Contents 1 Introduction 1 Preliminaries Singly structured matrices Doubly structured matrices 9.1 Matrices that are H-selfadjoint and G-selfadjoint...

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1 Technische Universitat Chemnitz Sonderforschungsbereich 9 Numerische Simulation auf massiv parallelen Rechnern Christian Mehl, Volker Mehrmann, Hongguo Xu Canonical forms for doubly structured matrices and pencils Preprint SFB9/- Preprint-Reihe des Chemnitzer SFB 9 SFB9/- June

2 Contents 1 Introduction 1 Preliminaries Singly structured matrices Doubly structured matrices 9.1 Matrices that are H-selfadjoint and G-selfadjoint Matrices that are H-selfadjoint and G-skew-adjoint Singly and doubly structured pencils Conclusions 5 Appendix Authors: Christian Mehl Fakultat fur Mathematik TU Chemnitz 91 Chemnitz Germany mehl@mathematik.tu-chemnitz.de Volker Mehrmann Fakultat fur Mathematik TU Chemnitz 91 Chemnitz Germany mehrmann@mathematik.tu-chemnitz.de Hongguo Xu Department of Mathematics Case Western Reserve University 19 Euclid Avenue Cleveland, Ohio, 1, USA hxx@po.cwru.edu

3 Canonical forms for doubly structured matrices and pencils Christian Mehl Volker Mehrmann y Hongguo Xu z June, Abstract In this paper we derive canonical forms under structure preserving equivalence transformations for matrices and matrix pencils that have a multiple structure, which is either an H-selfadjoint or H-skew-adjoint structure, where the matrix H is a complex nonsingular Hermitian or skew-hermitian matrix. Matrices and pencils of such multiple structures arise for example in quantum chemistry in Hartree-Fock models or random phase approximation. Keywords. Indenite inner product, selfadjoint matrix, skew-adjoint matrix, matrix pencil, canonical form. AMS subject classication. 15A1, 15A, 15A5. 1 Introduction Canonical forms for matrices and matrix pencils have been studied for more than hundred years since work of Jordan, Kronecker and Weierstra, see [5]. In recent years, motivated Fakultat fur Mathematik, Technische Universitat Chemnitz, D-91 Chemnitz, Germany mehl@mathematik.tu-chemnitz.de. Large part of this author's work was performed while he was visiting the College of William and Mary, Department of Mathematics, P.O.Box 895, Williamsburg, VA , USA, and while he was supported by Deutsche Forschungsgemeinschaft, Me 19/1-1, Berechnung von Normalformen fur strukturierte Matrizenbuschel. y Fakultat fur Mathematik, Technische Universitat Chemnitz, D-91 Chemnitz, Germany mehrmann@mathematik.tu-chemnitz.de. Supported by Deutsche Forschungsgemeinschaft within SFB9, Project: Algebraische Zerlegungsmethoden z Department of Mathematics, Case Western Reserve University, Cleveland, OH 1-58, USA hxx@po.cwru.edu 1

4 from applications in control theory as well as quantum physics and quantum chemistry, there is a revived interest in such canonical forms for matrices and pencils that have algebraic structures, like Lie groups or Lie algebras. While the possible invariants have been characterized already some time ago [], the emphasis in the new results lies in structure preserving equivalence transformations, see e.g., [1, 1, 1, 15, 1]. In this paper we derive canonical forms under structure preserving equivalence transformations for matrices and matrix pencils with multiple structure. Denition 1 Let H C nn be a nonsingular Hermitian or skew-hermitian matrix, and let X C nn. 1. X is called H-selfadjoint if X H = HX.. X is called H-skew-adjoint if X H = HX. Canonical forms for pairs (A H), where H is Hermitian or skew-hermitian nonsingular and A is H-selfadjoint or H-skew-adjoint are well-known in literature (see, e.g., [, 11]). These forms are obtained under transformations of the form (A H)! (P 1 AP P HP ) where P is nonsingular. Here, we are interested in canonical forms for matrix triples (A H G), where G and H are Hermitian or skew-hermitian nonsingular and A is doubly structured with respect to G and H, i.e., A is H-selfadjoint or H-skew-adjoint and at the same time G-selfadjoint or G-skew-adjoint. We are also interested in the pencil case, i.e., we will also consider pencils %A B, where both A and B are doubly structured with respect to H and G. The main motivation for our interest in these types of matrices and pencils arises from quantum chemistry. Response function models lead to the problem of solving the generalized eigenvalue problem with a matrix pencil of the form C Z E F E A := (1) Z C F E where E F C Z C nn E = E F = F C = C Z = Z, see [8, 1]. Furthermore, there are important special cases in which the pencil has even further structure. For example, the simplest response function model is the time-dependent Hartree-Fock model, also called the random phase approximation (RPA). In this case, C is the identity and Z is the zero matrix, see [8, 1]. Thus, the generalized eigenvalue problem (1) reduces to the problem of nding the eigenvalues of the matrix E F L = () F E

5 where E F are as in (1). For stable Hartree-Fock ground state wave functions, it is furthermore known that E F and E + F are positive denite, see [8]. In other applications, however, like in multicongurational RPA [8], it is not even guaranteed that the matrix E in (1) is nonsingular. It is easy to see that the matrices E, A in (1) and L in () are doubly structured. With In In G = J = I n I n In H = I n we have that E is I-selfadjoint and H-skew-adjoint, A is I-selfadjoint and H-selfadjoint, while L is G-selfadjoint and J-skew-adjoint. When designing structure preserving numerical methods for large scale structures eigenvalue problems sometimes diculties in the convergence of the methods were observed in [, ] that have to do with the invariants of these pencils under structure preserving equivalence transformations, see also [1]. It is another motivation for our work to derive canonical forms that allow a better understanding of those properties of the pencils that lead to these diculties. We will derive the canonical form for matrix triples (A H G) under structure preserving transformations of the form (A H G)! (P 1 AP P HP P GP ) where P is nonsingular. This preserves the (skew-)hermitian structure of H and G and also the structure of A with respect to H and G. Based on the classical results, see Section, we clearly have canonical forms for (A H), (A G) or the pencil %H G, and hence the invariants of the pairs (A H) and (A G), as well as the invariants of the pencil %H G under congruence are invariants of the triple (A H G). It is our goal to obtain a canonical form that displays simultaneously the Jordan structure of A and the invariants of the canonical forms of (A H) and (A G). In general this is a very dicult problem, such a form may not even exist. Consider the following example. Example Consider matrices A = G = and H = Then A is G-selfadjoint and H-selfadjoint. But it is impossible to simultaneously decompose A, H, and G further into smaller block diagonal forms. This follows from the obvious fact that the pencil %G H can not be further decomposed. On the other hand, A has the Jordan canonical form : 5 :

6 Hence, both (A G) and (A H) are decomposable into smaller blocks (see Theorems 9 and 11 below). Due to this diculty, we restrict ourselves to an important special case. In most applications, the matrices H and G, that induce the structure, are contained in the set I S = I n m Im Im m n N : () I m I n m I m If this is the case then the pencil %H G is nondefective. Denition Let %A B C nn be a matrix pencil. We say that %A B is nondefective, if there exists nonsingular matrices P Q C nn, such that both P AQ and P BQ are diagonal. We will show that if G H are Hermitian nonsingular, such that the pencil %H G is nondefective, then a canonical form for the triple (A H G) exists, which is also unique except for the permutation of blocks. In particular, this canonical form includes the Jordan structure of A, and also the canonical forms of the pairs (A H) and (A G) and the pencil %H G can easily be read o. The paper is organized as follows. After providing some preliminary results in Section, we review canonical forms for matrices that are structured with respect to only one Hermitian matrix in Section. In Section we then discuss doubly structured matrices and in Section 5 we discuss canonical forms for structured pencils of the form A B, where both A and B are singly or doubly structured matrices. Preliminaries Throughout the paper, we use the following notation: By (A) we denote the spectrum of the matrix A. J p () denotes pp upper triangular Jordan block with eigenvalue. By sign(t) we mean the sign of a real number t Rnfg. A = A 1 : : : A m stands for the block diagonal matrix A with diagonal blocks A 1,..., A m. Furthermore, we introduce the following p p matrices. Z p := Dp := ( 1) ( 1) p+1 5

7 and F p := ( 1) ( 1) p+1 Note that F p R pp is symmetric if p is odd and skew-symmetric if p is even, whereas Z p and D p are symmetric for all p. We list some properties of these matrices and the matrix J p () which can be easily veried, and will be used in the following. 5 Lemma Let p N. 1. Z p = I p D p = I p F p = ( 1) p+1 I p.. F p Z p = D p = ( 1) p+1 Z p F p D p F p = Z p = ( 1) p+1 F p D p :. D p Z p = F p = ( 1) p+1 Z p D p F p Z p F p = Z p.. Z 1 p J p ()Z p = J p (). 5. D 1 p J p ()D p = J p ().. F 1 p J p ()F p = J p (). Another important result that will frequently be used throughout the paper is the following well-known result, [5]. Lemma 5 Let A B X be square matrices, such that the spectra of A and B are disjoint. If AX = XB, then X =. Finally, we review the canonical forms for regular Hermitian pencils, i.e., pencils %H G, where both H and G are Hermitian and det(%h G). This result goes back to results from Weierstra (see [19]) and Kronecker (see [1]). Theorem Let %H G be a regular Hermitian pencil. Then there exists a nonsingular matrix P C nn such that P (%H G)P = (%H 1 G 1 ) : : : (%H l G l ) () where the blocks %H j G j have one and only one of the following forms: 1. Blocks associated with paired nonreal eigenvalues,, where Im() > : %H j G j = % Ir I r 5 J r () J r () : (5)

8 . Blocks associated with real eigenvalues and sign " f1 1g: %H j G j = %"Z r "Z r J r () = %" ". Blocks associated with the eigenvalue 1 and sign " f1 1g: %H j G j = %"Z r J r () "Z r = %" " : () 5 : () Moreover, the decomposition () is unique up to a block permutation that exchanges blocks H i G i. Proof. For a full proof, see [18], Lemmas 1.{. There, the result is shown without the additional condition Im() > for the blocks associated with nonreal eigenvalues, but applying a permutation, we may always place the block that is associated with the eigenvalue in the (1 )-block position of the form in (5). If %H G is nondefective then we immediately have the following corollary. Corollary Let %H G be a nondefective Hermitian pencil, where both H and G are nonsingular. Then there exists a nonsingular matrix P C nn such that P (%H G)P = (%H 1 G 1 ) : : : (%H l G l ) where the spectra of %H j G j and %H l G l are disjoint for j = l, and where each block %H j G j has either only one pair of complex conjugate eigenvalues or only one single real eigenvalue. Moreover, the block %H j G j has one and only one of the following forms. 1. Blocks with nonreal eigenvalues,, where Im > and r N: : %H j G j = % Ir I r. Blocks with real eigenvalue, where r p N: Ip %H j G j = I r p Ir I r Ip : I r p Moreover the decomposition is unique up to a block permutation.

9 Proof. Since the size of every Jordan blocks equals one, the result follows directly from Theorem after proper permutations such that equal eigenvalues are combined in a single block. Remark 8 Following from Theorem and Corollary it is obvious that if C nr is an eigenvalue of %H G then so is and both eigenvalues have the same Jordan structures. Singly structured matrices In this section, we review the well-known canonical forms for H-selfadjoint matrices and H-skew-adjoint matrices, where H always denotes a complex nonsingular Hermitian n n matrix. Theorem 9 Let A C nn be H-selfadjoint. Then there exists a nonsingular matrix P C nn, such that P 1 AP = A 1 : : : A k and P HP = H 1 : : : H k (8) where A j and H j are of the same size and the pair (A j H j ) has one and only one of the following forms: 1. Blocks associated with real eigenvalues: where R, p N, and " f1 1g. A j = J p () and H j = "Z p (9). Blocks associated with a pair of nonreal eigenvalues: Jp () Zp A j = and H J p () j = Z p (1) where C nr with Im() > and p N. Moreover, the form (P 1 AP P HP ) of (A H) is uniquely determined up to the permutation of blocks. Proof. See, e.g., []. Even though (8) is unique only up to a permutation of blocks, we call it the a canonical form of the pair (A H).

10 Remark 1 In some instances it will turn out be useful to use a slightly dierent form for the blocks of type (1) in (8). Multiplying the matrices from both sides by I p Z p, one nds that (1) takes the form A j = Jp ( ) J p ( ) and H j = Ip I p : (11) Using the same transformation, we can also get back from the form (11) to the form (1). This transformation will frequently be used in the following and its application will be called the Z-trick. Apart from the eigenvalues of an H-selfadjoint matrix A, the parameters " that are associated with blocks to real eigenvalues are invariants of the pair (A H). The collection of these parameters is sometimes referred to as the sign characteristic (see, e.g., [] and [11]). To highlight that these parameters are related to the matrix H (we will soon have to deal with two structures), we will use the term H-structure indices in the following. Theorem 11 Let S C nn be H-skew-adjoint. Then there exists a nonsingular matrix P C nn, such that P 1 SP = S 1 : : : S k and P HP = H 1 : : : H k (1) where S j and H j are of the same size and each pair (S j H j ) has one and only one of the following forms: 1. Blocks associated with purely imaginary eigenvalues: where R, p N, and " f1 1g. S j = ij p () and H j = "Z p (1). Blocks associated with a pair of non purely imaginary eigenvalues: ijp () Zp S j = and H = (1) ij p () Z p where C nr with Im() > and p N. Moreover, the form (P 1 SP P HP ) of (S H) is uniquely determined up to a permutation of blocks. Proof. This follows directly from Theorem 9 considering the H-selfadjoint matrix is. Again, we will call the parameter " in (1) the H-structure index of the block S j in (1). Moreover, the form (1) will be called the canonical form of the pair (S H). Remark 1 From Theorems 9 and 11, it is easy to nd the following symmetries in the spectra of H-selfadjoint and H-skew-adjoint matrices. If = R is an eigenvalue of the H- selfadjoint matrix A then so is and both eigenvalues have the same Jordan structure. If = ir is an eigenvalue of the H-skew-adjoint matrix A then so is and both eigenvalues have the same Jordan structure. 8

11 Doubly structured matrices In this section we give canonical forms for matrices that are doubly structured with respect to Hermitian or skew-hermitian nonsingular matrices G and H. First, we note that by Theorem 9, Jordan blocks associated with real eigenvalues in the selfadjoint case (or purely imaginary eigenvalues in the skew-adjoint case) have structure indices with respect to G and/or H. We will call these indices the G- and H-structure indices of A, respectively. Moreover, we may always assume that G and H are Hermitian. Otherwise, we may consider ig or ih, respectively, having in mind the following remark. Remark 1 Let H C nn be nonsingular and Hermitian or skew-hermitian and let A C nn. Then the following conditions hold. 1. A is H-selfadjoint if and only if A is ih-selfadjoint.. A is H-skew-adjoint if and only if A is ih-skew-adjoint.. A is H-selfadjoint if and only if ia is H-skew-adjoint. Remark 1 implies in particular that we may assume that the structure on A induced by one of the matrices G and H, say H, is the structure of a selfadjoint matrix. In other words, we may assume that A is H-selfadjoint. Otherwise, we may consider ia. Hence, it remains to discuss the following cases. Matrices that are H-selfadjoint and G-selfadjoint (Section.1) Matrices that are H-selfadjoint and G-skew-adjoint (Section.) Finally, we always assume that the pencil %H G is nondefective. Remark 1 Instead of requiring that %H G is nondefective, we may as well consider the generalization of this case, that for the matrices A G H at least one of the three pencils %H G, %H HA and %G GA is nondefective. For example, if A is nonsingular and both H- and G-selfadjoint then we can consider the matrix triple (H 1 G H HA) for which H, HA are Hermitian and H 1 G is H and HA selfadjoint, since (H 1 G) = GH 1. Thus, if %H HA is nondefective then we can get the canonical form of this new triple. But once we have this, we can easily get the canonical form of the original triple (A H G). So our results will cover more general cases. 9

12 .1 Matrices that are H-selfadjoint and G-selfadjoint In this section we will derive a canonical form for matrices that are selfadjoint with respect to nonsingular Hermitian matrices H and G, such that the pencil %H G is nondefective. For the proof of our main result, the following lemma will be needed. Lemma 15 Let G H C nn be Hermitian and nonsingular. Let A C nn be H- selfadjoint and G-selfadjoint. Then there exists a nonsingular matrix P C nn such that P 1 AP = A 1 : : : A k P HP = H 1 : : : H k P GP = G 1 : : : G k where A j, H j and G j have corresponding sizes. Moreover, each pencil %H j G j has as spectrum either f j j g for some j C nr or f j g for some j R and the spectra of two subpencils %H j G j and %H l G l, j = l, are disjoint. Proof. By Theorem, there exists a nonsingular matrix Q C nn such that Q GQ = G1 G ~ Q HQ = H1 H ~ and Q 1 AQ = A11 A 1 A 1 A where the pencil %H 1 G 1 has as spectrum either f 1 1 g for some 1 C nr or f 1 g for some 1 R and such that the spectra of the pencils %H 1 G 1 and % H ~ G ~ are disjoint. Since A is H-selfadjoint and G-selfadjoint, we obtain that A H 11 1 A ~ 1 H H1 A 11 H 1 A 1 and A 1 H 1 A ~ H A G 11 1 A ~ 1 G A G 1 1 A ~ G Since with G also ~ G is nonsingular, this implies = = ~H A 1 ~ H A G1 A 11 G 1 A 1 ~G A 1 G ~ : A A 1 ~ H ~ G 1 = H 1 A 1 ~ G 1 = H 1 G 1 1 G 1A 1 ~ G 1 = H 1 G 1 1 A 1 : Since the pencils %H 1 G 1 and % ~ H ~ G have disjoint spectra, we obtain that A 1 = and therefore A 1 = H 1 1 A 1 ~ H =. The rest of the proof now follows by induction. Theorem 1 Let G H C nn be Hermitian and nonsingular such that the pencil %H G is nondefective. Let A C nn be H-selfadjoint and G-selfadjoint. Then there exists a nonsingular matrix P C nn such that P 1 AP = A 1 : : : A k P GP = G 1 : : : G k (15) P HP = H 1 : : : H k 1

13 where the blocks A j, G j, H j have corresponding sizes and are of one and only one of the following forms. Type (1): A j = J p () H j = "Z p and G j = "Z p where R, p N, " f1 1g, and Rnfg. The H-structure index of A j is " and the G-structure index of A j is sign("). Type (): Jp () A j = J p () H j = Zp Z p and G j = Zp Z p where R, p N, and C, Im() >. The H-structure indices of A j are 1 1 and the G-structure indices of A j are 1 1. Type (): Jp () A j = J p () H j = Zp Z p and G j = Zp Z p where C nr, p N, and C nfg, where Im(). Moreover, the canonical form (15) is unique up to permutation of blocks. Proof. By Lemma 15 we may assume that the pencil %H G has the eigenvalues either for some C nr or for some R. Case 1: R. Since the pencil %H G is nondefective, by Corollary there exists a nonsingular matrix P C nn such that P Im Im (%H G)P = % I n m I n m i.e., in particular that G is a scalar multiple of H. Applying Theorem 9, we nd that there exists a nonsingular matrix Q C nn such that (Q 1 AQ Q HQ) is in canonical form (8). Since G = H, we obtain that A, H, and G can be reduced simultaneously to block diagonal form with diagonal blocks of types 1 and. Case : C nr. In this case, we obtain from Corollary that there exists a nonsingular matrix P C nn such that P (%H G)P = % Im Im I m I m 11

14 where m = n and Im() >. Let A = A11 A 1 A 1 A be partitioned conformably. Then we obtain from A H = HA and A G = GA that A 1 = A 1 and A 1 = A 1 : Since =, this implies that A 1 =. In an analogous way we show that A 1 =, and moreover, we have A = A 11 by symmetry. Let Q 1 be such that Q 1 A 1 11Q 1 is in Jordan canonical form and set Q1 Q = P Q : 1 Then we obtain ~A = Q 1 AQ = Q Im HQ = I m Q 1 1 A 11Q 1 Q 1 A 11 Q 1 and Q GQ = Jp () J p () H ~ Ip = I p Im I m After a proper block permutation, we obtain that A, H, and G can be reduced simultaneously to block diagonal form with diagonal blocks of the forms : and ~ G = Ip I p respectively, where p N and C. The result then follows by applying the Z-trick, see Remark 1. Uniqueness: Suppose that A1 A = H = A ~A = H1 G1 G = H G ~A1 ~H1 A ~ H ~ ~G1 = H ~ G ~ = G ~ are in canonical form, where H, G, H, ~ G ~ are Hermitian nonsingular, A is H-selfadjoint and G-selfadjoint and A ~ is H-selfadjoint ~ and G-selfadjoint ~ and all matrices have corresponding block structures. If P 1 AP = A, ~ (A1 ) = ( A1 ~ ) and (A ) = ( A ~ ), such that the spectra of A 1 and A are disjoint, then it follows immediately that P has a corresponding block diagonal structure. Analogously, assuming that the spectra of %H 1 G 1 and % H ~ G ~ (and of %H G and % H1 ~ G1 ~, respectively) are disjoint and that P HP = H ~ and P GP = G, ~ where P is nonsingular, we obtain again that P has a corresponding block diagonal structure. Indeed, partitioning P = P11 P 1 P 1 P and P = 1 Q11 Q 1 Q 1 Q and

15 conformably with H, we obtain that This implies that G 11 P 1 = Q 1 ~ G and H 11 P 1 = Q 1 ~ H : H 1 11 G 11P 1 = P 1 ~ H 1 ~ G and from that, we obtain P 1 =, since the spectra of H 1 11 G 11 and ~ H 1 ~ G are disjoint. Analogously, we show P 1 =. Hence, it is sucient to prove the uniqueness for the case that A has only one pair of eigenvalues and that %G H has only a pair of eigenvalues. But then, the uniqueness is clear, since we obtain from Theorem 9 the uniqueness of the canonical form for the pair (A H). Note that the structure G is then uniquely dened by the invariant with Im(). In both cases it is easy to verify that the H and G-structure indices of each block are as claimed in the theorem.. Matrices that are H-selfadjoint and G-skew-adjoint In this section we present a canonical form for a matrix A that is H-selfadjoint and G-skewadjoint, where H and G are Hermitian nonsingular matrices, such that the pencil %H G is nondefective. By Remark 1, the eigenvalues of A satisfy more symmetry properties. If C is an eigenvalue of A then, because A is G-skew-adjoint, so is having the same Jordan structure as. On the other hand, A is H-selfadjoint and thus, with and also and are eigenvalues of A having the same Jordan structures as. Thus, the eigenvalues of A occur in quadruples f g, where all these eigenvalues have the same Jordan structures. If is real or purely imaginary, this set is equal to f g, and if =, this set is just fg. The following lemmas will be needed for constructing the canonical form. Lemma 1 Let G H C nn be Hermitian and nonsingular. Furthermore, let A C nn be H-selfadjoint and G-skew-adjoint. Then there exists a nonsingular matrix P C nn such that P 1 AP = A 1 : : : A k P HP = H 1 : : : H k P GP = G 1 : : : G k where A j, H j and G j have corresponding sizes. Moreover, each matrix A j has the spectrum f j j j j g and the spectra of two matrices A j and A l, where j = l, are disjoint. 1

16 Proof. By using the eigenvalue properties of A mentioned above, one can nd a matrix Q C nn such that Q GQ = G11 G 1 G Q HQ = 1 G H11 H 1 H and Q 1 A1 AQ = 1 H A ~ where A 1 has the spectrum f g for some 1 C, such that the spectra of A 1 and ~ A are disjoint. Then we obtain from A H = HA and A G = GA that A 1 H 1 = H 1 ~ A and A 1 G 1 = G 1 ~ A By construction, the spectra of A 1 and ~ A are disjoint. This implies H 1 = and G 1 =. The proof then follows by induction. Lemma 18 Let G H C nn be Hermitian and nonsingular. Furthermore, let A C nn be H-selfadjoint and G-skew-adjoint. Then there exists a nonsingular matrix P C nn such that P 1 AP = A 1 : : : A k P HP = H 1 : : : H k P GP = G 1 : : : G k where A j, H j and G j have corresponding sizes. The spectrum of each pencil %H j G j is contained in f j j j j g for some j C and the spectrum of %H l G l is disjoint from the set f j j j j g if j = l. Proof. The proof proceeds analogously to the proof of Lemma 15 using the equations A H = HA and A G = GA. Note that in contrast to the eigenvalue of A, the eigenvalues of the pencil %H G need not occur in quadruples f j j j j g. If j is an eigenvalue of %H G then from Theorem, we only know that also j is an eigenvalue, but j and j need not be. However, to get corresponding block diagonal forms of A, G, H, we have to group j and j together with j and j if they are also eigenvalues of %H G. In view of Lemma 18, it is sucient to consider pencils %H G whose spectrum is contained in f g. Therefore, a discussion of properties of such pencils will be helpful. Lemma 19 Let G, H C nn be nonsingular and Hermitian such that the pencil %H G is nondefective. (i) If the spectrum of %H G is contained in f g, where Rnfg, then H 1 GH 1 G = I n : 1

17 (ii) If the spectrum of %H G is contained in f g, where C nr, then there exists a matrix P such that for ~ H = P HP, ~ G = P GP and ~ A = P 1 AP, Moreover, ~H 1 G ~ H ~ 1 ~G I = m : (1) I m A1 ~A = A 1 Proof. (i) We consider the problem in two cases. Case (1): Im() =. Since the pencil %H G is nondefective and has only the eigenvalues R, by Corollary there exists a nonsingular matrix P C nn and numbers p q r s N such that I p I p H = P I q I r 5 P and G = P I q I r 5 P: I s I s This implies H 1 GH 1 G = P 1 ( I n )P = I n. Case (): Re() =. Since the pencil %H G is nondefective and has only the eigenvalues ir, by Corollary there exists a nonsingular matrix P C nn such that Im P and G = P P H = P I m : I m I m where m = n N. This implies H 1 GH 1 G = P 1 ( I n )P = I n. (ii) By Corollary there exists a nonsingular matrix P such that where m = n % ~ H ~ G = %P HP P GP = % Im I m N and = I p ( I m p ), p m. We then obtain that ~H 1 ~ G = (1) and hence we have (1). Note that ~ A is ~ H-selfadjoint and ~ G-skew-adjoint. This implies that ~A( ~ H 1 ~G) = ~ H 1 ~A ~ G = ( ~ H 1 ~G) ~ A: Since in this case =, from the block form (1) we get ~ A = A1 A. Since ~ A is ~H-selfadjoint, we obtain that A = A 1. 15

18 Theorem Let G H C nn be Hermitian nonsingular such that the pencil %H G is nondefective. Furthermore, let A C nn be H-selfadjoint and G-skew-adjoint. Then there exists a nonsingular matrix P C nn such that P 1 AP = A 1 : : : A k P GP = G 1 : : : G k (18) P HP = H 1 : : : H k where, for each j, the blocks A j, G j, H j have corresponding sizes and are of one and only one of the following forms. Type (1a): H j = A j = Z p Z p Z p Z p J p () J p () J p () J p () 5 and G j = 5 (19) Z p Z p Z p Z p where C with Re()Im() >, p N and Rnfg, Re() Im(). Type (1b): Jp () A j = J p ( ) Zm H j = " ( G jj ) Z j = m Z m Z m 5 () (1) where >, p N and Rnfg, Re() Im(). The H j -structure index of is " and the H j -structure index of is "( jj ). Type (1c): Jp () A j = i J p ( ) H j = Zm Z m Zm G j = "jj ( jj () ) Z m where >, p N and Rnfg, Re() Im(). The G j -structure index of is " and the G j -structure index of is "( jj ). Type (1d1): A j = J p () H j = "Z p and G j = ~"F p () where Rnfg, Re() Im(), and p N is odd if R and even if ir. Moreover, the eigenvalue = has the H j -structure index " and the G j -structure index sign(~") if R and sign( i~") if ir. 1

19 Type (1d): Jp () A j = J p () H j = Zp Z p and G j = F p F p () where Rnfg, Re() Im(), and p N is even if R and odd if ir. Moreover, the eigenvalue = has the H j -structure indices +1 1 and the G j -structure indices Type (a): H j = A j = Z p Z p Z p Z p J p () J p () J p () J p () 5 and G j = 5 (5) Z p Z p Z p Z p where C with Re()Im() >, p N, and C with Re()Im() >. Type (b): H j = A j = Z p Z p Z p Z p J p () J p () J p () J p () 5 and G j = 5 () 5 () Z p Z p Z p Z p 5 (8) where >, p N, and C with Re()Im() >. The H j -structure indices of are +1 1 and the H j -structure indices of are Type (c): H j = A j = i Z p Z p Z p Z p J p () J p () J p () J p () 5 and G j = 1 5 (9) Z p Z p Z p Z p 5 ()

20 where >, p N, and C with Re()Im() >. The G j -structure indices of are +1 1 and the G j -structure indices of are Type (d): Jp () A j = J p () H j = Zp Z p G j = " F p ( 1) p+1 F p (1) where p N, " f+1 1g, and C with Re()Im() >. Moreover, the eigenvalue = has the H j -structure indices +1 1 and the G j -structure indices In the blocks of type (1a){(1d) the subpencil %H j G j has only real or purely imaginary eigenvalues and in the blocks (a){(d) the subpencil %H j G j has only eigenvalues that are neither real nor purely imaginary. Moreover, the canonical form (18) is unique up to permutation of blocks. Proof. In view of Lemma 18, we may assume that the spectrum of the pencil %H G is contained in f g for some C nfg, Re() Im(), and it is sucient to distinguish the following two cases. Case (1): Re()Im() =. In view of Lemma 1, we may distinguish the following four subcases. Subcase (1a): The spectrum of A is f g, where Re()Im() >. Since A is H-selfadjoint and G-skew-adjoint, it follows from Remark 1 that,, and have the same Jordan structures. Applying Theorem 9, the Z-trick, and a block permutation, we may assume that A and H have the following forms. A = J () J () J () J () 5 H = I m I m I m I m 5 () where m = n N and J () is an (m m) matrix in Jordan canonical form only having the eigenvalue. Then, the equation A G = GA and the fact that,,, and are pairwise distinct, imply that G necessarily has the form G = G G G G 5 () where G G C mm. By Lemma 19, we obtain that H 1 GH 1 G = I n. This implies in particular that G G = I m : () 18

21 Note that the equation A G = GA also implies that J () G = G J (), i.e., G commutes with J (). Hence, setting Q := we obtain that Q 1 AQ = A, Q HQ = H and Q GQ = 1 G 1 I m 1 G 1 I m I m 1 G G 1 G G I m 5 5 : (5) Then it follows from () and (5) that the triple (A H G) can be reduced to blocks of the form given by (19) and (), by applying a proper block permutation and the Z-trick. Subcase (1b): The spectrum of A is f g, where >. Theorem 11 implies that and have the same Jordan structures. Moreover, applying Theorem 9, we may assume that A, H, and G have the following forms. where A = A1 H = A 1 H1 and G = H A 1 = J p1 () : : : J pk () H 1 = " 1 Z p1 : : : " k Z pk H = ~" 1 Z p1 : : : ~" k Z pk G1 G G G and G j C mm for m = n. Observing that A G = GA, we obtain that G 1 = G =, since =, and A 1 G = G A 1. Moreover, H 1 GH 1 G = I n implies that H 1 G 1 H 1 G = I m = H 1 G H 1G 1 : () Setting Im Q = 1 H 1 G then from (), A G 1 = G A 1, Zp 1 J p () Z p = J p () (Lemma ), and the block forms of H and A 1, we obtain that Q 1 A1 A1 AQ = G H = A 1 H 1 G A 1 H1 H1 Q HQ = Q GQ = = 1 j j G H 1 H H 1 G 1 G H 1 G 1 G H 1 G 19 ( jj ) H 1 H1 = H 1 : and

22 Thus, it follows from a proper block permutation that we may assume that Jp () A = H = J p () Hence, setting "Zp "( and G = jj ) Z p ~Q = " 1 I m I m we nd that ~ Q 1 A ~ Q, ~ Q H ~ Q, and ~ Q G ~ Q have the desired forms. "Z p "Z p Subcase (1c): The spectrum of A is f g, where ir, Im() >. The matrix ia is G-selfadjoint, H-skew-adjoint and has only a pair of real eigenvalues. Noting that the spectrum of %G H is contained in f 1 1 g, we can reduce the problem to Case (1b), i.e., it is sucient consider the case that ia, G, and H have the form (1). ia = Jp () J p ( ) where ~" f+1 1g. Setting G = Q = we obtain that Q 1 ( ia)q = ia, Q Zm HQ = Z m ~"Zm ~"( jj H = ) Z m 1 I m 1 I m and Q GQ = ~"jj Subcase (1d): The spectrum of A is fg. 1 Z m 1 Z m Zm ~"( jj : ) Z m It follows from Lemma in the appendix that the triple (A H G) can be reduced to blocks of the forms () or (). Case (): Re()Im() =. By Corollary we may assume that the pencil %H G is already in the form %H G = % Im I m where m = n N and = diag(i p I m p ), 1 p m and, furthermore, we have that (1). Then Lemma 19 implies that A has the form A1 A = A : 1 Note, that by Lemma 19 a similarity transformation on A with a corresponding block diagonal matrix and simultaneous congruence transformations on H, G will not change :

23 the block structure of A and the identity (1), but it does change the block forms in H and G. Hence we can apply similarity transformations on A 1 and at the same time keep the relation (1). Again, we will consider the following four subcases. Subcase (a): The spectrum of A is f g, where Re()Im() >. Again, the eigenvalues, and have the same Jordan structures. Moreover, there exists a nonsingular matrix Q1 Q = Q C nn 1 such that Q 1 AQ = A 11 A A 11 A 5 and Q HQ = H where A 11 C kk has the eigenvalues and and A C ( n k)( n k) has the eigenvalues and. Partitioning Q GQ conformably, i.e., Q GQ = G 1 G G G G 1 G G G we obtain from the equation A G = GA and the fact that A 11 and A have no common eigenvalues that G = G =. Thus, after a proper block permutation, we may consider two smaller subproblems. The rst one is A11 ~A = A 11 and (1) implies that ~ H = Ik I k 5 ~H 1 G ~ H ~ 1 ~G I = k : I k and ~ G = G1 G 1 Hence, after applying a similarity transformation on A 11, we may assume that ~ A, ~ G, and ~H are in the forms () and (), where (G 1 ) = I. The remainder of the proof then proceeds analogously to Subcase (1a). The second subproblem with respect to A can be transformed in the same way. Subcase (b): The spectrum of A is f g, where >. We obtain from Theorem 11 that and have the same Jordan structures. Hence, both A 1 and A 1 must have the eigenvalues and with the same Jordan structures. Thus, there exists a nonsingular matrix Q1 Q = Q 1 1

24 such that Q 1 AQ = J () J () J () J () 5 and Q HQ = H where k = n and J () is an k k matrix in Jordan canonical form associated with only one eigenvalue. Partitioning Q GQ conformably, i.e., Q GQ = G 1 G G G G 1 G G G we obtain from A G = GA and the fact that J () and J () have no common eigenvalues that G 1 = G = and J () G = G J (), J () G = G J (). Moreover, we still have (1), which implies that G G = I. Thus we may assume that A, G, and H are in the forms () and (), where G G = I. The remainder of the proof then proceeds analogously to Subcase (1a). 5 Subcase (c): The spectrum of A is f g, where ir. The proof proceeds analogously to the proof of Subcase (1c). Subcase (d): The spectrum of A is fg. This case follows from Lemma in the appendix and by applying the Z-trick. Uniqueness: Analogous to the proof of Theorem 1, it is sucient to prove uniqueness for the case that the spectrum of A is f g for some C and that the spectrum of %H G is contained in f g for some C. Again, the canonical form for the pair (A H) is unique. In any case except for the case that = and = R, the matrix G is then uniquely determined by the invariants with Re() Im() (and signs " or ~" in some cases that are uniquely determined by the canonical form for the pair (A G)). Only in the case = and = R, we have an additional invariant " that is not an invariant of the canonical form for the pair (A G). In this case, the uniqueness follows from Lemma in the appendix. In all cases (1a){(d) it is easy to verify that the H and G-structure indices of each block are as claimed in the theorem. 5 Singly and doubly structured pencils In this section, we discuss canonical forms for matrix pencils %A B, where both A and B are matrices that are singly or doubly structured with respect to some indenite inner

25 product. It turns out that the case of structured pencils can be reduced to the matrix case. This is done in the following theorem. Theorem 1 Let the matrices G H C nn be nonsingular and Hermitian or skew- Hermitian, i.e., G = G G and H = H H where G H f1 1g. Furthermore, let %A B C nn be a regular pencil such that A H = " A HA A G = A GA B H = " B HB B G = B GB () where " A " B A B f1 1g. Then there exists nonsingular matrices P Q C nn that P 1 In1 M (%A B)Q = % N I n H11 Q HP = Q GP = H G11 G such where M H 11 G 11 C n 1 n 1 and N H G C n n. Moreover, M and N are in Jordan canonical form, N is nilpotent and the following conditions are satised. H 11 = H " A H 11 G 11 = G A G 11 M H 11 = " A " B H 11 M M G 11 = A B G 11 M H = H " B H G = G B G N H = " A " B H N N G = A B G N: Proof. Let P Q C nn be such that the pencil P 1 In1 (%A B)Q = N M I n (8) is in Kronecker canonical form (see []), where M, N are in Jordan canonical form and N is nilpotent. Then () and (8) imply in particular that Q H(%" A A " B B) = Q (%A B M )H = % P H: From this and (8) we obtain that Q In1 HP N Q HP M I n In1 N I n = Q In1 HAQ = " A N P HQ M = Q HBQ = " B P HQ: I n

26 Setting Q H11 H HP = 1 H 1 H and H11 H 1 This implies, in particular, that and noting that P HQ = H (Q HP ), we nd that H 1 N H N = H " A H 11 H 1 N H 1 N H H11 M H 1 M = H 11 M H 1 H 1 M H H " B H 1 H H 1 = H " B M H 1 = " A " B M H 1 N = (" A " B ) k (M ) k H 1 N k for every k N: Since N is nilpotent, it follows that H 1 = and thus, also H 1 = H " A N H 1 =. Moreover, H 11 = H " A H 11 and H = H " B H, and H N = H " A N H = " A " B N H, H 11 M = H " B M H 11 = " A " B M H 11. Analogously we show that Q GP has the structure claimed in the theorem. This concludes the proof. We note that M is a doubly structured matrix with structures induced by H 11 and G 11 and that N is a nilpotent doubly structured matrix with structured induced by H and G, where H 11, G 11, H, and G are all Hermitian or skew-hermitian. Therefore, Theorem 1 gives a general description about how to obtain the canonical forms for the pencil case from the canonical forms in the matrix case that are given in the previous sections. We only have to further reduce M and N by applying the results from Section. Note that Theorem 1 does not require the pencil %H G to be nondefective. However, canonical forms for the matrix case are known for this case only. Theorem 1 also describes the case of singly structured pencils. In this case one may choose H = G, " A = A, and " B = B. Thus, Theorem 1 gives a general description how to obtain canonical forms for singly and doubly structured pencils from the canonical forms in the matrix case. For obvious reasons, we do not give a list of the canonical forms for all possible cases, but only one example to illustrate the eect of Theorem 1. Theorem Let H C nn be Hermitian and nonsingular and let %A B C nn be a regular pencil such that A and B are H-selfadjoint. Then there exists nonsingular matrices P Q C nn such that P 1 (%A B)Q = % Q HP = A 1... A k H 1... H k 5 B 1... : B k where the blocks A j, B j, and H j have corresponding sizes and are of one and only one of the following forms: 5 5

27 1. Blocks associated with real eigenvalues: where p N, R, and " f1 1g. A j = I p B j = J p () and H j = "Z p. Blocks associated with a pair of nonreal eigenvalues: Jp () A j = I p B j = and H J p () j = Z p where p N and C nr.. Blocks associated with the eigenvalue 1: where p N and " f1 1g. A j = J p () B j = I p and H j = "Z p Moreover, this form is uniquely determined up to permutation of blocks. Proof. This follows directly from Theorem 1 and Theorem 9. Note that with the assumptions and notation of Theorem the pencil H(%A B) = %HA HB is a Hermitian pencil. It turns out that Theorem is a generalization of Theorem. Indeed, the pencil Q HP P 1 (%A B)Q is a Hermitian pencil in canonical form. Conclusions We have derived canonical forms for matrices and matrix pencils that are doubly structured in the sense that they are H-selfadjoint (or H-skew-adjoint) and at the same time G-selfadjoint (or G-skew-adjoint), where we have assumed that G H are nonsingular Hermitian (or skew Hermitian) and G H is a nondefective pencil. The general case that G or H are singular, or that the pencil G H is defective is still an open problem. Also the associated real canonical forms, which appear to be much more dicult, are open. In view of the applications in eigenvalue computations, it is also important to restrict the transformation matrices to be unitary (or orthogonal in the real case). This case will be covered in a forthcoming paper, that will also address numerical methods, in particular for the classes of pencils arising in quantum chemistry that we have discussed in the introduction. 5

28 References [1] G. Ammar, C. Mehl, and V. Mehrmann. Schur-like forms for matrix Lie groups, Lie algebras and Jordan algebras. Linear Algebra Appl., 8:11{9, [] D.Z. Djokovic, J. Patera, P. Winternitz and H. Zassenhaus, Normal forms of elements of classical real and complex Lie and Jordan algebras, J. Math. Phys., :1-1, 198. [] U. Flaschka. Eine Variante des Lanczos-Algorithmus fur groe, dunn besetzte symmetrische Matrizen mit Blockstruktur. Dissertation, Universitat Bielefeld, Bielefeld, FRG, 199. [] U. Flaschka, W.-W. Lin, and J.-L. Wu. A kqz algorithm for solving linear-response eigenvalue equations. Linear Algebra Appl., 15:9{1, 199. [5] F. Gantmacher. Theory of Matrices, volume 1. Chelsea, New York, [] F. Gantmacher. Theory of Matrices, volume. Chelsea, New York, [] I. Gohberg, P. Lancaster, and L. Rodman. Matrices and Indenite Scalar Products. Birkhauser Verlag, Basel, Boston, Stuttgart, 198. [8] A. Hansen, B. Voigt, and S. Rettrup. Large-scale RPA calculations of chiroptical properties of organic molecules: Program RPAC. International Journal of Quantum Chemistry, XXIII.:595{11, 198. [9] R. Horn and C. Johnson. Topics in matrix analysis. Camebridge University Press, Camebridge, [1] L. Kronecker. Algebraische Reduction von Scharen bilinearer Formen, 189. In Collected Works III (second part), pages 11{155, Chelsea, New York, 198. [11] P. Lancaster and L. Rodman. Algebraic Riccati Equations. Clarendon Press, Oxford, [1] P. Lancaster and M. Tismenetsky. The Theory of Matrices. Academic Press, Orlando, nd edition, [1] W.-W. Lin, V. Mehrmann, and H. Xu. Canonical forms for Hamiltonian and symplectic matrices and pencils. Linear Algebra Appl., 1{:9{5, [1] C. Mehl. Compatible Lie and Jordan algebras and applications to structured matrices and pencils. Dissertation, Fakultat fur Mathematik, TU Chemnitz, 91 Chemnitz (FRG), 1998.

29 [15] C. Mehl. Condensed forms for skew-hamiltonian/hamiltonian pencils. SIAM J. Matrix Anal. Appl., 1:5{, [1] V.Mehrmann, and H. Xu. Structured Jordan canonical forms for structured matrices that are Hermitian, skew Hermitian or unitary with respect to indenite inner products. Electron. J. Linear Algebra, 5:{1, [1] J. Olson, H. Jensen, and P. Jrgensen. Solution of large matrix equations which occur in response theory. J. Comput. Phys., :5{8, [18] R. Thompson. The characteristic polynomial of a principal subpencil of a Hermitian matrix pencil. Linear Algebra Appl., 1:15{1, 19. [19] K. Weierstra. Zur Theorie der bilinearen und quadratischen Formen. Monatsb. Akad. d. Wiss. Berlin, pages 1{8, 18. Appendix In the appendix we derive some technical Lemmas. Recall the Kronecker product (see, e.g., [9, 1]). Denition Let A = [a jk ] C mn and B C pq. Then A B := a 11 B : : : a 1n B.. a m1 B : : : a mn B 5 C mpnq : This product has the following basic properties (see, e.g., [9, 1]). Proposition Let A C C p 1 p, B D C q 1 q, E C p p, and F C q q. Then the following identities hold. 1. A (B + D) = A B + A D, (A + C) B = A B + C B.. (A B)(E F ) = (AE) (BF ):. A B is invertible if and only if A and B are invertible. In this case we have that (A B) 1 = A 1 B 1 :. (A B) T = A T B T, (A B) = A B : 5. A B = if and only if A = or B =.

30 We will frequently need the permutation matrix mn = [e 1 e n+1 : : : e (m 1)n+1 e e n+ : : : e (m 1)n+ e n e n : : : e mn ]: If A, B are m n and p q, respectively, then mp(a B) nq = B A: In the following we derive the canonical forms for doubly structured matrices that are nilpotent. This case is the most complicated case, since we have least symmetry in the spectrum. Therefore, we have to use a very technical reduction procedure. For the sake of briefness of notation, let J p denote the nilpotent Jordan block J p () of size p. O pq is the p q zero matrix. Lemma 5 Let Z p, D p, and F p be dened as in Section and let k l p q N, (p q). Then Z p Jp l = (Jp) l Z p D p Jp ld p = ( 1) l J l F p J k p Z p J k p Jq l O p qq D p Op qq F q Jq l O p qq = Op qq Z q J k+l q p = ( 1) p q Op qq F q J k+l = ( 1) p q Op qq D q F q F pjp l = ( 1) l (Jp) l F p : (9) : () q : (1) : () Denition Let A = (a jk ) nn C nn. Then the l-th lower anti-diagonal of A or, in short, the l-th anti-diagonal of A is dened by the elements a jk, where j + k = n l. Here, we allow l =. The -th anti-diagonal is also called the main anti-diagonal. If B = B ~ and C = ~C where ~ B and ~ C are square matrices, then the l-th anti-diagonal of ~ B and ~ C is called the l-th anti-diagonal of B and C, respectively. Analogously, we dene the l-th block anti-diagonal for square and non-square block matrices. Lemma Let G H C nn be Hermitian nonsingular such that the pencil %H G is nondefective and such that its spectrum is contained in f g, where Rnfg and Re() Im(). Furthermore, let A C nn be nilpotent, H-selfadjoint and G-skewadjoint. Then there exists a nonsingular matrix P C nn such that P 1 AP = A 1 : : : A k P GP = G 1 : : : G k () P HP = H 1 : : : H k 8

31 where the blocks A j, G j, H j have corresponding sizes and, for each j, are of one and only one of the following forms. Type (1d1): A j = J p () H j = "Z p and G j = ~"F p () where p N is odd if R and even if ir. Type (1d): Jp () A j = J p () H j = where p N is even if R and odd if ir. Zp Z p G j = F p F p (5) Proof. Applying Theorem 9, we may assume that (A H) is in canonical form, i.e., collecting blocks of same size and representing them by means of the Kronecker product, we may assume that A = I m1 J p H = m1 Z p I mk J pk mk Z pk where p 1 > : : : > p k are the sizes of Jordan blocks and mj are signature matrices for j = 1 : : : k. Setting I m1 F p1 F = 5 I mk F pk we obtain from A G = GA and (9) that A and F G commute. Thus, the structure of G is implicitly given by the well-known form for matrices that commute with matrices in Jordan canonical form (see [5]). For the sake of better clarity, we will not work directly on A, H, and G, but rst apply a permutation. Setting = m1 p 1 : : : mk p k and updating A, H, G by 1 A, H, G, we may consider the following situation. A = J p1 I m1 J pk I mk where H jj := Z pj mj. Partitioning G = 5 and H = G 11 : : : G 1k.. G 1k : : : G kk 9 H 11 H kk 5 : () 5 ()

32 conformably and using the well-known structures of matrices that commute with matrices in Jordan canonical form [5], we obtain that G qq = = px q 1 l= (F pq J l p q ) G (l) qq : : : : : : G () qq. G () qq G (1) qq. G () qq G (1) qq.. ( 1) pq+1 G () qq : : : : : : : : : ( 1) pq+1 G (pq 1) qq 5 (8) for q = 1 : : : k, where G qq (l) mq mq C and G qr = px r 1 l=1 Opq p rp r F pr J l p r G (l) qr (9) for 1 q < r k, with G (l) qr C mq mr. We will stepwise reduce the matrix G, while keeping the forms of A and H. Step (1): We rst show that G () jj is nonsingular for j = 1 : : : k. Since the pencil %H G is nondefective and has only the eigenvalues, where Rnfg, we obtain from Lemma 19 that GH 1 G = H: Comparing the j-th diagonal blocks on both sides, this implies in particular that H jj = G 1j H 11G 1j + : : : + G jj H jj G jj + : : : + G jk H kk G jk : (5) Because of the structure of the blocks G qr, it follows that all the block anti-diagonals of G qr H qqg qr and G qr H rr G qr are zero for q < r, and hence, comparing the main block anti-diagonals on both sides of (5), we obtain that Since F pj Z pj F pj Z pj mj = (F pj G () jj )(Z pj mj )(F pj G () jj ) = Z pj, this implies that = (F pj Z pj F pj ) (G () jj m j G () jj ) G () jj m j G () jj = 1 m j (51) and thus, G () jj is nonsingular.

33 Step (): Elimination of G 1 : : : G 1k Assume that we already have G (l 1) 1j = for all j = : : : k and G (l) 1j = for j = : : : r 1, where l and r. We then show how to eliminate G 1r, (l) while keeping the forms of A and H. Let X = r I r X r1 have a block form analogous to G, where zero blocks of the matrix are indicated by blanks and, moreover, X 1r = J l p r O p1 p rp r X 1r 1 ( 1)p 1 p r+1 (G () 11) 1 G (l) 1r X r1 = O prp 1 p r J l p r 1 ( 1)l+1 (G () rr) (G (l) 1r) : Substep (a) X is chosen such that it commutes with A. I 5 ^= Substep (b) In the updated matrix ~ G := X GX we have ~ G (l) 1r =. : : : Indeed, it is easy to see that ~ G is again a matrix of the form (), (8), and (9). The (1 r)-block of ~ G satises ~ G1r = G11X1r + X r1 G 1r X 1r + G 1r + X r1 G rr: (5) From the structure of G and X, we nd immediately that the rst l 1 block anti-diagonals of all the summands of the right hand side of (5) are zero. Furthermore, the l-th block anti-diagonal of G1r ~ has the form (F p1 G () 11) + Op1 p rp r F pr J l p r + Oprp 1 p r (Jp l r ) J l p r O p1 p rp r G (l) 1r 1 ( 1)l+1 G (l) = 1 ( 1)p 1 p r+1 Jp F l r p1 + 1 ( 1)l+1 Oprp 1 p r (J l p r ) O p1 p rp r 1 ( 1)p 1 p r+1 (G () 11) 1 G (l) 1r 1r(G () rr) 1 G (l) 1r + F pr G (l) 1r = 1 (F pr G () rr) Op1 p rp r F pr Jp l r G (l) 1r 1 5 C A

34 using (9) and (1). Substep (c) In the updated matrix G ~ := X GX we still have G ~ (l 1) 1j = for all j = : : : k and G ~ (l) 1j = for j = : : : r 1. Indeed, the elements of the rst block row of ~ G have the form G 1q + X r1 G qr for 1 < q < r and G 1q + X r1 G rq for r < q: From the block structure of G 1q, G rq, G qr, and X r1, we obtain that the rst p q p r + l block anti-diagonals in X r1 G qr and the rst p r p q + l 1 block anti-diagonals in X r1 G rq are zero. Substep (d) We show that the matrix ~ H := X HX is block diagonal. The only changes outside the block diagonal can have happened to the (1 r)-block ~ H 1r and the (r 1)-block ~ Hr1 = ~ H 1r. The (1 r)-block has the form ~H 1r = (Z p1 m1 )X 1r + X 1r(Z pr mr ) (5) = 1 Op1 p rp r Z pr J l p r (5) ( 1) p 1 p r+1 m1 (G () 11) 1 G (l) 1r + ( 1) l+1 G (l) 1r(G () rr) 1 mr (55) using (9) and (). On the other hand, we have GH 1 G = H. Noting that H 1 = H and comparing the (1 r) blocks of both sides, we obtain that = G 11 H 11 G 1r + r 1 X q= G 1q H qq G qr! + G 1r H rr G rr + kx q=r+1 G 1q H qq G rq! : (5) Clearly, the rst l 1 anti-diagonals of all the summands in (5) are zero. We now consider the l-th block anti-diagonal. We note that G 11 H 11 G 1r and G 1r H rr G rr are the only summands that have a nonzero l-th block anti-diagonal. For the terms G 1q H qq G qr, 1 < q < r this follows from the fact that the l-th block anti-diagonal of G 1q is already zero. For G 1q H qq G rq, q > r this can be seen as follows. If we write the j-th block antidiagonal of G 1q H qq G rq in the form S j T j, then we obtain p q p 1 p q S j = p q F pq Jp j Z pq h p r p q p q i 1 p q C F p q A q = p r p q p q p 1 p r p r p q p q F pq Jp j q Z pq F p q 5 :