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1 Electronic Transactions on Numerical Analysis Volume 2 pp September 1994 Copyright 1994 ISSN ETNA etna@mcskentedu ON THE PERIODIC QUOTIENT SINGULAR VALUE DECOMPOSITION JJ HENCH Abstract The periodic Schur decomposition has been generally seen as a tool to compute the eigenvalues of a product of matrices in a numerically sound way In a recent technical report it was shown that the periodic Schur decomposition may also be used to accurately compute the singular value decomposition (SVD) of a matrix This was accomplished by reducing a periodic pencil that is associated with the standard normal equations to eigenvalue revealing form If this technique is extended to the periodic QZ decomposition then it is possible to compute the quotient singular value decomposition (QSVD) of a matrix pair This technique may easily be extended further to a sequence of matrix pairs thus computing the periodic QSVD Key words singular value decomposition periodic Schur decomposition periodic QR algorithm periodic QZ algorithm QSVD SVD AMS subject classifications 15A18 65F05 65F15 1 Introduction In a recent technical report 8 a method was proposed by which to compute the singular value decomposition (SVD) of a matrix via the periodic Schur decomposition This method applied the periodic QR algorithm 1 6 to the matrices in the periodic linear system of equations (11) x 2 Ax 1 x 3 A T x 2 thereby computing the orthogonal matrices and the non-negative definite diagonal matrix that comprise the SVD of the matrix A While numerically not as efficient as the standard SVD algorithm it was shown that the periodic QR algorithm could compute the SVD of a matrix with comparable accuracy In fact if the standard periodic QR algorithm is modified to perform a single (real) implicit shift instead of the standard double (complex-conjugate) implicit shift the result is essentially the Golub Kahan SVD algorithm 3 applied to both A and A T doubling the normal operations count The technical report further showed that it was possible to compute the singular values of a matrix operator defined by a quotient of matrices Here we speak of the operator Ξ : X 1 X 2 X 1 R n X 2 R n defined by the equations (12) Ex 2 Fx 1 where E R n n and F R n n This method accurately extracted the singular values of Ξ by examining the eigenvalues of a related operator In this paper we develop the idea further by relating the results to the quotient singular value decomposition (QSVD) The QSVD reduces the matrices E and F in (12) to (diagonal) singular value revealing form More precisely via the Received January Accepted for publication September Communicated by P M VanDooren Institute of Information and Control Theory Academy of Sciences of the Czech Republic Pod vodárenskou věží Praha 8 (hench@utiacascz) This research was supported by the Academy of Sciences of the Czech Republic and by a Fulbright research grant from the Institute of International Education New York 138

2 JJ Hench 139 QSVD it is possible to compute a non orthogonal matrix X non negative definite diagonal matrices Φ and Θ and orthogonal matrices U and V such that X T EU Φ (13) X T FV Θ Furthermore if E is nonsingular the ordering of the diagonal elements of Φ and Θ may be chosen to reflect the ordering of the singular values in the SVD of the explicitly formed operator ˆΞ E 1 F ie θ i θ i+1 (14) φ i φ i+1 In this paper as in 7 we will show that it is possible to compute the matrices X Φ Θ U and V via the periodic QZ decomposition Further we will show how this technique may be extended to compute the periodic QSVD of a sequence of matrix pairs First however we review from 9 the periodic QZ decomposition 2 The Periodic QZ Decomposition The periodic QZ decomposition simultaneously triangularizes by orthogonal equivalences a sequence of matrices associated with the generalized periodic eigenvalue problem Consider the linear algebraic map Π k :X k X k+p where X k R n and X k+p R n and x k+p Π k x k is defined by the set of linear equations (21) E k x k+1 F k x k E k+1 x k+2 F k+1 x k+1 E k+p 1 x k+p F k+p 1 x k+p 1 It is possible to operate on the matrices E k and F k with orthogonal matrices Q k and Z k so as to triangularize the system associated with the generalized periodic eigenvalue problem (22) More precisely the following products (23) E k x k+1 F k x k E k+1 x k+2 F k+1 x k+1 E k+p 2 x k+p 1 F k+p 2 x k+p 2 λe k+p 1 x k F k+p 1 x k+p 1 Q T k E k Z k+1 Q T k F k Z k Q T k+1 E k+1z k+2 Q T k+1 F k+1z k+1 Q T k+p 2 E k+p 2Z k+p 1 Q T k+p 2 F k+p 2Z k+p 2 Q T k+p 1 E k+p 1Z k+p Q T k+p 1 F k+p 1Z k+p 1 have the properties that Q T k E kz k+1 is a quasi upper triangular matrix H k andthe remaining matrices Q T k+l E k+lz k+l+1 and Q T k+l F k+lz k+l are upper triangular matrices H k+l and T k+l respectively with Z k+p Z k This permits the periodic pencil

3 140 On the periodic quotient singular value decomposition in (22) to be expressed as the triangularized periodic pencil that is associated with the linear algebraic map Γ k :Y k Y k+p where Y k R n and Y k+p R n and where y k+p Γ k y k as follows: (24) H k y k+1 T k y k H k+1 y k+2 T k+1 y k+1 H k+p 2 y k+p 1 T k+p 2 y k+p 2 µh k+p 1 y k T k+p 1 y k+p 1 with (25) x l Z T l y l Clearly since the systems in (22) and (24) are (orthogonally) equivalent their eigenvalues are equal ie Λ(Π l )Λ(Γ l ) Further the triangularized system of linear equations in (24) is written in an eigenvalue revealing form; this allows the eigenvalues of the period map defined by (24) to be related to the diagonal elements of its constituent matrices Proceeding let t ijk denote the ijth element of the matrix T k and similarly let h ijk denote the ijth element of the matrix H k Also let the matrices T ik and H ik be 2 2 matrices centered on the diagonals of T k and H k respectively with the upper left hand entries being t iik and h iik if the system defined by T ik and H ik is associated with a 2 2 bulge in the quasi upper triangular H 1 Then the eigenvalues of the operator Γ l may be written as (26) Λ(Γ l ) p λ i j1 { {λi λ i+1 } Λ( H 1 ip t iij /h iij if h iij 0λ i R T 1 ip H T } i1 i1 ) λ i λ i+1 C { λ i if t iij h iij 0} { λ i C if t iij h iij 0} i {1n} l {1p} From the definition above the eigenvalues fall into four categories Since it is necessary to refer to these categories throughout the paper we provide a table of nomenclature in the following definition Definition 1 Let E and F in (12) be the scalars E α and F β Then the adjectives 1 enomorphic and medotropic may be used to describe the four categories of possible eigenvalues as shown in Table 1 3 Quotient Singular Value Decomposition In this section we show how the QSVD may be related to the periodic QZ decomposition We start by restricting our attention to the case when E and F in (12) are nonsingular In such a case it is 1 We introduce here the neologisms enomorphic and medotropic These terms are derived from Greek meaning of the form of one and changed by zero respectively The former term reflects the fact that an enomorphic number is equivalent to one for a finite scaling The latter term emphasizes the fact that a medotropic number has a zero in its rational representation

4 JJ Hench 141 Table 21 Table of Eigenvalue Categories α β λ categories β α 0 β 0 α finite determinate enomorphic α 0 β 0 0 finite determinate medotropic α 0 β 0 non-finite determinate medotropic α 0 β 0 C non-finite indeterminate medotropic possible to write the matrices (31) ˆ 1 F T E T E 1 F ˆ 2 FF T E T E 1 ˆ 3 E 1 FF T E T ˆ 4 E T E 1 FF T ˆ T 2 where the matrices ˆ i are nonsingular It turns out that the eigenvectors of these matrices are closely related to the QSVD which we demonstrate formally in the following lemmas and theorems Lemma 31 Let ˆ 1 ˆ 2 ˆ 3 and ˆ 4 be defined by (31) with E R n n and F R n n nonsingular There exist matrices M 1 M 2 M 3 andm 4 such that the following sets of relations hold: (32) S M 1 1 M 1 2 M 1 3 M 1 4 ˆ 1 M 1 ˆ 2 M 2 ˆ 3 M 3 ˆ 4 M 4 where S is real diagonal positive definite and has arbitrarily ordered diagonal elements and D 1 M2 1 F M 1 D 2 M3 1 E 1 M 2 (33) D 3 M4 1 E T M 3 D 4 M1 1 F T M 4 where the matrices D i are diagonal and positive definite Proof Since ˆ 1 is symmetric we can find an orthogonal M 1 which diagonalizes ˆ 1 in (31) and (32) with arbitrarily ordered eigenvalues as above Let (34) F M 1 M 2 D1 where the matrix D 1 is chosen to be an arbitrary positive definite diagonal matrix and where the matrix M 2 is nonsingular Inserting the identity matrix M 1 2 M 2 between E 1 and F in the equation for S in (31) yields the expression (35) M1 1 F T E T E 1 1 M2 M 2 FM 1 S M1 1 F T E T E 1 M2 D1

5 142 On the periodic quotient singular value decomposition Since S and D 1 are diagonal M 1 1 F T E T E 1 M2 must be diagonal as well Noting that diagonal matrices commute we write (36) M 1 2 FM 1M 1 1 F T E T E 1 M2 S This implies that it is possible to write M 2 M 2 and D 1 D 1 Repeating this procedure for all of the matrices D i in (33) completes the proof In Lemma 31 the matrices M 1 2 and M 1 diagonalize F ; however we would like to diagonalize this matrix without forming an inverse This indeed is possible as we demonstrate in the next lemma Lemma 32 Suppose M 2 and M 4 diagonalize ˆ 2 and ˆ 4 respectively with the same eigenvalue ordering as in (32) Then M T 4 and M T 2 diagonalize ˆ 2 and ˆ 4 respectively Further if the eigenvalues of the matrices ˆ i are distinct the matrices M 2 and M 4 may be related by the equation (37) M T 2 M 4 L with L diagonal Proof SinceM 2 diagonalizes ˆ 2 M 4 diagonalizes ˆ 4 ands S T S M 1 4 E T E 1 FF T M 4 S T M T 2 E T E 1 FF T M T 2 With the eigenvalues of S distinct columns of the the matrices M2 T and M 4 are right eigenvectors associated with the eigenvalues of S Since eigenvectors of distinct eigenvalues are equal up to a scaling then M2 T M 4 L with L nonsingular and diagonal The lemmas above demonstrate a number of remarkable properties of the sets of matrices M i and ˆ i namely that the matrices M i diagonalize the matrices ˆ i that the matrices M i diagonalize the matrices E E T F and F T individually and that under certain conditions the inverses of all of the M i s may be computed by appropriately weighting the columns of related M j s These properties allow us to demonstrate our main result Theorem 33 Let the matrices E R n n and F R n n be nonsingular with the eigenvalues of the matrices i in (31) distinct Further let M 1 M 2 M 3 and M 4 be defined as in (32) and (33) There exists a diagonal signature matrix P with ±1 ±1 (38) P ±1 such that U X V Φ and Θ associated with the QSVD (13) of Ξ (12) may be written as (39) U M 3 X M 4 P V M 1 Φ PM4 T EM 3 Θ PM4 T FM 1

6 JJ Hench 143 Proof Since ˆ 1 is symmetric it is possible to choose M 1 to be orthogonal Further the matrices D 1 D 2 andd 3 may be chosen such that the columns of M 2 M 3 and M 4 are unit normalized The fact that the columns of M 3 are unit normalized and that the matrix ˆ 3 is symmetric imply that M 3 is orthogonal Thus D 1 M2 1 1 D2 1 M2 1 3 with M 1 and M 3 are orthogonal and with D 1 and D 2 diagonal and positive definite Since the singular values of Ξ are distinct then via Lemma 32 LD 1 M4 T FM 1 LD2 1 M4 T EM 3 This implies that M4 T FM 1 and M4 T EM 3 are diagonal Setting P sgn(l) completes the proof Remark 1 In the proof of the above theorem the condition that the singular values appearing on the diagonal of Φ and Θ be ordered as in (14) was not imposed However any ordering of the singular values in the QSVD may be imposed without loss of generality 3 In general we would like to be able to prove Theorem 33 when E and F are not restricted to be nonsingular with the singular values of Ξ in (12) distinct Even inthecasewheree and F are nonsingular the procedure outlined in the proof of Lemma 31 is undesirable from a numerical point of view since it requires forming the matrices ˆ i explicitly Fortunately the periodic QZ decomposition allows us to view the matrices ˆ i as period maps i of certain related periodic systems thereby eliminating the need for inversions and multiplications by non-orthogonal matrices Three key properties of the periodic QZ decomposition are employed to generalize Theorem 33 to the case where E and F are singular The first property allows the reduction of the matrices comprising a periodic system of equations to quasi triangular (eigenvalue revealing) form without forming the period map explicitly Parenthetically this property implies that the periodic QZ decomposition triangularizes each of the period maps i separately The second property allows the eigenvalues appearing along the diagonals of the triangularized period map produced by the periodic QZ decomposition to be ordered arbitrarily The third property asserts that the first columns (rows) of the matrices Z i (Q T i ) of the periodic QZ decomposition are right (left) eigenvectors of the period maps i By rotating each of the eigenvalues of the system triangularized by the periodic QZ decomposition in turn to the upper left hand corner it is possible to assemble the eigenvector matrices M i associated with the ˆ i s in (32) and (33) and thereby to compute all of the matrices related to the QSVD This may be done irrespective of the rank of E and F Proceeding we propose a modified form for the periodic QZ decomposition which is more closely related to the QSVD than the standard periodic QZ decomposition Lemma 34 Let Q and Z be orthogonal matrices with the product QZ being upper-triangular Then QZ P where P is a diagonal signature matrix Proof By definition P is orthogonal which implies P T P I Since P is upper triangular as well P T P I implies that P is diagonal Finally since the modulus of the eigenvalues of an orthogonal matrix must be one and since P is real then P is a diagonal matrix with p ii ±1

7 144 On the periodic quotient singular value decomposition Next the operator 1 is defined in terms of a periodic system Eventually it will be used to compute the QSVD of Ξ The next lemma shows that its periodic QZ decomposition has a special form Lemma 35 Let the operator 1 : X 1 X 4 be the operator defined by the equations (310) Ex 2 Fx 1 E T x 3 x 2 x 4 F T x 3 with E R n n and F R n n nonsingular There exist orthogonal matrices Q U V andw such that (311) Q T EU H 1 Q T FV T 1 U T E T W H 2 V T F T W T 3 where H 1 H 2 T 1 andt 3 are upper triangular Proof Via the periodic Schur decomposition there exist orthogonal matrices Q 1 Q 2 Q 3 Z 1 Z 2 andz 3 such that (312) Q T 1 EZ 2 H 1 Q T 1 FZ 1 T 1 Q T 2 ET Z 3 H 2 Q T 2 Z 2 T 2 Q T 3 Z 1 H 3 Q T 3 F T Z 3 T 3 where H 2 H 3 T 1 T 2 andt 3 are upper triangular Since the operator 1 ˆ 1 is symmetric the eigenvalues of the operator will be real and therefore H 1 is uppertriangular as well By Lemma 34 Q T 2 Z 2 P 1 and Q T 3 Z 1 P 2 where the diagonal elements of P 1 and P 2 are ±1 This implies that the columns of Q 2 and Z 2 differ only by sign This is equally the case for Q 3 and Z 1 By setting Q Q 1 U Q 2 Z 2 P 1 V Q 3 Z 1 P 2 andw Z 3 we complete the proof Remark 2 Since the operator 1 ˆ 1 with E and F nonsingular the orthogonal matrix V resulting from the (modified) periodic QZ decomposition that diagonalizes the operator is the matrix V associated with the QSVD in (13) modulo the sign of the columns Similarly the orthogonal matrix U resulting from the periodic QZ decomposition is the matrix U associated with the same QSVD 3 In the following lemma we show how the remaining matrices X ΦandΘmay be computed via the periodic QZ decomposition Lemma 36 Let Ξ be defined as in (12) with E R n n and F R n n nonsingular Further let the eigenvalues of the matrices i in (31) be distinct Then the matrices X U V Φ andθ associated with the QSVD in (13) may be computed via the (modified) QZ decomposition in (311) Proof Via Lemma 35 it is possible to find orthogonal matrices Q Ū V and W that diagonalize 1 with a particular eigenvalue ordering Let λ i be the eigenvalue associated with the ith column of V Let the matrix V j be the orthogonal matrix that results from interchanging the first column of V and the jth column Since the operator 1 is diagonalized by the matrix V V j also diagonalizes the operator 1 with the jth eigenvalue in the upper left hand corner Now define the matrices Q j U j andw j as the matrices along with V j that triangularize matrices E F E T and F T in 1 with the jth eigenvalue in the upper left hand corner Also define the vector x j to be the first column of W j Since 4 is triangularized by W j with the

8 JJ Hench 145 jth eigenvalue in the upper left hand corner then x j is a right eigenvector of 4 and via Lemma 32 x T j is a left eigenvector of 2 Thus the matrix X x 1 x n diagonalizes 4 from the right and X T diagonalizes 2 from the left with the same eigenvalue ordering This implies via Lemma 31 that X T along with matrices U and V diagonalize E and F albeit with the diagonal elements not necessarily positive There exists however a diagonal signature matrix P as in (38) a matrix X P X and orthogonal matrices U and V such that X T U andv diagonalize E and F with positive diagonal elements In the proof above we rely on the fact that the matrices E and F are nonsingular and with the singular values distinct In the following theorem we provide a proof for the existence of the QSVD with E and F possibly singular Theorem 37 Let E R n n and F R n n and the operator Ξ be defined as in (12) There exist a non orthogonal matrix X non negative definite diagonal matrices Φ and Θ and orthogonal matrices U and V such that (313) X T EU Φ X T FV Θ Proof Let {E (k) } and {F (k) } be a sequence of nonsingular n n matrices that converge pointwise to E and F respectively with the eigenvalues of the associated matrix ˆ (k) distinct Let the matrices Q (k) U (k) V (k) W (k) H 1(k) H 2(k) T 1(k) and T 2(k) be the matrices produced by the (modified) periodic QZ decomposition in (311) Further let X (k) Φ (k) andθ (k) be produced by the procedure in the proof in Lemma 36 For any given E (k) and F (k) the periodic QZ decomposition produces bounded Q (k) U (k) V (k) W (k) The boundedness of Q (k) U (k) V (k) W (k) E (k) and F (k) clearly imply the boundedness of X (k) Φ (k) andθ (k) Finally since these matrices are all elements of a compact metric space the Bolzano-Weierstrass Theorem ensures that the bounded sequence {(U (k) X (k) V (k) Φ (k) Θ (k) )} has a converging subsequence k i ie lim {(U (k i X k (ki)v (ki) Φ (ki) Θ (ki))} (U X V Φ Θ) i In the limit the matrices U and V remain orthogonal and the matrices Φ and are non negative definite and therefore are matrices associated with the QSVD in (13) Remark 3 This proof of the existence of the QSVD for E and F singular requires the use of a converging subsequence which is impractical from a computational point of view If the requirement that Φ and Θ is diagonal is relaxed to be block upper triangular where the non-diagonal blocks correspond to the non-distinct singular values then the matrices resulting from the procedure in the proof in Lemma 36 suffices In such a case this algorithm requires approximately 162n 3 flops assuming 2 implicit QZ steps per eigenvalue This compares with approximately 52n 3 flops for the standard QSVD algorithm 13 3 Remark 4 In 12 and 13 Van Loan suggests the use of the VZ algorithm 12 applied to the pencil (314) E T E λf T F to compute the singular values of Ξ in (12) This idea is similar to the idea of this paper in a number of ways First the matrix pencil in (314) is closely related to

9 146 On the periodic quotient singular value decomposition the operator 4 Second the VZ decomposition is nearly identical to the modified periodic QZ decomposition proposed in Lemma 35 To our knowledge however the eludication of the relationships between the eigenvectors of the operators i and the matrices U X andv and the method of extracting the eigenvectors from the orthogonal matrices that comprise the modified QZ decomposition is novel to this paper 3 4 Periodic Quotient Singular Value Decomposition In this section we examine the operator Π 1 : X 1 X p+1 X 1 R n X p+1 R n whereπ 1 is defined by the equations (41) E 1 x 2 F 1 x 1 E p x p+1 F p x p with E i R n n and F i R n n In this section we intend to show that it is possible to implicitly reduce the matrices comprising the operator Π 1 to diagonal form as was the case for the non periodic operator Ξ in (12) As in the previous section we restrict our attention first to the case where the matrices E i and F i are nonsingular Proceeding we write definitions for the matrices ˆΓ i for i 14p where the matrices ˆΓ i are analogous to matrices ˆ i in the non periodic case ˆΓ 1 F1 T E1 T F2 T E2 T Fp T E T p ˆΓ 2 F 1 F1 T E1 T F2 T Fp T E T p (42) ˆΓ 4p E1 T F2 T E2 T F3 T Fp T E T p E 1 p E 1 p F pep 1 1 F p 1 E1 1 F 1 F pep 1 1 F p 1 F 2 E 1 1 Ep 1F pep 1 1 p 1 F 1 F1 T The matrices ˆΓ i defined as above provide a mechanism by which the existence of a periodic QSVD may be proved Lemma 41 Let ˆΓ 1 ˆΓ 2 ˆΓ 4p be defined by (42) with E i R n n and F i R n n nonsingular for i 1p There exist matrices M i for i 14p such that the following sets of relations hold: (43) S M1 1 ˆΓ 1 M 1 M2 1 ˆΓ 2 M 2 M 1 4p ˆΓ 4p M 4p where S is real diagonal positive definite and (44) D 1 M2 1 F M 1 D 2 M3 1 E 1 M 2 D 4p 1 M4p 1 E T M 4p 1 D 4p M1 1 F T M 4p where the matrices D i are diagonal and positive definite Proof The proof is trivial extension of the proof of Lemma 31 with F 1 substituting for F in (34) and with the ˆΓ i playing the role of ˆ i throughout

10 JJ Hench 147 Proceeding along the lines of the previous section we generalize Lemma 32 to the periodic case Lemma 42 Suppose M r diagonalizes ˆΓ r for r 22p and r 2p+24p with the same eigenvalue ordering as in (43) Then M T 4p r+2 diagonalizes ˆΓ r for r 2 2p and r 2p +2 4p Further if the eigenvalues of the matrices ˆΓ i are distinct then the matrices M r and M 4p r+2 may be related by the equation (45) M T 4p r+2 M rl r with L r diagonal Proof The key element of the proof for Lemma 32 was the observation that ˆ 2 ˆ T 4 In the periodic case ˆΓ r ˆΓ T 4p r+2 for r 22p and r 2p+24p The remainder of the proof follows from this observation Next we prove the existence of the periodic QSVD in the case where the matrices E i and F i are nonsingular Theorem 43 Let the matrices E i R n n and F i R n n in (42) be nonsingular with the eigenvalues of the matrix ˆΓ 1 distinct There exist non orthogonal matrices X 1 X p and Y 2 Y p positive definite diagonal matrices Φ 1 Φ p and Θ 1 Θ p and orthogonal matrices U and V such that (46) X1 T E 1Y 2 Φ 1 X1 T F 1V Θ 1 X2 T E 2Y 3 Φ 2 X2 T F 2Y 2 Θ 2 Xp 1E T p 1 Y p Φ p 1 Xp 1F T p 1 Y p 1 Θ p 1 Xp T E pu Φ p Xp T F py p Θ p Further the constituent matrices of the periodic QSVD may be related to the matrices M i in (43) and (44) which diagonalize the matrices ˆΓ i in the following way: X 1 M 4p P 4p V M 1 X 2 M 4p 2 P 4p 2 Y 2 M 4p 1 P 4p 1 X 3 M 4p 4 P 4p 4 Y 3 M 4p 3 P 4p 3 X p M 2p+2 P 2p+2 Y p M 2p+3 P 2p+3 (47) U M 2p+1 Φ 1 P 4p M4pE T 1 M 4p 1 P 4p 1 Θ 1 P 4p M4pF T 1 M 1 Φ 2 P 4p 2 M4p 2E T 2 M 4p 3 P 4p 3 Θ 2 P 4p 2 M4p 2F T 2 M 4p 1 P 4p 1 Φ p 1 P 2p M2p T E p 1M 2p+3 P 2p+3 Θ p 1 P 2p M2p T F p 1M 2p+1 Φ p P 2p+2 M2p+2 T E pm 2p+1 Θ p P 2p+2 M2p+2 T F pm 2p+3 P 2p+3 where the matrices P i are diagonal signature matrices as in (38) Proof The proof is a straightforward extension of the proof of Theorem 33 By the substitution of the results of Lemmas 41 we can assure that the matrix S in (43) and the matrices D i s (44) are diagonal and positive definite Since ˆΓ 1 and ˆΓ 2p+1 are symmetric M 1 and M 2p+1 may be chosen to be orthogonal With M 1 and M 2p+1 orthogonal and S positive definite Lemma 41 and Lemma 42 imply that there exist matrices P i such that (47) holds completing the proof

11 148 On the periodic quotient singular value decomposition As in the previous section we show that the matrices in the periodic QSVD may be constructed from the matrices resulting from the periodic QZ decomposition of the matrices comprising a related operator Γ 1 : X 1 X 2p+2 X i R n where Γ 1 is defined by the equations (48) E 1 x 2 F 1 x 1 E p x p+1 F p x p Ep T x p+2 x p+1 Ep 1x T p+3 Fp T x p+2 E1 T x 2p+1 x 2p+2 F2 T x 2p F1 T x 2p+1 To prove the existence of the periodic QSVD it is necessary to generalize the lemmas and theorems of the previous section Lemma 44 Let the operator Γ 1 be defined as in (48) There exist orthogonal matrices Q 1 Q 2p 1 W 2 W 2p U and V such that the matrices H k and T k are upper triangular Q T 1 E 1W 2 H 1 Q T 1 F 1V T 1 (49) Q T p E p U H p Q T p F p W p T p U T Ep T W p+1 H p+1 Q T p+1 ET p 1 W p+2 H p+2 Q T p+1 F p T W p+1 T p+2 Q T 2p 1E1 T W 2p H 2p Q T 2p 1F2 T W 2p 1 T 2p V T F1 T W 2p T 2p+1 Proof The proof is a trivial extension of that in Lemma 35 Lemma 45 Let the operator Γ 1 be defined as in (48) with the matrices E i R n n and F i R n n nonsingular and with the eigenvalues of Γ 1 distinct The matrices X i U i V i Φ i and Θ i associated with the periodic QSVD in (46) may be computed via the (modified) QZ decomposition in (49) Proof The proof for the existence of the periodic QSVD with the matrices E i and F i nonsingular is an extension of Lemma 36 with the matrices X k and Y k constructed analogously Lemma 44 ensures that there exist orthogonal matrices Q 1 Q 2p 1 Ū V and W2 W 2p that diagonalize Γ 1 with a particular eigenvalue ordering Let λ i be the eigenvalue of associated with the ith column of V Let the matrix V j be the orthogonal matrix that results from interchanging the first column of V and the jth column Since the operator Γ 1 is diagonalized by the matrix V V j also diagonalizes the operator with the jth eigenvalue in the upper left hand corner Now define the matrices Q l j U l andw l j as the matrices along with V l that triangularize constituent matrices of Γ 1 with the jth eigenvalue in the upper left hand corner Also define the vector x l j to be the first column of W 2p l+1 j Via Lemma 42 and in analogy with Lemma 36 x T l j is a left eigenvector of the operator Γ 4p 2l Similarly define the vector ỹ l j to be the first vector of W l+1 j The vector ỹ l j is a right eigenvector of the operator Γ 2l 2 Thus the matrices X l x l 1 x l n and

12 JJ Hench 149 Ỹ l ỹ l 1 ỹ l n are matrices of eigenvectors of Γ 4p 2l and Γ 2l 2 respectively and therefore diagonalize the constituent matrices of the operator Γ 1 via Lemma 41 albeit with the diagonal elements not necessarily positive There exists however diagonal signature matrices P xl and P yl as in (38) such that the matrices X l X l P xl and Y l ỸlP yl diagonalize the constituent matrices of the operator Γ 1 with the diagonalized matrices positive definite Finally Theorem 37 is generalized for the periodic case Theorem 46 Let the operator Π be defined as in (41) There exist non orthogonal matrices X 1 X p and Y 2 Y p non negative definite diagonal matrices Φ 1 Φ p and Θ 1 Θ p and orthogonal matrices U and V such that the relations in (46) hold Proof The proof is a trivial extension of Theorem 37 5 Numerical Examples In this section we give some numerical examples to illustrate the points discussed in the previous sections Example 1 Consider the operator Ξ defined by (12) where the matrices E and F are E F Since E is nonsingular it is possible to write the map Ξ ˆΞ directly: 2 1 ˆΞ 1 0 The singular value decomposition of ˆΞ Û ˆΣ ˆV T where Û ˆΣ ˆV The matrices U V Q W H i andt i that result from the periodic QZ decomposition of 1 are: U V Q W H 1 Q T EU H 2 U T E T W T 1 Q T FV T 3 V T F T W

13 150 On the periodic quotient singular value decomposition With a two by two system it is possible to construct X x 1 x 2 directly from Q and W : x 1 w 1 x 2 q 2 yielding X Computing Φ and Θ yields Φ X T EU Θ X T FV The product Σ φθ Φ 1 Θis Σ φθ which is equal to ˆΣ as expected Example 2 Consider again the operator Ξ where E F Since E is singular it is not possible to write the map ˆΞ directly However it is possibletocomputetheqsvdofthesystemthematricesu V Q W H i andt i that result from the periodic QZ decomposition of are: U V Q W H 1 Q T EU H 2 U T E T W T 1 Q T FV T 3 V T F T W As before we construct X x 1 x 2 directly from Q and W : x 1 w 1 x 2 q 2

14 JJ Hench 151 making X Computing Φ and Θ yields Φ X T EU Θ X T FV Note that the system has an infinite and a zero singular value and no enomorphic eigenvalues Example 3 Consider the operator Π in (41) defined by matrices E i and F i where E 1 E F 1 F Since the matrices E i are nonsingular it is possible to write the map ˆΠ directly: 1 2 ˆΠ 1 1 The singular value decomposition of ˆΠ Û ˆΣ ˆV T where Û ˆΣ ˆV The matrices U V Q k W k H k and T k that result from the periodic QZ decomposition of the pencil of Γ are: U V Q Q Q W W W H 1 Q T 1 E 1W H 2 Q T 2 E 2U

15 152 On the periodic quotient singular value decomposition H 3 U T E2 T W H 4 Q T 3 E1 T W T 1 Q T 1 F 1 V T 2 Q T 2 F 2 W T 4 Q T 3 F2 T W T 5 V T F1 T W With a two by two system it is possible to construct X 1 x 11 x 12 X 2 x 21 x 22 and Y 2 y 21 y 22 directly from the Q k s and W k s: x 11 w 41 x 12 q 12 x 21 w 31 x 22 q 22 y 11 w 21 y 12 q 32 yielding X X and Y Computing Φ 1 Φ 2 Θ 1 andθ 2 yields Φ 1 X1 T E 1 Y Θ 1 X1 T F 1 V Φ 2 X2 T E 2 U Θ 2 X2 T F 2 Y The product Σ φθ Φ 1 2 Θ 2Φ 1 1 Θ 1 is Σ φθ which is equal to ˆΣ as expected

16 JJ Hench Conclusion In this paper the relationship between the QSVD and the periodic QZ decomposition is elaborated Specifically the periodic QZ algorithm may be viewed as a method by which the QSVD may be computed as accurately albeit half as efficiently than the standard algorithm Nevertheless by using this technique the periodic QSVD of a sequence of matrix pairs may be readily computed In this paper we have discussed the QSVD; however other canonical forms such as those discussed in previous papers or the computation of principal angles and vectors 4 may be cast in the periodic QZ framework This demonstrates the versatility of the QZ decomposition; it provides a theoretical basis for computing eigenvalue or singular value revealing matrix decompositions REFERENCES 1 A Bojanczyk GH Golub and P van Dooren The periodic Schur decomposition algorithms and applications in Proc SPIE Conf vol pp B De Moor and P van Dooren Generalizations of the singular value and QR decompositions SIAM J Matrix Anal Appl 13 (1992) pp GH Golub and W Kahan Calculating the singular values and pseudo-inverse of a matrix SIAM J Matrix Anal Appl B2 (1965) pp GH Golub and CF Van Loan Matrix Computations (2nd edition) The Johns Hopkins University Press Baltimore Maryland (1989) 5 MT Heath AJ Laub CC Paige and RC Ward Computing the singular value decomposition of a product of two matrices SIAM J Sci Stat Comput 7 (1986) pp JJ Hench Numerical Methods for Periodic Linear Systems PhD thesis University of California Santa Barbara Electrical and Computer Engineering Dept September The quotient singular value decomposition and the periodic QZ decomposition Tech Report 1775 Institute of Information and Control Theory Academy of Sciences of the Czech Republic Pod vodárenskou věží 4 Praha 8 October Special decompositions via the periodic Schur and QZ decompositions Tech Report 1774 Institute of Information and Control Theory Academy of Sciences of the Czech Republic Pod vodárenskou věží 4 Praha 8 September JJ Hench and AJ Laub Numerical solution of the discrete-time periodic Riccati equation IEEE Trans Automat Control 39(1994) CB Moler and GW Stewart An algorithm for generalized matrix eigenvalue problems SIAM J Numer Anal 10 (1973) pp CC Paige and M Saunders Towards a generalized singular value decomposition SIAMJ Numer Anal 18 (1981) pp CF Van Loan A general matrix eigenvalue algorithm SIAM J Numer Anal 12 (1975) pp Generalizing the singular value decomposition SIAM J Numer Anal 13 (1976) pp 76 83