitary similarity transformations, see 7 But if the matrix has an extra symmetry structure, for example if it is Hermitian, skew Hermitian or unitary, then the matrix is normal, and the Jordan form and the Schur form coincide Consequently, complete eigenstructure information can be obtained via a numerically stable procedure 4, 7, 3 for matrices in these classes 3

4 We may expect that between the general case and these special cases there are more refined condensed forms for matrices from the classes defined above For the classes defined via the skew-symmetric matrix J such forms have been discussed in detail in the context of the solution of algebraic Riccati equations 8, 3 We will review these results in Section 3 In Section 4 we will then discuss analogous condensed forms for the classes defined by the symmetric matrix Σ p,q In Section 5 we then discuss condensed forms for matrices which lie in the intersection of two of the classes defined above Our main motivation to work on this topic arose from a class of matrices that occur in quantum chemistry,, 5 In linear response theory one needs to compute eigenvalues and eigenvectors of matrices of the form A B, A, B C n,n, A A H, B B H () B A Such matrices are clearly in H n C n,n A difficulty in computing the eigenstructure of such matrices was observed in,, where the structure-preserving methods sometimes had convergence difficulties As we will show, these difficulties arise from the fact that the reduction to a structured Schur form is not always possible, essentially because nonzero vectors may have zero length in the indefinite form defined by the matrix Σ p,q Moreover, we will see that the eigenvalues and invariant subspaces of the matrix are already available in this situation, even though the matrix has not yet been reduced to a triangular-like structure Preliminaries In this section we give some preliminary results Proposition 4 Let M O p,q and Mx λx with x Then λ is also an eigenvalue of M, and M H (Σ p,q x) λ (Σ p,q x) If x H Σ p,q x then λ Let M A p,q and Mx λx with x Then λ is also an eigenvalue of M, and M H (Σ p,q x) λ(σ p,q x) If x H Σ p,q x then λ λ 3 Let M C p,q and Mx λx with x Then λ is also an eigenvalue of M, and M H (Σ p,q x) λ(σ p,q x) If x H Σ p,q x then λ λ Proof The proof follows directly from the definitions of the Lie group, Lie algebra, and Jordan algebra Similar results are also known for the Lie group of symplectic matrices and the corresponding Lie algebra and Jordan algebra, see eg 3, There exist a vector space isomorphism between the Lie algebras and the associated Jordan algebras: 4

5 Proposition 5 The map A ia is a vector space isomorphism between the Lie algebra A p,q (or H n ) and the associated Jordan algebra C p,q (or SH n, respectively) Proof The proof follows directly from the definitions We will use similarity transformations that retain the structure to transform the matrices from the Lie groups, Lie algebras and Jordan algebras to a condensed form from which the eigenvalues can be read off in a simple way These are symplectic similarity transformations for Sp n, H n and SH n and the Σ p,q -unitary matrices for O p,q, A p,q and C p,q In order to use such transformations for numerical computation, we would prefer that these transformation matrices also be unitary, since then the methods can be implemented as numerically backwards stable procedures These classes are characterized as follows Proposition 6 Unitary symplectic matrices, ie, matrices in Sp n O n, are of the form U U, U U with U U H + U U H I n and U U H U U H Matrices in O p,q O n (n p + q) have the form Q, Q where Q O p and Q O q Matrices in all the Lie groups and their intersections can be generated as products of elementary matrices in these classes, see eg, 5, 3 Unfortunately, the class of matrices O p,q O n is not big enough to perform the reduction to the condensed forms As an extra class of elementary Σ p,q -unitary transformations, the hyperbolic rotations H p (c, s) are needed These matrices are equal to the identity matrix except for the submatrix c s in rows and columns and p +, given by, with c s c s 3 J-Schur-like forms In this section we recall some of the known results concerning Schur-like forms for matrices in the classes defined by J We will call these J-Schur-like forms Theorem 7 i) Let M H n Then there exists a symplectic matrix Q Sp n such that Q T T MQ, () 5 T H

6 where T, T C n,n, T is upper triangular and T is Hermitian, if and only if every purely imaginary eigenvalue λ of M has even algebraic multiplicity, say k, and any basis X k C n,k of the maximal invariant subspace for M corresponding to λ satisfies Xk H JX k c I k I k (3) Here c denotes congruence ii) Let M SH n Then there exists a symplectic matrix Q Sp n such that Q T T MQ T H, (4) where T, T C n,n, T is upper triangular and T is skew Hermitian, if and only if every real eigenvalue λ of M has even algebraic multiplicity, say k, and any basis X k C n,k of the maximal invariant subspace for M corresponding to λ satisfies (3) iii) Let M Sp n Then there exists a symplectic matrix Q Sp n such that Q MQ T T T H, (5) where T, T C n,n, T is upper triangular and T is Hermitian, if and only if every unimodular eigenvalue λ of M has even algebraic multiplicity, say k, and any basis X k C n,k of the maximal invariant subspace for M corresponding to λ satisfies (3) Proof This result was first stated and proved in A simpler proof based on canonical forms under symplectic similarity transformations has recently been given in 4 Note that there are also more refined Jordan-like forms for matrices in H n, SH n and Sp n, which do not have a triangular structure, see It follows from a result in 6 that the symplectic matrices Q in each part of Theorem 7 can be chosen to be unitary symplectic However, matrices in the J classes exist for which the forms () (5) can be achieved only via non-symplectic transformations or not at all Consider for example the Hamiltonian (and symplectic) matrix J I n I n Since J is invariant under symplectic similarity transformations, we cannot achieve a J-Schur-like form under these transformations But in the case that n m is even, there exists a J-Schur-like form under a unitary but non-symplectic similarity transformation via and P : P H I m P I m I m ii m ii m I m ii m I m I m ii m ii m ii m ii m ii m 6

7 In the case that n is odd, no J-Schur-like form can be obtained for J, since the matrices T and T H in () have the same purely imaginary eigenvalues (resp the matrices T and T H in (5) have the same unimodular eigenvalues), but the algebraic multiplicities of the eigenvalues i and i of J are odd See 4 for a detailed discussion 4 Σ p,q -Schur-like forms In this section we present results analogous to Theorem 7 concerning Schur-like forms for matrices in the classes defined by Σ p,q under similarity transformations from the group O p,q We will call these forms Σ p,q -Schur-like forms Of course, if p or q, then the matrices in question (unitary, Skew-Hermitian, and Hermitian matrices) are all normal, and their Schur forms under unitary similarity transformations are diagonal So, unless explicitly mentioned, we will assume that p and q are both positive (and m p + q ) We describe the Σ p,q -Schur-like forms in terms of a block matrix with the partitioning Q MQ p q p A C, (6) q D B p p p 3 p 4 p s q q q 3 q 4 q s p A p A A 3 A 4 A s C C 3 C 4 C s p 3 A 3 A 33 C 3 p 4 A 4 A 44 A 4s C 4 C 44 C 4s p s A s A s4 A ss C s C s4 C ss, (7) q B q D D 3 D 4 D s B B 3 B 4 B s q 3 D 3 B 3 B 33 q 4 D 4 D 44 D 4s B 4 B 44 B 4s q s D s D s4 D ss B s B s4 B ss where s j p j p, s k q k q, and such that For odd i, p i, q i, and the blocks A ii and B ii are each either diagonal or void; for even i, p i q i, provided that s > 7

8 Theorem 8 i) Let M O p,q, then there exists Q O p,q such that Q MQ p q p A C q D B is in the form (7) For odd indicies i, the blocks A ii O pi and B ii O qi are either void or diagonal (with eigenvalues of modulus ) For the even indicies i, provided that s >, the blocks A ii, B ii are both Furthermore and A i+,i A s,i A i,i + C i,i D i,i + B i,i λ, A i,i D i,i B i,i C i,i λ, C i+,i C s,i, B i+,i B l,i D i+,i Ai,i+ A i,s Di,i+ D i,s, D l,i, (8) (9) Bi,i+ B i,l Ci,i+ C i,l Moreover the eigenvalues of M are the eigenvalues of the matrix obtained by deleting all the off-diagonal blocks in A, B, C, D ii) Let M A p,q Then there exists Q O p,q such that A C Q MQ C H, () B with B B H, A A H, and A, B, C structured as in (7), where for the blocks with odd numbered indices, A ii A pi and B ii A qi are diagonal with purely imaginary eigenvalues or void and, provided that s >, the even numbered blocks A ii, B ii are both Furthermore A i,i + C i,i C i,i + B i,i λ () Again the eigenvalues of M are the eigenvalues of the matrix obtained by deleting all the off-diagonal blocks in A, B, and C iii) Let M C p,q Then there exists Q O p,q such that Q A C MQ C H, () B with B B H, A A H, and A, B, C structured as in (7), where for the blocks with odd numbered indices, A ii C pi, B ii C qi are diagonal with real eigenvalues or void and, provided that s >, the even numbered blocks A ii, B ii are both Furthermore A i,i + C i,i C i,i + B i,i λ (3) 8

9 Again the eigenvalues of M are the eigenvalues of the matrix obtained by deleting all the off-diagonal blocks in A, B and C Proof i) The proof proceeds via induction on the dimension m p + q The case m is trivial, since in this case p or q Assume that p and q are both positive x Let λ be an eigenvalue of M, and let x be an associated eigenvector, with x C p and x C q, and let Q O p, Q O q be such that x Q H x α e, Q H x α e p+, where α x, α x are real and nonnegative and e i denotes the i-th unit vector, eg 4 If α and α are both nonzero, then we cannot eliminate another element using a matrix in O p,q O m So if we wish to retain that the matrix remains in the group, we have to use hyperbolic rotations If α α, then a hyperbolic transformation can be applied to further reduce the transformed vector We then have to consider the three cases α > α, α α and α < α The case that both parameters are cannot happen, since x c s If α > α, then there exists a hyperbolic rotation O s c, such that c s α s c α β with β (α α) / > If α < α, then there exists a hyperbolic rotation such that c s α, s c α with β (α α) / > In the third case, α α, no hyperbolic rotation exists that eliminates either of the two elements Then we set c, s Having chosen c and s, set Q Q H : H p (c, s) Q H O p,q It follows that Q x β, β T ; α > α, β T ; α < α, α α T ; α α 9

10 In the first case, we see that Q MQ e λe, so that λ w M : Q H MQ M (4) and since M O p,q, we obtain λ, w and M O p,q In the second case, we have so the transformed matrix takes the form Q MQ e p+ λe p+, M Q MQ M M 3 w H λ w H 3 M 3 M 33, (5) and since M O p,q, we obtain λ, w, w 3 and M M M 3 O M 3 M p,q 33 In the third case no further reduction of the vector Q x α e + α e p+ is possible with a hyperbolic rotation In this case we have m w H m,p+ w4 H M Q MQ y M y 3 M 4 m p+, w3 H m p+,p+ w34 H (6) y 4 M 4 y 43 M 44 and M(α e + α e p+ ) (α e + α e p+ )λ From α α we obtain that and m + m,p+ λ, m p+, + m p+,p+ λ (7) y 4 + y 43, y + y 3 (8) By Proposition 4 we have that Σ p,q (e + e p+ ) e e p+ is a left eigenvector associated with the eigenvalue λ Hence we obtain m m p+, λ, m,p+ + m p+,p+ λ (9) and w w 3, w 4 w 34 () m m We immediately obtain that the submatrix,p+ has the eigenvalues λ, λ m p+, m p+,p+

11 If we consider the unitary matrix U, () then we obtain from the identities (7) through () that λ U MU H M M 4 λ () M 4 M 44 As a consequence we obtain that the spectrum of M is equal to the union of the spectra m m of,p+ and M m p+, m M M 4 Furthermore, from (8) and () it is p+,p+ M 4 M 44 easy to see that M O p,q The proof now follows by induction In each of the above three cases, we perform a similarity transformation based on the given eigenvector, and produce a matrix in a smaller group whose eigenvalues are the remaining eigenvalues of the original matrix By induction, there is a Σ p,q -unitary matrix V such that V M V is in the form of (7) (9) Partitioning V compatibly with (4), (5) or (6), and embedding the partitioned matrix into a Σ p,q -unitary matrix Q in the obvious fashion, we see that Q MQ has the desired form, where Q Q Q The proof for ii) is analogous while iii) follows from ii) using Proposition 5 Remark 9 The class of transformation matrices O p,q is too small to bring every matrix in O p,q, (in A p,q or C p,q ) to upper triangular form via similarity Consider for example the matrix M : 5 3 O, with the eigenvalues 5 and There exists no upper triangular matrix in O, that is similar to M, since in this case all the eigenvalues have to be unimodular On the other hand, the form (7) displays all the eigenvalue information The columns of the part of the matrix that cannot be eliminated have length in the indefinite scalar product defined as < u, v > p,q u H Σ p,q v We also see that, in contrast with the symplectic case, we cannot obtain the condensed form using transformations in O p,q O m, again since this class is too small to perform the necessary reductions

12 Remark In some cases we can reduce the form (7) further if subparts in the offdiagonal blocks of the top and bottom part do not have equal length But since we can never eliminate in these blocks completely and since the eigenvalues are displayed, we may as well avoid further reduction Remark Since we want the proof of Theorem 8 to be constructive, we start every step of the induction by choosing an arbitrary eigenvector Therefore, the parameters p i and q i in (7) are not uniquely determined Analogous results can also be obtained in the case of real matrices in all three classes To obtain these results, we always combine complex conjugate pairs of eigenvalues and the associated eigenvectors We have presented structured condensed forms from which the eigenvalues can be read off in a simple way These results simplify considerably in the case that the matrices have multiple structures We will discuss Schur forms for such matrices in the next section 5 Schur-like forms for multi-structured matrices In some applications one needs the computation of eigenvalues of matrices that have more than one structure In this section we present Schur-like forms for matrices from intersections of two of the classes introduced in Section 5 Intersections of two Σ p,q classes Let us first consider the intersections of classes defined by Σ p,q and Σ p, q, where p + q p + q m and, wlog, p > p Directly from the definitions we obtain the following structures Proposition i) Matrices in A p,q A p, q have the form with A A C A H A 3 C C H C H B A A C A H A 3 C B C H C H B A p,q and B A q q ii) Matrices in C p,q C p, q have the form A A C A H A 3 C B C H C H B (3) (4)

13 with A A C A H A 3 C C H C H B C p,q and B C q q iii) Matrices in O p,q O p, q have the form with A A C A A C D D B A A C A A C B D D B O p,q and B O q q iv) Matrices in A p,q C p, q have the form C C C H C H B B H (5) (6) The intersection of a group with one of the algebras does not show any obvious structure In cases i) iii), the problem reduces to two smaller problems, each having a single structure for which the results of Section 4 apply For case iv) the computation of the eigenvalues reduces to the computation of the singular values of the matrix C H C H B An important special case that arises in particle physics 8, 9, is the case p q and p In this case the matrices have the form C C H, (7) where C C p p Again, the eigenvalues can be determined via the singular values of C 5 Intersections of a J class and a Σ p,q class, p n In this subsection we consider matrices with a multiple structure related to both J and Σ p,q, with p + q n and wlog p > n We obtain the following obvious structures which follow directly from the definitions Proposition 3 i) Matrices in H n A p,q have the form A C A C C A C A 3, (8)

14 A C with A C H C A (n q) A (n q) and H C A q A q,q ii) Matrices in H n C p,q have the form A C A C C A, (9) C A A C with A C H C A (n q) C (n q) and H C A q C q,q iii) Matrices in SH n A p,q have the form A C A C C A C A A C A C with SH C A (n q) A (n q) and iv) Matrices in SH n C p,q have the form A C with SH C A (n q) C (n q) and C A A C A C C A C A A C C A, (3) SH q A q,q, (3) SH q C q,q No obvious simplified structure occurs in intersections of A p,q or C p,q with Sp n or of H n or SH n with O p,q In all four cases of Proposition 3 the treatment of the smaller matrices is relatively easy The first submatrices in each of the cases are discussed in detail in 8 while the second submatrices will be discussed in the next section 53 Intersections of a J class and a Σ p,q class, p q n In this section we discuss the most important multi-structured case in applications, p q n Theorem 4 i) Let { U V M O n,n Sp n V U } : UU H V V H I n, UV H V U H 4

15 Then there exists a unitary matris ˆQ O n,n Sp n, such that the eigenvalues of ˆQ M ˆQ Û ˆV ˆV Û (3) are given by u ii ± v ii, where u ii and v ii denote the i-th diagonal element of Û and ˆV, respectively Furthermore we have (u ii + v ii )(u ii v ii ) ii) Let { } A B M A n,n H n : A A H, B B H B A Then there exists a unitary matrix Q O n,n Sp n, such that the eigenvalues of A B Q MQ (33) B A are given by a ii ± b ii, where a ii and b ii denote the i-th diagonal element of A and B, respectively, Furthermore b ii is real and a ii is purely imaginary iii) Let { } A B M C n,n SH n : A A H, B B H B A Then there exists a unitary matrix Q O n,n Sp n, such that the eigenvalues of A B Q MQ (34) B A are given by a ii ± b ii, where a ii and b ii denote the i-th diagonal element of A and B, respectively Furthermore a ii is real and b ii is purely imaginary U V x Proof i) Let M O V U n,n Sp n, and let, with x x, x C n, be an eigenvector associated with the eigenvalue λ of M If x x, then it follows immediately that x + x x + x is also an eigenvector of M associated with the eigenvalue λ Thus, we may assume wlog that x x (In the case x x the proof follows analogously by changing some signs) Let Q O n be such that Q H (x ) αe (n), where α If we form M Q H Q H Q M Q u z H u v z H v y u M u y v M v v z H v u z H u y v M v y u M u, 5

16 where u, v C and M u, M v C (n ) (n ), then we see that e (n) +e (n) n+ is an eigenvector of M associated with the eigenvalue λ, ie, we have u v λ v u u v The eigenvalues of are u v u ±v Furthermore we have y u +y v Similarly, by Proposition 4 the vector e (n) e (n) n+ is a left eigenvector of M associated with the eigenvalue λ, so that zu H zv H Mu M as well Hence we obtain v O M v M n,n Sp (n ) and u u v O v u, Sp Since the latter matrix is symplectic, we have If we consider then we obtain S MS (u ii + v ii )(u ii v ii ) S : I n I n, u + v zu H v zv H M u y v M v u v M v y u M u u v ie, the spectrum of M is equal to the union of the spectra of v u Mu M v The rest of the proof follows by induction M v M v The proof of ii) is analogous to the proof of i) noting that A is skew-hermitian and that B is Hermitian The proof of iii) follows from ii) and Proposition 5 The intersections of algebras with groups again does not give any particular structure, so it remains to discuss the final class which motivated our interest in analyzing multistructured matrices Matrices from the set { } A B H n C n,n : A A H, B B H (35) B A arise in linear response theory in quantum chemistry,, 5 By definition, similarity transformations with matrices from O n,n Sp n will preserve the structure The elements of H n have the eigenvalue-symmetry λ, λ while we find the eigenvalue-symmetry λ, λ for the matrices from C n,n Therefore the eigenvalues of M H n C n,n will occur in 6, and

17 quadruples λ, λ, λ, λ unless λ is real or purely imaginary So we expect 4 4-blocks to occur in Schur-like forms for these matrices Before we formulate the main result let us state some properties of the matrices under consideration A B x Proposition 5 Let M H B A n C n,n, let x, with x x, x C n x, be an eigenvector of M associated with the eigenvalue λ of M and let y : Then y is an eigenvector of M associated with the eigenvalue λ x x H x H is a left eigenvector of M associated with the eigenvalue λ 3 If λ is not purely imaginary, then x H x x H x 4 M(x + y) λ(x y) and M(x y) λ(x + y) ix + x 5 is an eigenvector of the matrix ix + x ˆM B A : associated with the A B eigenvalue iλ Proof,, 4 and 5 are easy to verify Furthermore we have by that λ(x H x x H x ) λ x H x H x x x H x H A B x B A x λ x H x H x x ie, (λ + λ)(x H x x H x ) λ(x H x x H x ), Theorem 6 i) For each M H n C n,n, there exists ˆQ Sp n O n,n such that the matrix A A A k B B B k A A A k B B B k ˆQ M ˆQ A k A k A kk B k B k B kk, B B B k A A A k B B B k A A A k B k B k B kk A k A k A kk with A ij, B ij C n i n j and n i, n j {, }, has the following properties: 7

18 The eigenvalues of M are the eigenvalues of the matrix obtained by deleting all the off-diagonal blocks in A A ij and B B ij Aii B If n i, then the eigenvalues of ii are ± A ii B ii In particular B ii A ii these eigenvalues are both real or both purely imaginary 3 If n i, let m a denote the (, ) element of A ii and m b denote the (, ) element Aii B of B ii Then the eigenvalues of ii are λ, λ, λ and λ, where B ii A ii λ (m a + m b )(m a m b ) { A B ii) Let M SH n A n,n : A A B A H, B B } H Then there exists ˆQ Sp n O n,n, such that the matrix ˆQ M ˆQ A A A k B B B k A A A k B B B k A k A k A kk B k B k B kk B B B k A A A k B B B k A A A k B k B k B kk A k A k A kk, with A ij, B ij C n i n j and n i, n j {, }, has the following properties: The eigenvalues of M are the eigenvalues of the matrix obtained by deleting all the off-diagonal blocks in A A ij and B B ij Aii B If n i, then the eigenvalues of ii are ± A ii B ii In particular B ii A ii these eigenvalues are both purely imaginary or both real 3 If n i, let m a and m b denote the (, )-element of A ii and B ii, respectively Then Aii B the eigenvalues of ii are λ, λ, λ and λ, where B ii A ii λ (m a + m b )(m a m b ) A B Proof i) Let M with A B A H A, B H B C n n x and let, x with x, x C n, be an eigenvector of M associated with the eigenvalue λ We will use 8

19 Q transformation matrices that are either of the form with Q O Q n or hyperbolic rotations H n (c, s) with c, s R We have to distinguish two cases: x and x are linearly dependent If we assume wlog that x, then there exists x γ C such that x γx (In the case x we consider and λ, according to Proposition 5) Let e (n) i denote the i-th unit vector in C n, and choose a unitary matrix Q O n such that Q H x αe (n), where α Then we obtain and hence we have M Q Q H M Q H x γαe (n), Q Q x m a y H a m b y H b y a M a y b M b m b y H b m a y H a y b M b y a M a where m a, m b C and M a, M b C (n ) (n ) Since e (n) + γe (n) n+ is an eigenvector of M associated with the eigenvalue λ, we have (a) If γ, ie, γ ±, then λ and m a ( γ ) ( + γ )λ and ( γ )y a m a + γm b and y a + γy b Hence, using we obtain S : S MS I n γ I n ya H m b yb H M a y b M b M b y a M a, ma m ie, the spectrum of M is the union of the spectra of b m b m a Ma M and b M b M a 9

20 (b) If γ, then γ and γ ma m are eigenvectors of b m b m a we obtain S : S MS are linearly independent and by Proposition 5 they associated with the eigenvalues λ and λ Forming γ I n γ I n, λ M a M b λ M b M a ma m ie, the spectrum of M is the union of the spectra of b Ma M and b m b m a M b M a ma m Since b H m b m C,, we find by symmetry that λ must be real or purely a imaginary If x and x are linearly independent, (ie, in particular n ), then this also holds for x + βx and x βx, where β (a) If λ is not purely imaginary then we have by Proposition 53 that x H x x H x Therefore, β x H x x H x R yields, (x + βx ) H (x βx ) x H x β x H x + β(x H x x H x ) We may assume wlog that β Otherwise we exchange x and x and consider the eigenvector x x associated with the eigenvalue λ, according to Lemma 5 There exists Q O n such that Q H (x + βx ) α e (n) and Q H (x βx ) α e (n), where α and α are real and positive Considering H Q Q M M Q Q m a m b M a M b m b m a M b M a with m a, m b C and M a, M b C (n ) (n ), we see that α e + α e + α β e n+ α β e n+ is an eigenvector of M associated with the eigenvalue λ: M(α e + α e + α β e n+ α β e n+),

21 H Q x + βx + x βx M Q x β + x x β + x H Q x M Q x H Q x λ Q x λ(α e + α e + α β e n+ α β e n+) ci s i If β, that is β >, then there are hyperbolic rotations i c s s c α α β α and c s s c α α β s i c i, i,, such that Since α, α, β R, we can choose c i and s i to be real Transforming M and the eigenvector associated with λ analogously, we have reduced the problem to the case (b) In particular it follows that λ is real or purely imaginary ma m a α ii If β, there exist no hyperbolic rotation as before Then let m a : mb m and m b : b m b m b3 using Proposition 54 we obtain: and In particular we have m a m a3 The relevant eigenvector is α e +α e +α e n+ α e n+ Thus, m a m a m b m b m a m a3 m b m b3 m b m b m a m a m b m b3 m a m a3 m a m a m b m b m a m a3 m b m b3 m b m b m a m a m b m b3 m a m a3 α α α α λ λ α α m a + m b α α λ and m a m b α α λ α α This implies λ (m a + m b )(m a m b )

22 If we form the matrix then we obtain S : s s n S : t t n H 4α α α α α α α α α α I n α α α α α α α α I n, α α α α α α α α I n α α α α α α α α I n and noting that according to Proposition 5 the columns s and s n+ are right eigenvectors of M associated with the eigenvalues λ resp λ and according to Proposition 5 the rows t H and t H n+ are left eigenvectors of M associated with the eigenvalues λ and λ, we obtain λ λ M S MS a M b λ, λ M b M a ma m ie, the spectrum of M is the union of the spectra of the 4 4 matrix b and m b m a Ma M b ma m and the eigenvalues of b are λ, λ, λ and λ M a M b m b m a ix + x (b) If λ iµ is purely imaginary, then it follows from Proposition 55 that ix + x B A is an eigenvector of associated with the eigenvalue µ Transforming this matrix as in the case (a), yields analogous results for M, since we have for all transformations A B U V O V U n,n Sp n that U V A B U H V H S T V U B A V H U H T S U V B A U H V H T S V U A B V H U H S T,

23 Note that in the case (a)ii we obtain the formula λ iµ i (m b + m a )(m b m a ) (m a + m b )(m a m b ) Ma M In all cases we have b H M b M k C k,k with k n or k n So the a proof follows by induction The proof for ii) follows directly from Proposition 5 Remark 7 In general the remaining 4 4-blocks in Theorem 6 cannot be divided further into two -blocks This is possible only if the eigenvalues are real or purely imaginary Remark 8 As we see from the proof of Theorem 6, the only time we need hyperbolic rotations is when we want to split certain 4 4-blocks into two -blocks That means that we are able to achieve a Schur-like form whose eigenvalues are displayed by at most 4 4-blocks by using only unitary transformations This result does not hold in the real case Remark 9 Not every matrix M A B B A Hn C n,n has the property that the eigenvalues of M can be obtained by deleting all the off-diagonal blocks in A and B This is a special property of the Schur-like form for these matrices Consider for example A and B In this case M is nonsingular, but the matrix obtained by deleting all the off-diagonal blocks in A and B is zero 6 Conclusions We have discussed Schur-like forms for matrices with one or more algebraic structures arising from a classical Lie group, Lie algebra or Jordan algebra In all cases we obtain a structured Schur-like form that displays all the eigenvalues In particular, we have obtained such Schur-like forms for multi-structured matrices which arise in quantum chemistry References G S Ammar, W B Gragg, and L Reichel Determination of Pisarenko frequency estimates as eigenvalues of an orthogonal matrix In F T Luk, editor, Advanced Algorithms and Architectures for Signal Processing II, pages SPIE, 987 E Artin Geometric Algebra Interscience Publishers, New York, H Braun and M Koecher Jordan-Algebren Springer-Verlag, Berlin, A Bunse-Gerstner Der HR Algorithmus zur numerischen Bestimmung der Eigenwerte einer Matrix Dissertation, Universität Bielefeld, Bielefeld, Germany, 978 3

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