Structured decompositions for matrix triples: SVD-like concepts for structured matrices

Transcription

1 Structured decompositions for matrix triples: SVD-like concepts for structured matrices Christian Mehl Volker Mehrmann Hongguo Xu July 1 8 In memory of Ralph Byers (1955-7) Abstract Canonical forms for matrix triples (A G Ĝ) where A is arbitrary rectangular and G Ĝ are either real symmetric or skew symmetric or complex Hermitian or skew Hermitian are derived These forms generalize classical product Schur forms as well as singular value decompositions An new proof for the complex case is given where there is no need to distinguish whether G and Ĝ are Hermitian or skew Hermitian This proof is independent from the results in 1 where a similar canonical form has been obtained for the complex case and it allows generalization to the real case Here the three cases ie that G and Ĝ are both symmetric both skew symmetric or one each are treated separately Keywords Matrix triples indefinite inner product structured SVD canonical form Hamiltonian matrix skew-hamiltonian matrix AMS subject classification 15A1 65F15 65L8 65L5 1 Introduction Let F denote either the complex field C or the real field R Consider a triple of matrices (A G Ĝ) with A Fm n G F m m and Ĝ Fn n where G and Ĝ are nonsingular and either Hermitian or skew-hermitian (in the complex case) or symmetric or skew-symmetric (in the real case) In this paper we derive canonical forms (A CF G CF ĜCF) under the transformation (A CF G CF ĜCF) := (X AY X GX Y ĜY ) (11) with nonsingular matrices X F m m and Y F n n (Here A denotes the conjugate transpose of a matrix A if F = C or the transpose if F = R) School of Mathematics University of Birmingham Edgbaston Birmingham B15 TT United Kingdom mehl@mathsbhamacuk Technische Universität Berlin Institut für Mathematik MA 4-5 Straße des 17 Juni Berlin Germany mehrmann@mathtu-berlinde Department of Mathematics University of Kansas Lawrence KS 6645 USA xu@mathkuedu Partially supported by the University of Kansas General Research Fund allocation # Part of the work was done while this author was visiting TU Berlin whose hospitality is gratefully acknowledged Partially supported by the Deutsche Forschungsgemeinschaft through the DFG Research Center Matheon Mathematics for key technologies in Berlin 1

2 The canonical form for the complex case is already known and has appeared in 1 although uniqueness of the canonical form had not been considered there The real case however has only been investigated in 5 so far for the special case that G = I m I m and Ĝ = I n and in 6 where a numerical method was derived for this case In this paper based on stair-case-like decompositions we give a new and independent proof of the canonical form in the complex case These stair-case decompositions have analogues in the real case which allow a generalization of the results for the complex case to cover the real case as well The difficulties encountered in the treatment of the real case in full generality stem from the fact that one has to distinguish the cases that G and Ĝ are either both symmetric or both skew-symmetric or that one is symmetric and the other skew-symmetric In the complex case in contrast there is no need to distinguish between Hermitian and skew-hermitian matrices G and Ĝ because multiplication with the imaginary unit ı easily converts an Hermitian matrix into a skew-hermitian matrix and vice versa A corresponding transformation can be performed on the canonical form so that all cases are covered by presenting the canonical form for the case that G and Ĝ are both Hermitian A third case besides the real case and the complex case with Hermitian and skew-hermitian G and Ĝ is obtained if one assumes that G and Ĝ are complex symmetric or complex skewsymmetric and if one replaces the conjugate transpose in (11) by the transpose This case has been investigated in 18 So together with this paper the complete set of canonical forms for real and complex matrix triples of the form (11) is available The study of the described canonical forms is motivated by the goal of unifying the solution procedures for eigenvalue problems associated with structured matrices from Lie and Jordan algebras related to indefinite inner products Consider for example a signature matrix Iπ1 Σ π1 ν 1 = π I 1 + ν 1 = m ν1 A matrix H C m m is called Σ π1 ν 1 -Hermitian if (Σ π1 ν 1 H) = Σ π1 ν 1 H ie if Σ π1 ν 1 H is Hermitian The matrix Σ π1 ν 1 H then possesses a factorization Σ π1 ν 1 H = AΣ π ν A where Σ π ν is another signature matrix and A R m n with n = π + ν This means that H has a factorization H = Σ π1 ν 1 AΣ π ν A If for the triple (A Σ π1 ν 1 Σ π ν ) we can determine a suitable canonical form (A CF G CF ĜCF) = (X AY X Σ π1 ν 1 X Y Σ π ν Y ) then this will allow us to determine the eigenstructure of H because X 1 HX = (X Σ π1 ν 1 X) 1 (X AY )(Y Σ π ν Y ) 1 (Y A X) = G 1 CFA CF Ĝ 1 CFA CF Simultaneously the eigenstructure for the Σ π ν -Hermitian matrix Ĥ = Σ π ν A Σ π1 ν 1 A is obtained because Y 1 Σ π ν A Σ π1 ν 1 AY = Ĝ 1 CFA CFG 1 CFA

3 In general the canonical form (11) of the matrix triple (A G Ĝ) will allow us to simultaneously determine the eigenstructures of the two structured matrices H = G 1 AĜ 1 A Ĥ = Ĝ 1 A G 1 A (1) Structured matrices with such product representations cover all the structured matrices from the Lie and Jordan algebras (see 3) associated with the sesquilinear forms < x y > G = x Gy < x y >Ĝ= x Ĝy (13) Furthermore the form (11) can be interpreted as a generalization of the singular value decomposition 8 of a matrix A C m n ie the decomposition A CF := U AV = σ 1 σr σ 1 σ r > with unitary matrices U V Indeed the SVD can be considered as a canonical form for the matrix triple (A I m I n ) under the transformation (A I m I n ) (A CF I m I n ) = (X AY X I m X Y I n Y ) (14) Here the equation for the first components of the two matrix triples in (14) is the actual singular value decomposition while the equations for the second and third components just force the transformation matrices to be unitary The canonical form then displays the eigenstructure of the I n -selfadjoint matrix A A and the I m -selfadjoint matrix AA because the nonzero singular values σ 1 σ r are just the square roots of the nonzero eigenvalues of A A and AA When generalizing the concept of the singular value decomposition to analogous factorizations for linear maps L : C n C m where the spaces C n and C m are equipped with indefinite inner products given by invertible Hermitian matrices G C m m and Ĝ Cn n one may consider to apply a transformation A X AY to a matrix representation of L where X and Y are matrices that are unitary with respect to the sesquilinear forms (13) ie where X GX = G Y ĜY = Ĝ However if one allows general changes of bases in the spaces C n and C m ie changes that affect the indefinite inner products as well then this corresponds exactly to the transformation as in (11) and the canonical forms will appear to be less complicated Generalizations of the singular value decomposition in the sense of this paper have been studied earlier in the literature probably starting with 9 1 The generalized singular value decomposition defined there corresponds to (11) for the case that for all three matrices A CF H CF := G 1 CFA CF Ĝ 1 CFA CF and ĤCF := Ĝ 1 CFA CFG 1 CFA CF a diagonal representation can be chosen The general complex case (allowing also non-diagonal represenations) was then discussed in 1 In 14 the canonical forms of the matrices X X and XX were investigated where X = H 1 XH denotes the adjoint of a matrix X C n n with respect to the indefinite inner product induced by the nonsingular Hermitian matrix H C n n This question is motivated from the theory of polar decompositions in indefinite inner product spaces It is 3

4 said that a matrix X C n n allows an H-polar decomposition if there exists an H-selfadjoint matrix B ie a matrix satisfying B H = HB and an H-unitary matrix U ie a matrix satisfying U HU = H such that X = UB It was shown in 19 that X allows an H-polar decomposition if and only if the two matrices X X and XX have the same canonical forms as H-selfadjoint matrices Setting A = X G = H 1 and Ĝ = H we find that X X = Ĝ 1 A G 1 A = Ĥ and XX = AĜ 1 A G 1 = GHG 1 and thus the canonical forms of X X and XX can be read off from the canonical form for the matrix triple (A G Ĝ) = (X H 1 H) Consequently many of the results from 14 can be recovered from the results in this paper Recently the relation of the spectra of X X and XX has been investigated in terms of infinite dimensional indefinite inner product spaces (also known as Krein spaces) in A canonical form closely related to the form obtained under the transformation (11) has been developed in 1 where transformations of the form (B C) (X 1 BY Y 1 CX) B C m n C C n m have been considered Then a canonical form is constructed that reveals the Jordan structures of the products BC and CB In our framework this corresponds to a canonical form of the pair of matrices (G 1 A Ĝ 1 A ) rather than for the triple (A G Ĝ) When focussing on matrix triples our approach is more general because the canonical form for the pair (G 1 A Ĝ 1 A ) can be easily read off from the canonical form for (A G Ĝ) but not vice versa The approach in 1 on the other hand focusses on different aspects and allows to consider pairs (B C) where the ranks of B and C are distinct This case is not covered by the canonical forms obtained in this paper or in 18 as the considered pairs of matrices always have the same rank The paper is organized as follows In Section we review the definitions of matrices having structures with respect to indefinite inner products and provide some auxiliary results In Section 3 we present preliminary factorizations that are essential tools for the derivation of the canonical forms In Section 4 we then derive the canonical forms for the complex case and for the real case when G and Ĝ are both symmetric In Section 5 we study the case that one of G Ĝ is real symmetric and the other is real skew-symmetric In Section 6 we present the canonical forms for the case that both G Ĝ are real skew-symmetric Throughout the paper we use F to denote the field of real or complex matrices ie F = R or F = C R (R + ) is the set of real negative (positive) numbers and C (C + ) is the open left (right) half complex plane The n n identity and n n zero matrices are denoted by I n and O n respectively The m n zero matrix is denoted by O m n and e j is the jth column of the identity matrix or equivalently the jth standard basis vector of F n Moreover we introduce I π Σ πνδ = Iπ In I ν Σ πν = Σ πν = J I n = O ν I n δ The transpose and conjugate transpose of a matrix A are denoted by A T and A respectively We use A 1 A k to denote a block diagonal matrix with diagonal blocks A 1 A k If A = a ij F n m and B F l k then A B = a ij B F nl mk denotes the Kronecker 4

5 product of A and B For a real symmetric or complex Hermitian matrix A we call (π ν δ) the Sylvester inertia index with π ν δ being the number of positive negative and zero eigenvalues of A respectively For a square matrix A σ(a) denotes the spectrum of A We use R n = 1 1 J n (λ) = λ 1 λ 1 λ to denote the n n reverse identity or the n n upper triangular Jordan block associated with the eigenvalue λ respectively and a b 1 b a 1 a b a b b a J n (a b) = I n + J b a n () I = 1 1 a b b a for blocks associated with complex conjugate eigenvalues in the real Jordan form of a real matrix Matrices structured with respect to sesquilinear forms Our general theory will cover and generalize results for the following classes of matrices Definition 1 Let G F n n be invertible and let H K F n n be such that (GH) = GH and (GK) = GK 1) If F = C and G is Hermitian or skew-hermitian then H is called G-Hermitian and K is called G-skew-Hermitian ) If F = R and G is symmetric then H is called G-symmetric and K is called G-skewsymmetric 3) If F = R and G is skew-symmetric then H is called G-Hamiltonian and K is called G-skew-Hamiltonian G-Hermitian and G-symmetric matrices are often called G-selfadjoint matrices as they are selfadjoint with respect to the indefinite inner product induced by G In this paper we prefer the notions G-Hermitian and G-symmetric in order to clearly distinguish between the real and the complex case Observe that transformations of the form (H G) (P 1 HP P GP) P F n n invertible preserve the structure of H with respect to G ie if for example H is G-Hermitian then P 1 HP is P GP-Hermitian as well Clearly each complex Hermitian or real symmetric 5

6 invertible matrix G is congruent to Σ πν for some π ν and each real skew-symmetric invertible matrix G is congruent to J n for some n Thus we may always restrict ourselves to the case that either G = Σ πν or G = J n In the latter case we refer to J n -Hamiltonian or J n -skew- Hamiltonian matrices simply as Hamiltonian or skew-hamiltonian matrices respectively G-(skew-)Hermitian G-(skew-)symmetric and G-(skew-)Hamiltonian matrices have been intensively studied in the literature In particular canonical forms for such matrices have been derived in many places We review these well-known canonical forms in the following Theorem (Canonical form for G-Hermitian matrices ) Let G C n n be Hermitian and invertible and let H C n n be G-Hermitian Then there exists an invertible matrix X C n n such that X 1 HX = H c H r X GX = G c G r where H c = H c1 H cmc G c = G c1 G cmc H r = H r1 H rmr G r = G r1 G rmr and where the diagonal blocks have the following forms: 1) blocks associated with pairs (λ j λ j ) of nonreal eigenvalues of H: Jξj (λ H cj = j ) Rξj G J ξj ( λ j ) cj = R ξj = R ξj where Imλ j > and ξ j N for j = 1 m c ; ) blocks associated with real eigenvalues: H rj = J ηj (α j ) G rj = s j R ηj where α j R s j { 1 1} and η j N for j = 1 m r H has the (not necessarily pairwise distinct) non-real eigenvalues λ 1 λ mc λ 1 λ mc and (not necessarily pairwise distinct) real eigenvalues α 1 α mr Remark 3 Besides the eigenvalues the signs s 1 s mr associated with the real eigenvalues are additional invariants of G-Hermitian matrices The collection of these sign is called the sign characteristic of H sometimes also called Krein signature 13 For details on the sign characteristics we refer to 7 and the references therein The real version of Theorem is as follows: Theorem 4 (Canonical form for real G-symmetric matrices ) Let G R n n be symmetric and invertible and let H R n n be G-symmetric Then there exists an invertible matrix X R n n such that where X 1 HX = H c H r X T GX = G c G r H c = H c1 H cmc G c = G c1 G cmc H r = H r1 H rmr G r = G r1 G rmr and where the diagonal blocks have the following forms: 6

7 1) blocks associated with pairs (λ j λ j ) of nonreal eigenvalues of H: H cj = J ξj (a j b j ) G cj = R ξj where b j = Im λ j > a j = Reλ j and ξ j N for j = 1 m c ; ) blocks associated with real eigenvalues: H rj = J ηj (α j ) G rj = s j R ηj where α j R s j { 1 1} and η j N for j = 1 m r H has the (not necessarily pairwise distinct) non-real eigenvalues λ 1 λ mc λ 1 λ mc and (not necessarily pairwise distinct) real eigenvalues α 1 α mr The corresponding canonical form for G-skew-Hermitian matrices immediately follows from Theorem because a matrix K is G-skew-Hermitian if and only if H = ık is G- Hermitian In the real case however the trick of multiplying by the imaginary unit ı is not an option and a canonical form has to be derived separately We also need additional notation We denote Ξ n = ( 1) ( 1) n 1 Γ n = Ξ n R n = ( 1) ( 1) n 1 Theorem 5 (Canonical form for G-skew-symmetric matrices ) Let G R n n be symmetric and invertible and let K R n n be G-skew symmetric Then there exists an invertible matrix X R n n such that where X 1 KX = K c K r K ı K z X T GX = G c G r G ı G z K c = K c1 K cmc G c = G c1 G cmc K r = K r1 K rmr G r = G r1 G rmr K ı = K ı1 K ımı G ı = G ı1 G ımı K z = K z1 K zmo +m e G z = G z1 G zmo +m e and where the diagonal blocks have the following forms: 1) blocks associated with quadruples (λ j λ j λ j λ j ) of nonreal non purely imaginary eigenvalues of K: Jξj (a K cj = j b j ) R G J ξj (a j b j ) cj = R 4ξj = ξj R ξj where a j = Reλ j > b j = Im λ j > and ξ j N for j = 1 m c ; ) blocks associated with pairs (α j α j ) of real nonzero eigenvalues of K: Jηj (α K rj = j ) Rηj G J ηj (α j ) rj = R ηj = R ηj where α j > and η j N for j = 1 m r ; 7

8 3) blocks associated with pairs (ıβ j ıβ j ) of purely imaginary nonzero eigenvalues of K: J K ıj = ρj (β j ) Rρj G J ρj (β j ) ıj = s j R ρj where β j > s j { 1 1} and ρ j N for j = 1 m ı ; 4) blocks associated with the eigenvalue λ = of K: K zj = J ζj () G zj = t j Γ ζj where ζ j N is odd t j { 1 1} for j = 1 m o and Jζj () Rζj K zj = G J ζj () zj = R ζj where ζ j N is even for j = m o + 1 m o + m e K has the (not necessarily pairwise distinct) eigenvalues ±λ 1 ±λ mc ±λ 1 ±λ mc ±α 1 ±α mr ±ıβ 1 ±ıβ mı and the additional eigenvalue provided that m o +m e > If G is skew-hermitian then the canonical form for G-Hermitian (G-skew-Hermitian) matrices follows directly from Theorem because ıg is Hermitian and a matrix H is G- Hermitian (G-skew-Hermitian) if and only if H is ıg-hermitian (ıg-skew-hermitian) The real case once again has to be treated separately Theorem 6 (Canonical form for G-Hamiltonian matrices ) Let G R n n be skew-symmetric and invertible and let H R n n be G-Hamiltonian Then there exists an invertible matrix X R n n such that where X 1 HX = H c H r H ı H z X T GX = G c G r G ı G z H c = H c1 H cmc G c = G c1 G cmc H r = H r1 H rmr G r = G r1 G rmr H ı = H ı1 H ımı G ı = G ı1 G ımı H z = H z1 H zmo +m e G z = G z1 G zmo +m e and where the diagonal blocks have the following forms: 1) blocks associated with quadruples (λ j λ j λ j λ j ) of nonreal non purely imaginary eigenvalues of H: Jξj (a H cj = j b j ) R G J ξj (a j b j ) cj = ξj R ξj where a j = Reλ j > b j = Im λ j > and ξ j N for j = 1 m c ; ) blocks associated with pairs (α j α j ) of real nonzero eigenvalues of H: Jηj (α H rj = j ) R G J ηj (α j ) rj = ηj R ηj where α j > and η j N for j = 1 m r ; 8

9 3) blocks associated with pairs (ıβ j ıβ j ) of purely imaginary nonzero eigenvalues of H: J H ıj = ρj (β j ) R G J ρj (β j ) ıj = s ρj j R ρj where β j > s j { 1 1} and ρ j N for j = 1 m ı ; 4) blocks associated with the eigenvalue λ = of H: Jζj () H zj = G J ζj () zj = R ζj R ζj where ζ j N is odd for j = 1 m o and H zj = Ξ ζj J ζj () G zj = t j Γ ζj where ζ j N is even and t j { 1 1} for j = m o + 1 m o + m e H has the (not necessarily pairwise distinct) eigenvalues ±λ 1 ±λ mc ±λ 1 ±λ mc ±α 1 ±α mr ±ıβ 1 ±ıβ mı and the additional eigenvalue provided that m o +m e > Theorem 7 (Canonical form for G-skew-Hamiltonian matrices ) Let G R n n be skew-symmetric and invertible and let K R n n be G-skew-Hamiltonian Then there exists an invertible matrix X R n n such that X 1 KX = K c K r X T GX = G c G r where K c = K c1 K cmc G c = G c1 G cmc K r = K r1 K rmr G r = G r1 G rmr and where the diagonal blocks have the following forms: 1) blocks associated with pairs (λ j λ j ) of nonreal eigenvalues of K: Jξj (a K cj = j b j ) R G J ξj (a j b j ) cj = ξj R ξj where a j = Reλ j R b j = Im λ j > and ξ j N for j = 1 m c ; ) blocks associated with real eigenvalues α j of K: Jηj (α K rj = j ) G J ηj (α j ) rj = R ηj R ηj where α j R and η j N for j = 1 m r K has the (not necessarily pairwise distinct) non-real eigenvalues a 1 ±ıb 1 a mc ±ıb mc and the (not necessarily pairwise distinct) real eigenvalues α 1 α mr (possibly including zero) In the following we need some results concerning the existence of structured square roots of structured matrices This question has been deeply investigated in the literature mostly in the context of polar decompositions and necessary and sufficient conditions for the existence of square roots have been developed see 1 4 We do not quote the results in full generality but only consider the following special cases 9

10 Theorem 8 Let G F n n be Hermitian and nonsingular and let H F n n be G- Hermitian nonsingular and such that σ(h) R = Then there exists a square root S F n n of H that satisfies σ(s) C + This square root is unique and is a real polynomial in H (ie a polynomial in H whose coefficients are real) In particular S is G-Hermitian Proof Comparing the canonical forms of G-Hermitian and G-symmetric matrices it is easily seen that any pair (G H) C n n C n n where G is a nonsingular Hermitian matrix and H is G-Hermitian can be transformed into a real pair (G r H r ) = (P GP P 1 HP) by some complex nonsingular transformation matrix P C n n (This corresponds to the well-known fact that a matrix H is G-Hermitian for some Hermitian G if and only if H is similar to a real matrix see 7) Therefore it is sufficient to consider the real case Then by the discussion in Chapter 64 in 11 we obtain that a square root S of H with σ(s) C + exists is unique and can be expressed as a polynomial (with real coefficients) in H Clearly this polynomial stays invariant under the transformation with the transformation matrix P in the complex case It is then straightforward to check that a real polynomial in H is again G-Hermitian In the case of a skew-symmetric real bilinear form we have a similar result The proof follows exactly the same line as the proof of the preceding theorem Theorem 9 Let G R n n be skew-symmetric and nonsingular and let K R n n be G-skew-Hamiltonian nonsingular and such that σ(k) R = Then there exists a square root S R n n of K that satisfies σ(s) C + This square root is unique and is a real polynomial in K In particular S is G-skew-Hamiltonian One might ask whether G-Hamiltonian matrices have G-Hamiltonian or G-skew-Hamiltonian square roots but this is never the case because squares of such matrices must always be G-skew-Hamiltonian On the other hand each real G-skew-Hamiltonian matrix K will have a G-Hamiltonian square root 4 but this square root cannot be a polynomial in K because such a polynomial would be G-skew-Hamiltonian again After reviewing some of the basic canonical forms in this section we introduce some basis factorizations in the next section 3 Preliminary factorizations In the following sections we aim to compute canonical forms via some type of staircase algorithm A key factorization needed in the steps of this algorithm is presented in the following lemma Proposition 31 Let B F m n m n and let π ν be integers such that π + ν = m Suppose that rankb = n and that the inertia index of the Hermitian matrix B Σ πν B is (π ν δ ) Then π + ν + δ = n and there exists an invertible matrix X F m m such that X B = B π 1 + ν 1 δ n X Σ πν X = Σ π1 ν 1 Σ π ν where B F n n is nonsingular π 1 = π π δ and ν 1 = ν ν δ I δ I δ 1

11 Proof By assumption there exists a nonsingular matrix Y F n n such that Y B Σ πν BY = Σ π ν δ Let B 1 F m π be the matrix formed by the leading π columns of BY and partition it as B11 B 1 = B 11 F π π B 1 F ν π B 1 Then from B 1 Σ πνb 1 = I π we have that B 11B 11 B 1B 1 = I π (31) Since B11 B 11 and B1 B 1 are positive semidefinite it follows that rankb 11 = rank(i π + B1 B 1) = π and therefore π π by Sylvester s Law of Inertia Hence there exists a unitary matrix U 1 F π π such that U 1B 11 = where T 1 F π π is invertible Since B 11 B 11 = T 1 T 1 we obtain that (31) is equivalent to T1 I π (B 1 T 1 1 ) (B 1 T 1 1 ) = T 1 T 1 1 Then the matrix I ν (B 1 T1 1 )(B 1 T1 1 ) is positive definite because it easily follows from 5 that it has the same eigenvalues as I π (B 1 T1 1 ) (B 1 T1 1 ) with a possible exception for the eigenvalue λ = 1 Thus we have the factorization I ν (B 1 T 1 1 )(B 1 T 1 1 ) = T 1 T 1 (3) for some invertible T 1 F ν ν Let U1 X 1 = I ν T 1 (B 1 T1 1 ) T 1 I π π B 1 T 1 With (31) and (3) it is easily verified that I π π X1B 1 = π π ν Then since Σ πν = I m the last relation implies that X 1Σ πν X 1 = Σ πν X 1 Σ πν X 1 = X 1 Σ πν Σ πν Σ πν X 1 = X 1 Σ πν X 1Σ πν X 1 Σ πν X 1 and thus Σ πν X 1 Σ πν X 1 = I m or equivalently X 1 Σ πν X 1 = Σ πν Also recall that B 1 consists of the first π columns of BY Thus partitioning X 1BY = Iπ B 1 B 11

12 where B is (m π ) (n π ) we obtain from that Σ π ν δ = Y B Σ πν BY = (X1BY ) Σ πν (X1BY ) Iπ Iπ Iπ B 1 = B Σ π π ν B B 1 B 1 = B Σ π π ν B Iν = Σ ν δ = O δ Letting B F (m π ) ν be the matrix consisting of the leading ν columns of B we obtain that B Σ π π νb = I ν By a procedure analogous to the one used for B 1 above we can determine a nonsingular matrix X F (m π ) (m π ) such that XB = I ν π π ν ν ν X Σ π π νx = Σ π π ν which also shows that ν ν With X 3 = X 1 (I π X ) we then have I π X3BY = B 13 I ν B 3 X 3Σ πν X 3 = Σ πν B 33 which also implies X 3 Σ πν X3 = Σ πν and thus (X3 BY ) Σ πν (X3 BY ) = Σ π ν δ Then it easily follows that B13 B13 B 3 = and = Σ π πν ν = B13B 13 B33B 33 (33) B 33 Let P 1 be the permutation matrix that interchanges the middle two block-rows of X3 BY by pre-multiplication and set X 4 = X 3 P1 Then X 4BY = I π I ν B 13 B 33 X 4Σ πν X 4 = B 33 Σπ ν Σ π π ν ν Now both B 13 and B 33 have full column rank because otherwise by (33) it is not difficult to show that B 13 and B 33 would have a common null space But this is not possible because then X4 BY as well as B would have rank less than n contradicting the assumption Since B 13 F (π π ) δ and B 33 F (ν ν ) δ we have that π π + δ and ν ν + δ Observe that (33) implies that the positive definite factors in the polar decompositions of B 13 and B 33 coincide ie we have B 13 = Ũ3W and B 33 = Ũ4W for some Ũ3 F (π π ) δ Ũ 4 F (ν ν ) δ where U 3 U 4 have orthonormal columns and W = (B13 B 13) 1/ = (B33 B 33) 1/ F δ δ is nonsingular Extending Ũ3 and Ũ4 to unitary matrices U 3 F (π π ) (π π ) U 4 F (ν ν ) (ν ν ) we obtain that U 3 B 13 = W U 4 B 33 = 1 W

13 Setting X 5 = X 4 (I π +ν U3 U 4 ) we obtain that I π +ν X5BY W = W X 5Σ πν X 5 = Σ π ν Σ π π ν ν Let P be the permutation matrix that interchanges the 3rd and 4th block row of X5 BY by pre-multiplication and let X 6 = X 5 P Then X 6BY = I π +ν W W X 6Σ πν X 6 = Σ π ν Σ δ δ Σ π1 ν 1 where π 1 = π π δ and ν 1 = ν ν δ Then setting Iδ I Z = δ I δ I δ and X 7 = X 6 (I π +ν Z I π1 +ν 1 ) it is easily verified that Z W W = Z Iδ Σ W δ δ Z = I δ and thus we have I π +ν X7BY = W Iδ X 7Σ πν X 7 = Σ π ν I δ Σ π1 ν 1 Let P 3 be the permutation matrix that changes the order the block rows of X7 BY to the order by pre-multiplication and set X = X 7 P3 Then X BY = I π +ν W X Σ πν X = Σ π1 ν 1 I δ Σ π ν I δ The desired factorization then follows by multiplying with Y 1 from the right and setting B = (I π +ν W)Y 1 Proposition 3 Let B R m n and suppose that rank B = n rank B T J m B = n (note that the rank of a real skew-symmetric matrix is even) and let δ = n n denote the dimension of the null space of B T J m B Then there exists an invertible matrix X R m m such that X T B = B n 1 δ n X T J m X = J n1 where B C n n is nonsingular and n 1 = m n δ I δ J n I δ Proof The proof follows the same lines as in the complex case (or more precisely as in the case of a complex skew-symmetric bilinear form induced by J m ) see 18 for details 13

14 4 Canonical form for G Ĝ Hermitian In this section we investigate the matrix triple (A G Ĝ) for the case that both G Ĝ are Hermitian and nonsingular We first consider the simpler case that A is square and nonsingular Theorem 41 Let A C n n be nonsingular and let G Ĝ Cn n be Hermitian and nonsingular Then there exist nonsingular matrices X Y C n n such that X AY = A c A r X GX = G c G r Y ĜY = Ĝc Ĝr (41) and for the Ĝ-Hermitian matrix Ĥ = Ĝ 1 A G 1 A C n n and for the G-Hermitian matrix H = G 1 AĜ 1 A C n n we have that Y 1 ĤY = Ĥc Ĥr X 1 HX = H c H r (4) The diagonal blocks in these decompositions have the following forms: 1) blocks associated with pairs (µ j µ j ) of nonreal eigenvalues of Ĥ and H: Jξ1 (µ 1 ) A c = G c = Ĝ c = Ĥ c = H c = J ξ1 ( µ 1 ) Rξ1 R ξ1 Rξ1 R ξ1 J ξ1 (µ 1 ) Jξ 1 ( µ 1 ) J ξ1 (µ 1 ) Jξ 1 ( µ 1 ) Jξmc (µ m c ) J ( µ ξm c m c ) R ξm c R ξm c R ξm c R ξm c J ξm c (µ m c ) J ξ ( µ m m c c ) J ξm c (µ m c ) J ξ m c ( µ m c ) where µ j C arg µ j ( π/) and ξ j N for j = 1 m c ; ) blocks associated with real eigenvalues α j of H and Ĥ: A r = J η1 (β 1 ) J ηm r (β m r ) G r = s 1 R η1 s mr R ηm r Ĝ r = ŝ 1 R η1 ŝ mr R ηm r Ĥ r = s 1 ŝ 1 J η 1 (β 1 ) s mr ŝ mr J η m r (β m r ) H r = s 1 ŝ 1 ( J η1 (β 1 ) ) smr ŝ mr ( J ηm r (β m r ) ) where β j > s j ŝ j {+1 1} and η j N for j = 1 m r Thus α j = β j > if s j = ŝ j and α j = β j < if s j ŝ j Moreover the form (41) is unique up to the simultaneous permutation of blocks in the right hand side of (41) Proof The proof will be performed in two steps Step 1) We first show that we may assume without loss of generality that Ĥ either has only one pair of conjugate complex nonreal eigenvalues (λ λ) or only one real eigenvalue α 14

15 Indeed in view of Theorem there exists a nonsingular matrix Y C n n such that Y 1 ĤY = Ĥ 1 Ĥ Y ĜY = Ĝ 1 Ĝ where Ĥ1 Ĝ1 C p p Ĥ Ĝ C (n p) (n p) σ(ĥ1) σ(ĥ) = and Ĥ1 either has only one eigenvalue that is real or only two eigenvalues that are conjugate complex Using G 1 AĤ = HG 1 A and the fact that G 1 A is nonsingular we find that H and Ĥ are similar Thus there exists a nonsingular matrix X C m m such that X 1 Ĥ1 HX = Ĥ1 Ĥ = Ĥ X GX = G1 G 1 G 1 G Here G has been partitioned conformably with H By assumption σ(ĥ1) = σ(ĥ 1 ) and thus σ(ĥ 1 ) σ(ĥ) = Then using that H is G-Hermitian ie Ĥ 1 G1 G 1 Ĥ G = X H GX = X G1 G GHX = 1 Ĥ1 1 G G 1 G Ĥ we obtain G 1 = because the Sylvester equation Ĥ 1 G 1 G 1 Ĥ = only has the trivial solution given that the spectra of the coefficient matrices Ĥ 1 and Ĥ do not intersect Next we will show that X AY decomposes in the same way as H Ĥ G and Ĝ To this end we partition (X AY ) 1 A11 A = 1 A 1 A conformably with G and Ĝ Then implies ĤA 1 = Ĝ 1 A G 1 = (G 1 AĜ 1 ) = (HA ) = A 1 H Ĥ1 A11 A 1 A11 A = 1 Ĥ 1 Ĥ A 1 A A 1 A Ĥ Using once again the fact that a Sylvester equation only has the trivial solution if the spectra of the coefficient matrices do not intersect we finally obtain that (X AY ) 1 A11 = A and thus X AY is block diagonal as well Repeating this argument several times we see that it remains to study triples (A G Ĝ) for which Ĥ has the restricted spectrum as initially stated Step ) By Step 1) we may assume without loss of generality that Ĥ either has only one pair of conjugate complex nonreal eigenvalues (λ λ) or only one real eigenvalue α We discuss these two cases separately 15

16 Case 1: σ(ĥ) = {λ λ} for some λ C Im λ > By Theorem 8 Ĥ has a unique Ĝ-Hermitian square root S Cn n satisfying σ(s) C + Then by Theorem there exists a nonsingular matrix Ỹ Cn n such that S CF := Ỹ 1 SỸ = Jξ1 (µ) Jξm (µ) J ξ1 ( µ) G CF := Ỹ ĜỸ = Rξ1 R ξ1 H CF := Ỹ 1 ĤỸ = J ξ1 (µ) J ξ 1 ( µ) J ξm ( µ) R ξm R ξm J ξm (µ) Jξ m ( µ) where µ = λ C arg µ ( π ) and ξ j N for j = 1 m Here the third identity immediately follows from Ĥ = S Since H and Ĥ are similar and since Ĥ has only a pair of conjugate complex nonreal eigenvalues we obtain from Theorem that the canonical forms of the pairs (H G) and (Ĥ Ĝ) coincide In particular this implies the existence of a nonsingular matrix X C n n such that H CF = X 1 H X = G CF = X G X = J ξ1 (µ) Jξ 1 ( µ) Rξ1 R ξ1 J Finally setting X = G 1 X and Y = A 1 G XS CF we obtain ξm (µ) J ξ m ( µ) R ξm R ξm X AY = X 1 G 1 AA 1 G XS CF = S CF X GX = X 1 G 1 GG 1 X = ( X G X) 1 = G 1 CF = G CF Y ĜY = S X CF GA ĜA 1 G XS CF = S CF X G X X 1 H 1 XSCF = S CFG CF (H CF ) 1 S CF = G CF S CF (H CF ) 1 S CF = G CF as desired where we have used that S CF is G CF -Hermitian and that S CF = H CF It is now easy to check that X 1 HX and Y 1 ĤY have the claimed forms Case : σ(ĥ) = {α} for some α R \ {} Observe that sign(α)ĥ has the only positive eigenvalue α Thus we can apply Theorems 8 and which yield the existence of a square root S C n n of sign(α)ĥ and a nonsingular matrix Ỹ Cn n such that S CF := Ỹ 1 SỸ = J η 1 (β) J ηm (β) Ỹ ĜỸ = ŝ 1R η1 ŝ m R ηm H CF := Ỹ 1 ĤỸ = sign(α)j η 1 (β) sign(α)jη m (β) where β = α η j N and ŝ j {+1 1} for j = 1 m Again using that H and Ĥ are similar we obtain from Theorem the existence of a nonsingular matrix X C n n such that H CF = X 1 H X = sign(α)j η 1 (β) sign(α)j η m (β) G CF := X G X = s 1 R η1 s m R ηm 16

17 for some s 1 s m {+1 1} Setting X = G 1 X and Y = A 1 G XS CF we obtain as in Case 1 that X AY = S CF X GX = G CF and Y ĜY = G CF S CF (H CF ) 1 S CF = sign(α)g CF We mention in passing that it is possible to link the sign characteristic (ŝ 1 ŝ m ) to the sign characteristic (s 1 s m ) but we refrain from doing so because the explicit knowledge of the parameters ŝ 1 ŝ m is irrelevant for the development of the canonical form for the triple (A G Ĝ) It is now straightforward to check that X 1 HX and Y 1 ĤY have the claimed forms Concerning uniqueness we note that the form (41) is uniquely determined by the canonical form of Ĥ as a Ĝ-Hermitian matrix and the restrictions arg µ j ( π/) and β > In the general situation that A is non square we have the following result Theorem 4 Let A C m n and let G C m m and Ĝ Cn n be Hermitian and nonsingular Then there exist nonsingular matrices X C m m and Y C n n such that X AY = A nz A z1 A z A z3 A z4 X GX = G nz G z1 G z G z3 G z4 (43) Y ĜY = Ĝnz Ĝz1 Ĝz Ĝz3 Ĝz4 Moreover for the Ĝ-Hermitian matrix Ĥ = Ĝ 1 A G 1 A C n n and for the G-symmetric matrix H = G 1 AĜ 1 A C m m we have that Y 1 ĤY = Ĥnz Ĥz1 Ĥz Ĥz3 Ĥz4 X 1 HX = H nz H z1 H z H z3 H z4 The diagonal blocks in these decompositions have the following forms: ) blocks associated with nonzero eigenvalues of Ĥ and H: A nz G nz Ĝnz have the forms as in (41) and Ĥnz H nz have the forms as in (4); 1) one block corresponding to n Jordan blocks of size 1 1 of Ĥ and m Jordan blocks of size 1 1 of H associated with the eigenvalue zero: A z1 = O m n G z1 = Σ π ν Ĝ z1 = Σˆπ ˆν Ĥ z1 = O n H z1 = O m where m n π ν ˆπ ˆν N {} and π + ν = m ˆπ + ˆν = n ; ) blocks corresponding to a pair of j j Jordan blocks of Ĥ and H associated with the eigenvalue zero: A z = G z = Ĝ z = Ĥ z = H z = γ 1 J () γ 1 R γ 1 R γ 1 γ J 4 () γ R 4 γ R 4 γ l γ l J l () R l γ l R l J γ () J4 γ () l Jl () γ 1 γ γ l J ()T J4 ()T Jl ()T where γ 1 γ l N {}; thus Ĥ z and H z both have each γ j Jordan blocks of size j j where exactly γ j blocks have sign +1 and γ j blocks have sign 1 for j = 1 l; 17

18 3) blocks corresponding to a j j Jordan blocks of Ĥ and a (j + 1) (j + 1) Jordan block of H associated with the eigenvalue zero: A z3 = m 1 I1 m m I l 1 Il l (l 1) m 1 G z3 = s (i) 1 R m s (i) m R l 1 3 s (i) l 1 R l Ĝ z3 = Ĥ z3 = m 1 H z3 = m 1 m 1 ŝ (i) 1 R 1 s (i) 1 ŝ(i) s (i) 1 ŝ(i) 1 J 1() m 1 J () T m m ŝ (i) R s (i) ŝ(i) J () s (i) ŝ(i) J 3() T m l 1 m l 1 m l 1 ŝ (i) l 1 R l 1 s (i) l 1ŝ(i) J l 1() s (i) l 1ŝ(i) J l() T where m 1 m l 1 N {} and for j = 1 l 1 we have that s (i) j ŝ (i) j {+1 1} if j is odd and s (i) j m j Jordan blocks of size j j with signs ŝ (i) j {+1 1} and ŝ (i) j = 1 and = 1 if j is even; thus Ĥz3 has if j is odd and signs s (i) j H z3 has m j Jordan blocks of size (j + 1) (j + 1) with signs ŝ (i) j s (i) j if j is even for i = 1 m j and j = 1 l 1; if j is even and if j is odd and signs 4) blocks corresponding to a (j + 1) (j + 1) Jordan blocks of Ĥ and a j j Jordan block of H associated with the eigenvalue zero: A z4 = n 1 G z4 = Ĝ z4 = n 1 Ĥ z4 = n 1 H z4 = n 1 I1 1 n n 1 s (i) 1 R 1 ŝ (i) 1 R s (i) 1 ŝ(i) s (i) 1 ŝ(i) n 1 J () n 1 J 1() T n n I 3 l 1 n s (i) R ŝ (i) R 3 s (i) ŝ(i) J 3() s (i) ŝ(i) J () T n l 1 Il 1 n l 1 n l 1 n l 1 (l 1) l s (i) l 1 R l 1 ŝ (i) l 1 R l s (i) l 1ŝ(i) J l() s (i) l 1ŝ(i) J l 1() T where n 1 n l 1 N {} and for j = 1 l 1 we have that s (i) j ŝ (i) j {+1 1} if j is even and s (i) j {+1 1} and ŝ (i) j n j Jordan blocks of size (j + 1) (j + 1) with signs s (i) j even and H z4 has n j Jordan blocks of size j j with signs s (i) j ŝ (i) j if j is even for i = 1 m j and j = 1 l 1; = 1 and = 1 if j is odd; thus Ĥ z4 has if j is odd and signs ŝ (i) j if j is if j is odd and signs For the eigenvalue zero the matrices Ĥ and H have γ j+m j +n j 1 respectively γ j +m j 1 +n j Jordan blocks of size j j for j = 1 l where m l = n l = and where l is the maximum of the index of Ĥ and the index of H (Here index refers to the maximal size of a Jordan block associated with the eigenvalue zero) 18

19 Furthermore the form (43) is unique up to simultaneous block permutation of the blocks in the block diagonal of the right hand side of (43) Proof Due to its very technical nature the proof is omitted here and presented in the Appendix Since the canonical form of Theorem 4 is quite complicated we present some examples to illustrate this form Example 43 Let A G Ĝ be given by A = G = Ĝ = Then the canonical form consists of one block of type ) with j = one block of type 3) with j = 1 and sign ŝ (1) 1 = 1 and two blocks of type 4) one with j = 1 and sign s (1) 1 = +1 and one with j = and sign ŝ (1) = 1 Observe that the signs only occur in the blocks of G or Ĝ respectively that have odd size The signs attached to the corresponding even sized blocks are always +1 Thus for example the signs corresponding to blocks of type 4) will always be found in G if j is odd and they can be read off Ĝ if j is even Example 44 It is important to note that rectangular matrices with a total number of zero rows or columns are allowed in the canonical form For example consider the two nonequivalent triples A 1 = 1 G 1 = 1 1 Ĝ 1 = and A = 1 G = 1 Ĝ = The first example is just one block of type 4) with sign s (1) 1 = 1 Indeed forming the products Ĥ 1 = Ĝ 1 1 A 1G 1 1 A = 1 H 1 = G 1 1 AĜ 1 1 A 1 = as predicted Ĥ 1 has only one Jordan block of size associated with the eigenvalue λ = and the sign s = 1 while H 1 has one Jordan block of size 1 associated with λ = and the sign s = 1 The situation is different in the second case Here we obtain Ĥ = Ĝ 1 A G 1 A = 1 H = G 1 AĜ 1 A = 1 ie Ĥ has two Jordan blocks of size 1 one associated with λ = and sign s 1 = 1 and a second one associated with λ = 1 and sign s = 1 while H has one Jordan block of size 19

20 1 associated with λ = 1 and sign s = 1 Here the triple (A G Ĝ) is in canonical form consisting of one block of type 1) and size 1 and of one block of type ): A = 1 G = 1 Ĝ = 1 1 We have the following real versions of Theorem 41 and Theorem 4 Theorem 45 Let A R n n be nonsingular and let G Ĝ Rn n be symmetric and nonsingular Then there exist nonsingular matrices X Y R n n such that X T AY = A c A r X T GX = G c G r Y T ĜY = Ĝc Ĝr (44) Moreover for the Ĝ-symmetric matrix Ĥ = Ĝ 1 A T G 1 A and for the G-symmetric matrix H = G 1 AĜ 1 A T we have that Y 1 ĤY = Ĥc Ĥr X 1 HX = H c H r (45) The diagonal blocks in these decompositions have the following forms: 1) blocks associated with pairs (µ j µ j ) of nonreal eigenvalues of Ĥ and H: A c = J ξ1 (a 1 b 1 ) J ξm c (a m c b mc ) G c = R ξ1 R ξm c Ĝ c = R ξ1 R ξm c Ĥ c = J ξ 1 (a 1 b 1 ) J ξ m c (a m c b mc ) H c = J ξ 1 (a 1 b 1 ) T J ξ m c (a m c b mc ) T where a j b j > µ j = a j + ıb j and ξ j N for j = 1 m c ; ) blocks associated with real eigenvalues α j of H and Ĥ: A r = J η1 (β 1 ) J ηm r (β m r ) G r = s 1 R η1 s mr R ηm r Ĝ r = ŝ 1 R η1 ŝ mr R ηm r Ĥ r = s 1 ŝ 1 J η 1 (β 1 ) s mr ŝ mr J η m r (β m r ) H r = s 1 ŝ 1 ( J η1 (β 1 ) ) T smr ŝ mr ( J ηm r (β m r ) ) T where β j > s j ŝ j {+1 1} and η j N for j = 1 m r Thus α j = β j > if s j = ŝ j and α j = β j < if s j ŝ j Furthermore the form (44) is unique up to the simultaneous permutation of blocks in the right hand side of (44) Proof The proof follows exactly the same lines as the proof of Theorem 41 (The key point here is that the square roots that are constructed analogously to the proof of Theorem 41 are real see Theorem 8)

21 Theorem 46 Let A R m n and let G R m m and Ĝ Rn n be symmetric and nonsingular Then there exist nonsingular matrices X R m m and R n n such that X T AY = A nz A z1 A z A z3 A z4 X T GX = G nz G z1 G z G z3 G z4 (46) Y T ĜY = Ĝnz Ĝz1 Ĝz Ĝz3 Ĝz4 Moreover for the Ĝ-symmetric matrix Ĥ = Ĝ 1 A T G 1 A C n n and for the G-symmetric matrix H = G 1 AĜ 1 A T C m m we have that Y 1 ĤY = Ĥnz Ĥz1 Ĥz Ĥz3 Ĥz4 X 1 HX = H nz H z1 H z H z3 H z4 Here the blocks A nz G nz Ĝnz Ĥnz and H nz have the forms as in (44) and (45) while A zk G zk Ĝzk Ĥ zk and H zk have the forms as in Theorem 4 for k = 1 4 Moreover the form (46) is unique up to the simultaneous permutation of blocks in the right hand side of (46) Proof The proof follows exactly the same lines as the proof of Theorem 4 Indeed the proof (as presented in the Appendix) makes use of the decompositions as presented in Proposition 31 which has a real version as well In the particular case that one of the Hermitian matrices is positive definite (say G) we obtain the following special case of Theorem 4 and Theorem 46 that can be interpreted as a generalization of both the Schur form for a Hermitian matrix as well as a generalization of the standard singular value decomposition Corollary 47 Let A F m n let G F m m be Hermitian and positive definite and let Ĝ F n n be Hermitian and nonsingular Then there exist nonsingular matrices X F m m and Y F n n such that X AY = β 1 O m n O n1 I n1 X GX = Y ĜY = β mr 1 I m I n1 = I m 1 ŝ 1 ŝ mr Σˆπ ˆν In1 I n1 where n = ˆπ + ˆν and β j > ŝ j { 1 1} for j = 1 m r Moreover ŝ 1 β 1 Y 1 Ĝ 1 A G 1 AY = In1 O n ŝ mr βm r X 1 G 1 AĜ 1 A X = ŝ 1 β 1 O m +n 1 ŝ mr β m r 1

22 Proof Because G is positive definite due to the inertia index relation in the canonical form of Theorem 4 G c as well as A c Ĝc must be void Furthermore η 1 = = η mr = 1 and s 1 = = s mr = 1 Concerning the blocks A zk G zk Ĝ zk the blocks for k = 1 may exist but G z1 has to be the identity matrix I m ; the blocks for k = and k = 3 must be void and the blocks for k = 4 may only exist when j = 1 In this case G z4 has to be I n1 and applying an appropriate permutation we can achieve the forms A z4 = O n1 I n1 Gz4 = I n1 Ĝ z4 = The proof for the real case is analogous In1 I n1 Remark 48 It should be noted that when G = I m then X is unitary and Corollary 47 gives the Schur form of the Hermitian matrix product AĜ 1 A Also it simultaneously displays the Jordan form of Ĝ 1 A A One should observe here the difference in the eigenstructures of AĜ 1 A and Ĝ 1 A A corresponding to the eigenvalue λ = (Indeed it is well known that two matrix products AB and BA have identical nonzero eigenvalues including identical algebraic geometric and partial multiplicities but the Jordan structure for the eigenvalue λ = may be different for both matrices see 5) If G = I m and Ĝ = I n then also Y is unitary and Corollary 47 becomes the standard singular value decomposition 5 Canonical form for G symmetric and Ĝ skew-symmetric In this section we determine the canonical form for the case that G is symmetric and Ĝ is skew-symmetric We only consider the real case because the corresponding complex case (ie G being Hermitian and Ĝ being skew-hermitian) can be easily derived from the canonical form in Theorem 4 by simply multiplying Ĝ with ı For the real case the situation is different and the canonical form becomes more complicated Again we start with the result for the case that A is square and nonsingular Theorem 51 Let A R n n be nonsingular let G R n n be symmetric and nonsingular and let Ĝ Rn n be skew-symmetric and nonsingular Then there exist nonsingular matrices X Y R n n such that X T AY = A c A r A ı X T GX = G c G r G ı Y T ĜY = Ĝc Ĝr Ĝı (51) Moreover for the Ĝ-Hamiltonian matrix Ĥ = Ĝ 1 A T G 1 A and for the G-skew-symmetric matrix H = G 1 AĜ 1 A T we have that Y 1 ĤY = Ĥ c Ĥ r Ĥ ı X 1 HX = H c H r H ı (5) The diagonal blocks in these decompositions have the following forms: 1) blocks associated with quadruples ((a j ± ıb j ) (a j ± ıb j ) ) of nonreal and non-purely

23 imaginary eigenvalues of Ĥ and H: Jξ1 (a A c = 1 b 1 ) Jξmc (a m c b mc ) J ξ1 (a 1 b 1 ) J (a ξm c m c b mc ) R G c = ξ1 R ξm c R ξ1 R ξm c R Ĝ c = ξ1 R ξm c R ξ1 R ξm c Jξ1 (a Ĥ c = 1 b 1 ) Jξmc J ξ1 (a 1 b 1 ) (a m c b mc ) J (a ξm c m c b mc ) Jξ1 (a H c = 1 b 1 ) T Jξmc J ξ1 (a 1 b 1 ) (a m c b mc ) T J (a ξm c m c b mc ) where a j > b j > and ξ j N for j = 1 m c ; ) blocks associated with pairs of real eigenvalues (αj α j ) of H and Ĥ: Jη1 (α A r = 1 ) Jηmr (α m r ) J η1 (α 1 ) J (α ηm r m r ) Rη1 R G r = ηm r R η1 R ηm r R Ĝ r = η1 R ηm r R η1 R ηm r Jη1 (α Ĥ r = 1 ) Jηmr J η1 (α 1 ) (α m r ) J (α ηm r m r ) Jη1 (α H r = 1 ) T Jηmr J η1 (α 1 ) (α m r ) T J (α ηm r m r ) where α j > and η j N for j = 1 m r ; 3) blocks associated with pairs of purely imaginary eigenvalues (ıβj ıβ j ) of H and Ĥ: Jρ1 (β A ı = 1 ) Jρmı (β m ı ) J ρ1 (β 1 ) J (β ρm ı m ı ) Rρ1 G ı = s 1 R ρ1 R Ĝ ı = s ρ1 1 R ρ1 J Ĥ ı = ρ1 (β 1 ) J ρ1 (β 1 ) J H ı = ρ1 (β 1 ) J ρ1 (β 1 ) Rρm ı s mı R ρm ı R s ρm ı mı R ρm ı T J (β ρm ı m ı ) T J (β ρm ı m ı ) J (β ρm ı m ı ) J (β ρm ı m ı ) where β j > s j {+1 1} and ρ j N for j = 1 m ı ; Furthermore the form (51) is unique up to the simultaneous permutation of blocks in the right hand side of (51) 3

24 Proof Analogous to the proof of Theorem 41 it can be shown that without loss of generality we may assume that σ(ĥ) = {λ λ λ λ} where λ C\{} We then distinguish the three different cases λ > λ < and λ R The proof then proceeds similar to the proof of Theorem 41 but instead of constructing a square root of Ĥ a square root of a related Ĝ-skew-Hamiltonian matrix S will be considered The proof for the cases λ > and λ R follows exactly the same lines as the proof of Theorem 51 in 18 and will not be reproduced here The proof for the remaining case differs slightly and will therefore be presented here in full detail Thus assume without loss of generality that σ(ĥ) = {ıλ ıλ} where λ > By Theorem 6 there exists a nonsingular matrix W R n n such that W 1 J ĤW = ρ1 (λ) J ρm (λ) J ρ1 (λ) J ρm (λ) W T R ĜW = ŝ ρ1 R 1 ŝ ρm R ρ1 m R ρm where ρ j N and ŝ j {+1 1} for j = 1 m Next we define the matrix S to be such that W 1 Jρ1 (λ) Jρm (λ) SW = J ρ1 (λ) J ρm (λ) Then S is Ĝ-skew-Hamiltonian and satisfies σ( S) R + Thus applying Theorem 9 we obtain that S has unique square root S C n n that satisfies σ(s) R + and that is a polynomial in S Consequently with S also its square root S is Ĝ-skew-Hamiltonian Let β = λ Then Theorem 7 implies the existence of a nonsingular matrix Ỹ Rn n such that S CF := Ỹ 1 SỸ = Jρ1 (β) Jρm (β) J ρ1 (β) J ρm (β) Ỹ T Rρ1 R ĜỸ = ρm R ρ1 R ρm (Note that the Jordan structure of S follows from the fact that S is a polynomial in S) Moreover using G 1 AĤ = HG 1 A and the fact that G 1 A is nonsingular we find that Ĥ and H are similar Thus by Theorem 5 we find that there is a nonsingular matrix X 1 such that X 1 1 H X J 1 = ρ1 (λ) J ρ1 (λ) X 1 TG X Rρ1 1 = s 1 R ρ1 J ρm (λ) J ρm (λ) Rρm s m R ρm where s 1 s m {+1 1} On the other hand since β = λ the matrix Jρ H CF := 1 (β) Jρ 1 (β) Jρ m (β) Jρ m (β) 1 is similar to X 1 H X 1 It is also obvious that H CF is X1 TG X 1 -skew symmetric Again by Theorem 5 there exists a nonsingular matrix X such that with setting X = X 1 X we have 4

25 that H CF := X 1 H X Jρ = 1 (β) Jρ 1 (β) G CF := X T G X Rρ1 = s 1 R ρ1 Jρ m (β) Jρ m (β) s m Rρm R ρm for some s 1 s m {+1 1} (It is actually possible to show that s j = s j for j = 1 m but we refrain from doing so as it is not necessary for the proof) Observe that S CF is G CF - symmetric and satisfies S CF (H CF ) 1 S CF = Iρ1 I ρ1 Iρm I ρm Using this identity and setting X = G 1 X T and Y = A 1 G XS CF we obtain X T AY = X 1 G 1 AA 1 G XS CF = S CF X T GX = X 1 G 1 GG 1 X T = ( X T G X) 1 = (G CF ) 1 = G CF Y T ĜY = S T CF X T GA T ĜA 1 G XS CF = S T CF X T G X X 1 H 1 XSCF = SCFG T CF (H CF ) 1 S CF = G CF S CF (H CF ) 1 S CF Rρ1 R = s 1 s ρm R ρ1 m R ρm It is now straightforward to check that Y 1 HY and X 1 ĤX have the claimed forms Concerning uniqueness we note that the form (51) is already uniquely determined by the Jordan structure the sign characteristic of H and by the restrictions on the parameters For the general non square case we have the following result Theorem 5 Let A R m n let G R m m be symmetric nonsingular and let Ĝ Rn n be skew-symmetric and nonsingular Then there exist nonsingular matrices X R m m and Y R n n such that X T AY = A nz A z1 A z A z3 A z4 A z5 A z6 X T GX = G nz G z1 G z G z3 G z4 G z5 G z6 (53) Y T ĜY = Ĝnz Ĝz1 Ĝz Ĝz3 Ĝz4 Ĝz5 Ĝz6 Moreover for the Ĝ-Hamiltonian matrix Ĥ = Ĝ 1 A T G 1 A R n n and for the G-skewsymmetric matrix H = G 1 AĜ 1 A T R m m we have that Y 1 ĤY = Ĥnz Ĥz1 Ĥz Ĥz3 Ĥz4 Ĥz5 Ĥz6 X 1 HX = H nz H z1 H z H z3 H z4 H z5 H z6 The diagonal blocks in these decompositions have the following forms: ) blocks associated with nonzero eigenvalues of Ĥ and H: A nz G nz Ĝnz have the forms as in (51) and Ĥnz H nz have the forms as in (5); 5

26 1) one block corresponding to n Jordan blocks of size 1 1 of Ĥ and m Jordan blocks of size 1 1 of H associated with the eigenvalue zero: A z1 = O m n G z1 = Σ π ν Ĝ z1 = J n Ĥ z1 = O n H z1 = O m where m n π ν N {} and m = π + ν ; ) blocks corresponding to a pair of j j Jordan blocks of H and eigenvalue zero: Ĥ associated with the A z = γ 1 J () γ 1 R γ 1 γ γ J 4 () γ l+1 γ l+1 J 4l+ () G z = R 4 R 4l+ 1 γ γ R l+1 R Ĝ z = l+1 1 R R l+1 γ 1 γ Ĥ z = ( Σ )J4 γ () l+1 ( Σ l+1l+1 )J4l+ () H z = γ 1 γ Σ 31 J4 ()T γ l+1 Σ l+l J 4l+ ()T where γ 1 γ l N {}; thus Ĥ z and H z both have each γ j Jordan blocks of size j j for j = 1 l + 1; moreover if j is odd then exactly γ j Jordan blocks of H z of size j j have sign s = +1 and exactly γ j blocks have sign s = 1 (even-sized Jordan blocks associated with zero of G-skew-symmetric matrices do not have signs) and if j is even then exactly γ j Jordan blocks of Ĥ z of size j j have sign s = +1 and exactly γ j blocks have sign s = 1 (odd-sized Jordan blocks associated with zero of Ĝ-Hamiltonian matrices do not have signs); 3) blocks corresponding to a j j Jordan block of Ĥ and a (j + 1) (j + 1) Jordan block of H associated with the eigenvalue zero: m I A z3 = m l Il 3 (l+1) l m G z3 = s () m l i R 3 s (l) i R l+1 m m R1 l Rl Ĝ z3 = R 1 R l Ĥ z3 = m ( s () i Σ 11 )J () m l ( s (l) i Σ ll )J l () H z3 = m s (l) i Σ 1 J 3 () T m l s (l) i Σ l+1l J l+1 () T where m m 4 m l N {}; thus Ĥ z3 has m j Jordan blocks of size j j with signs s (j) i and H z3 has m j Jordan blocks of size (j + 1) (j + 1) with signs s (j) i for i = 1 m j and j = 1 l; 6