A Note on Simple Nonzero Finite Generalized Singular Values

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1 A Note on Simple Nonzero Finite Generalized Singular Values Wei Ma Zheng-Jian Bai December Abstract In this paper we study the sensitivity and second order perturbation expansions of simple nonzero finite generalized singular values of a complex matrix pair which is analytically dependent on several parameters. Our results generalize the perturbation analysis given by Sun J. Comput. Math pp for simple nonzero singular values. Keywords Generalized singular value sensitivity analysis AMS subject classification. 65F15 15A18 1 Introduction The generalized singular value decomposition GSVD of two matrices of the same number of columns was proposed by Van Loan 25 and further studied by Paige and Saunders 13. The GSVD is very useful in many applications including constrained least squares problems 8 p. 58 and 25 weighted least squares problems 4 25 information retrieval 1 linear discriminant analysis 14 computing the Kronecker structure of matrix pencil A λb 12 discriminant analysis 11 ionospheric tomography 3 etc. Numerical methods and sensitivity analysis of the GSVD can be found for instance in Recently Chen and Li 7 presented the sensitivity of multiple nonzero finite generalized singular values of a real matrix pair which is analytically dependent on several parameters. In this paper we focus on the sensitivity issue and second order perturbation expansions of simple nonzero finite generalized singular values of a complex matrix pair analytically dependent on several parameters. We provide the explicit expressions for the first order partial derivatives of simple nonzero finite generalized singular values of the complex matrix pair and associated generalized singular matrix set. We also give the second order partial derivatives of simple nonzero finite generalized singular values of the complex matrix pair which is very useful for computing the second order Taylor expansions for simple nonzero finite generalized singular School of Mathematical Sciences Xiamen University Xiamen 3615 People s Republic of China mawei83424@yahoo.com.cn. School of Mathematical Sciences Xiamen University Xiamen 3615 People s Republic of China zjbai@xmu.edu.cn. The research of this author was partially supported by the National Natural Science Foundation of China grant the Natural Science Foundation of Fujian Province of China for Distinguished Young Scholars No. 21J62 and NCET. 1

2 values. Our results may be used to check the effectiveness and stability of GSVD-based methods for practical applications such as the method of particular solutions for solving planar eigenvalue problems 2 ionospheric tomography techniques 3 and discriminant analysis 11 etc. Our results generalize the perturbation analysis in 21 for simple non-zero singular values. Throughout this paper the following notation will be used. Let C m n and R m n stand for the set of all m n complex matrices and the set of all m n real matrices respectively. The symbols C n and R n denote the set of all complex n-vectors and the set of all real n-vectors respectively. Denote by A T and A the transpose and the conjugate transpose of a matrix A respectively. Let I n be the identity matrix of order n. The paper is organized as follows. In Section 2 we introduce some preliminary results on the GSVD. In Section 3 we discuss the sensitivity analysis and second order perturbation expansions of simple nonzero finite generalized singular values. The partial derivatives of a generalized singular vector set corresponding to the simple nonzero finite generalized singular values are also established. In Section 4 we define the sensitivity of simple nonzero finite generalized singular values and give some examples for computing the sensitivity and second order expansions of simple nonzero finite generalized singular values. Finally some conclusions and future work are presented in Section 5. 2 Preliminaries On the GSVD of a complex matrix pair we have the following result Lemma 2.1 Let A C m n and B C l n be such that ranka B n. Then there exist two unitary matrices U C m m V C l l and a nonsingular matrix Q C n n such that U Λ AQ and V l+r n r BQ Σ m r s n r s where s t denotes the s t null matrix and with Λ diagα 1... α r+s and Σ diagβ r+1... β n 1 α 1 α r > α r+1 α r+s > α r+s+1 α n β 1 β r < β r+1 β r+s < β r+s+1 β n 1 α 2 j + β 2 j 1 for 1 j n. ere {α j β j } n j1 are called the generalized singular values of the complex matrix-pair {A B}. For simplicity denote by σ{a B} the set of generalized singular values of {A B}. Suppose that α β is a simple nonzero finite generalized singular value of {A B} and σ 1 α/β. Then By Lemma 2.1 we can easily know that there exist two unitary matrices U u 1 U 2 C m m V v 1 V 2 C l l and a nonsingular matrix Q Q 2 C n n with u 1 C m v 1 C l and C n such that U σ1 AQ and V 1 BQ 1 Λ 2 Σ 2 2

3 where σ 1 1 / σ{λ 2 Σ 2 }. Therefore we get by 1 A σ 1 u 1 B v 1 u 1 u 1 v 1 v The vector set {u 1 v 1 } satisfying 2 is called a generalized singular vector set of {A B} corresponding to the generalized singular value σ Computing partial derivatives In this section we study the sensitivity issue of simple nonzero finite generalized singular values and the associated generalized singular vector set. Let p p 1... p N T Ap C m n and Bp C l n. In what follows without loss of generality we assume that p R N. By using the Implicit Function Theorem we establish the following results. Theorem 3.1 Let p R N Ap C m n Bp C l n and rankap Bp n. Suppose that ReAp ReBp ImAp and ImAp are all real analytic matrix-valued functions of p in some neighborhood B of the origin. If σ 1 1 σ 1 > is a simple nonzero finite generalized singular value of {A B} and there exist two unit vectors u 1 C m v 1 C l and a nonzero vector C n such that {u 1 v 1 } is an associated generalized singular vector set of {A B} then 1 there exists a simple generalized singular value σ 1 p 1 of {Ap Bp} such that σ 1 p is a real analytic function of p in some neighborhood B B of the origin σ 1 p > and σ 1 σ 1. 2 there exist u 1 p C m v 1 p C l and p C n with u 1 p u 1 p v 1 p v 1 p 1 such that Reu 1 p Imu 1 p Rev 1 p Imv 1 p Re p and Im p are all real analytic functions of p in B and {u 1 p v 1 p p} is a generalized singular vector set of {Ap Bp} corresponding to the generalized singular value σ 1 p 1 i.e. Ap p σ 1 pu 1 p and Bp p v 1 p where u 1 u 1 v 1 v 1 and. Proof: By the hypothesis there exist two unitary matrices U u 1 U 2 C m m V v 1 V 2 C l l and a nonsingular matrix Q Q 2 C n n such that U σ1 AQ and V 1 BQ 3 Λ 2 Σ 2 where σ 1 1 / σλ 2 Σ 2 and σ 1 >. Let Ãp Q Ap ã11 p ã ApQ 21 p ã 21 p à 22 p b11 Bp Q Bp p b21 p BpQ b21 p B22 p 4 5 3

4 where ã 11 p b 11 p R ã 11 σ 2 1 and b We now define the following vectorvalued functions fz w p ã 21 p + Ã22pz + wã 11 p + wã 21 p z 6 gz w p b 21 p + B 22 pz + w b 11 p + w b 21 p z 7 where f f 1... f n 1 T g g 1... g n 1 T z ζ 1... ζ n 1 T C n 1 w ω 1... ω n 1 T C n 1 p R N. Let f j ϕ j + iψ j ζ j ξ j + iη j g j ϕ j + i ψ j ω j ξ j + i η j j 1... n 1 and x ξ 1... ξ n 1 T R n 1 y η 1... η n 1 T R n 1 x ξ 1... ξ n 1 T R n 1 ỹ η 1... η n 1 T R n 1. It is clear that ϕ j x y x ỹ p ψ j x y x ỹ p ϕ j x y x ỹ p and ψ j x y x ỹ p are all real analytic functions of real variables x y x ỹ R n 1 and p B and ϕ j ψ j ϕ j ψj 8 for j 1... n 1. Notice that f 1... f n 1 g 1... g n 1 are all complex analytic functions of the complex variables ζ 1... ζ n 1 ω 1... ω n 1 for any p B. Thus we have 5 p.39 Theorem 8 det ϕ 1... ϕ n 1 ψ 1... ψ n 1 ϕ 1... ϕ n 1 ψ 1... ψ n 1 ξ 1... ξ n 1 η 1... η n 1 ξ 1... ξ n 1 η 1... η n 1 det f 1... f n 1 g 1... g n 1 2 ζ 1... ζ n 1 ω 1... ω n 1. This together with f1... f n 1 g 1... g n 1 I1 det det Ã22 I 1 B 22 ζ 1... ζ n 1 ω 1... ω n 1 zw ã 11 I n 1 b11 T I n 1 Λ T det 2 Λ 2 Σ T 2 Σ 2 σ1 2I detλ T 2 Λ n 1 I 2 σ1σ 2 T 2 Σ 2 n 1 implies that det ϕ 1... ϕ n 1 ψ 1... ψ n 1 ϕ 1... ϕ n 1 ψ 1... ψ n 1 ξ 1... ξ n 1 η 1... η n 1 ξ 1... ξ n 1 η 1... η n 1 4

5 where denotes the Kronecker product see for instance 9. Therefore by using the Implicit Function Theorem 18 Theorem 1.2 we know that the system of equations ϕ j x y x ỹ p ψ j x y x ỹ p ϕ j x y x ỹ p ψj x y x ỹ p for j 1... n 1 i.e. { fz w p has a unique real analytic solution gz w p 9 x xp y yp x xp ỹ ỹp i.e. z zp w wp in some neighbourhood B B of the origin where x y x ỹ i.e. z w 1 and deti n 1 wpzp p B. 11 Next we construct a simple generalized singular value of {Ap Bp} and an associated generalized singular vector set. From 11 it follows that the matrix 1 wp zp I n 1 is nonsingular for any p B. Thus we obtain by 4 and 5 for any p B 1 wp zp I n 1 Ãp 1 wp zp I n 1 a1 p A 2 p 12 and 1 wp where and zp I n 1 Bp 1 wp zp I n 1 a 1 p + Q 2 zp Ap Ap + Q 2 zp b1 p B 2 p ã 11 p + zp ã 21 p + ã 21 p zp + zp à 22 pzp b 1 p + Q 2 zp Bp Bp + Q 2 zp b 11 p + zp b21 p + b 21 p zp + zp B22 pzp with a 1 σ 2 1 and b 1 1. We observe that for sufficiently small B a 1 p > b 1 p > p B. ence we can define a positive valued function σ 1 : B R by σ 1 p a 1 p 1 2 b1 p 1 2 p B. 16 5

6 In addition let 1 p Q zp b 1 p 1 2 p B 17 u 1 p Ap p/σ 1 p v 1 p Bp p p B. 18 By 4 18 it is easy to know that the functions u 1 p v 1 p and p are such that Reu 1 p Imu 1 p Rev 1 p Imv 1 p Re p and Im p are all real analytic in B o with and Ap p σ 1 pu 1 p Bp p v 1 p u 1 p u 1 p v 1 p v 1 p 1 19 u 1 u 1 v 1 v 1. 2 By using the perturbation theorem for generalized singular values see for instance 16 it is easy to see that if the neighborhood B is small enough the generalized singular value σ 1 p 1 of {Ap Bp} such that σ 1 p 1 is simple and σ 1 p is a real analytic function of p in B and σ 1 σ 1. Theorem 3.2 Under the same assumptions as in Theorem 3.1 the following formulas for the simple generalized singular value σ 1 p 1 and the generalized singular vector set {u 1 p v 1 p p} defined in hold: σ1 p Re u 1 Ap σ 1 Re v 1 Bp 21 q1 p Ap Ap u1 p Q 2 Φ 1 Q 2 Bp Q 2 Φ 3 Q 2 Bp Re v1 u 1 + Q 2 Φ 2 U2 v 1 Q 2 Φ 4 V2 Bp 22 { Ap 1 q1 p + A σ 1 1 Ap Ap {U 2 Λ 2 Φ 1 Q 2 u 1 + U 2 Λ 2 Φ 2 U2 σ 1 1 Bp {U 2 Λ 2 Φ 3 Q 2 v 1 + U 2 Λ 2 Φ 4 V2 σ 1 { Ap + 1 σ 1 Re u 1 Ap σ1 p u 1 Bp } } } } u

7 v1 p q1 p Bp B + Ap Ap V 2 Σ 2 Φ 1 Q 2 V 2 Σ 2 Φ 3 Q 2 Bp + Bp u 1 + V 2 Σ 2 Φ 2 U2 Re v 1 V 2 Σ 2 Φ 4 V2 Bp v 1 Bp v σ 1 p Re u 2 Ap 1 σ 1 Re v 2 Bp Ap Ap Re q 1 σ 1 Bp Bp σ 1 Re q 1 u1 +Re Dk C u1 v1 D j + σ 1 Re σ 1 Re 1 σ 1 Re Re Re v1 u 1 u 1 +3σ 1 Re u 1 Sj C 1 D k + Sk C u1 1D j Ap Ap Ap v 1 Bp Re Re Re v 1 v 1 Re u 1 Ap Bp Bp v 1 Bp S j C 2 S k v1 25 for j k 1... N where Φ σ 2 1Σ T 2 Σ 2 Λ T 2 Λ 2 1 D j Φ 1 σ 1 Φ Φ 2 ΦΛ T 2 Φ 3 σ1φ 2 Φ 4 σ1φσ 2 T 2 Bp S j Ap Ap Bp 7

8 Q2 Φ 1 Q 2 Q 2 Φ 2 U2 C U 2 Φ 2 Q 1 2 σ 1 U 2 Λ 2 Φ 2 U2 C 2 Q2 Φ C 1 1 Q 2 Q 2 Φ 2 U2 V 2 Σ 2 Φ 1 Q 2 V 2 Σ 2 Φ 2 U2 Q2 Φ 3 Q 2 Q 2 Φ 4 V2 V 2 Σ 2 Φ 3 Q 2 V 2 Σ 2 Φ 4 V2 Proof: 1 By Theorem 3.1 see 19 and 2 we have. It follows from 26 and 19 that σ 1 p σ 1 p u 1 p Ap p p Ap u 1 p 26 1 v 1 p Bp p p Bp v 1 p. 27 u1 p σ 1 p u 1 p + u 1 p Ap p + u 1 p q1 p Ap and σ 1 p q1 p Ap Ap u 1 p + p u 1 p + σ 1 pu 1 p u1 p. 29 By27 and 19 we get v1 p v 1 p + v 1 p Bp p + v 1 p q1 p Bp 28 3 and q1 p Bp Bp v 1 p + p v 1 p + v 1 p v1 p. 31 Using and u 1 p u 1 p v 1 p v 1 p 1 we have σ 1 p 1 u 1 p Ap Ap p + p u 1 p σ 1p v 1 p Bp Bp p + p v 1 p + 1 u1 p Ap σ 1 pv 1 p Bp p 2 q1 p Ap u 1 p σ 1 pbp v 1 p Substituting p into 32 and using u 1 A σ 1v 1 B 1 n we obtain By Theorem 3.1 see and 17 we get Ap Ap p σ 2 1pBp Bp p 8

9 which yields σ 2 1 B B A A p Ap Ap A + A σ1 p 2σ 1 B B Bp Bp σ1 2 B + B. 33 This together with 3 1 and 17 gives rise to σ1 2Σ 2 Σ 2 Λ 2 Λ zp 2 σ1 Ap σ 1 Q { Bp σ1 2 Q u 1 + v 1 + Λ 2 1 Σ 2 Ap U V Bp σ1 p 2σ 1 } 1 and thus zp σ 2 1Σ T 2 Σ 2 Λ T 2 Λ 2 1 σ 1 Q 2 σ 2 1σ 2 1Σ T 2 Σ 2 Λ T 2 Λ 2 1 Q 2 Ap Bp u 1 + Λ 2 U2 v 1 + Σ 2 V2 Ap Bp. 34 Moreover we have by 15 and 3 b1 p 2Re v 1 Bp 35 and using 17 we get q1 p Combining and 36 yields From 19 we obtain u1 p zp Q Ap 1 q1 p + A σ 1 9 b1 p. 36 σ1 p u 1.

10 This together with and the relation AQ 2 U 2 Λ 2 yields By using 19 again we obtain v1 p q1 p Bp B + 37 which together with 22 and the relation BQ 2 V 2 Σ 2 gives rise to By 32 we have Re 2 σ 1 p u1 p +Re σ 1 Re σ 1 Re u 1 Ap v1 p v 1 Ap Bp u1 p +Re σ1 p v1 p σ 1 Re q1 p Bp A Re + Re q1 p q1 p u 2 Ap 1 σ1 p Bp Re v1 2 Bp σ 1 Re v1 q1 p B q1 p B + Re u 1 v 1 σ 1 Re Combining it with and 24 yields 25. Ap v 1 Bp q1 p q1 p We remark that as in we derive our sensitivity results based on the Implicit Function Theorem and the definition of the generalized singular value decomposition. Also if Ap R m n Bp R l n the first order partial derivatives related to the simple generalized singular value σ 1 p 1 and the generalized singular vector set {u 1 p v 1 p p} of {Ap Bp} in Theorem 3.2 are same as in 7 Theorem 2.1 under the assumption that the generalized singular value σ 1 1 of {A B} is simple. On the other hand suppose that m n. Let Bp I n for all p R N. By 3 there exist two unitary matrices U u 1 U 2 C m m V v 1 V 2 C n n and a nonsingular matrix Q V C n n such that U T σ1 AV and V BQ V 1 V 38 Λ 2 Σ 2 1.

11 σ 2... Λ 2 σ n Rm 1 n 1 Σ 2 I n 1 where σ 2... σ n and < σ 1 σ j for j 2... n. That is σ 1 is a simple nonzero singular value of A v 1 C n and u 1 C m are associated unit right and unit left singular vectors respectively. By using the unitarity of U we have Ap Ap v 1 UU { Ap u 1 Also it is easy to check that v 1 } v 1 u 1 + U 2 U2 Ap v I m 1 + Λ 2 σ 2 1I n 1 Λ T 2 Λ 2 1 Λ T 2 σ 2 1σ 2 1I m 1 Λ 2 Λ T Therefore by using Theorems 3.1 and and 4 it is easy to derive the same results as in 21 on the analyticity of simple nonzero singular values of a matrix analytically dependent on several parameters. Corollary 3.3 Let p R N and Ap C m n m n. Suppose that ReAp and ImAp are real analytic matrix-valued functions of p in some neighborhood B of the origin. If σ 1 > is a simple nonzero singular value of A and there exist two unit vectors u 1 C m and v 1 C n such that u 1 and v 1 are associated left and right singular vectors respectively i.e. there exist two unitary matrices U u 1 U 2 C m m and V v 1 V 2 C n n such that the first equality of 38 holds then 1 there exists a simple singular value σ 1 p > of Ap such that σ 1 p is a real analytic function of p in some neighborhood B B of the origin and σ 1 σ 1. 2 there exist unit vectors u 1 p C m and v 1 p C n such that Reu 1 p Imu 1 p Rev 1 p and Imv 1 p are all real analytic functions of p in B and u 1 p and v 1 p are the left and right singular vectors of Ap corresponding to the simple singular value σ 1 p i.e. Apv 1 p σ 1 pu 1 p and Ap u 1 p σ 1 pv 1 p where u 1 u 1 and v 1 v 1. Moreover the simple nonzero singular value σ 1 p is given by σ 1 p 1 ã 11 p + zp ã 21 p + ã 21 p zp + zp 2 Ã 22 pzp 1 + zp zp

12 for all p B where ã 11 p ã z1 p à 22 p are given by V Ap ã11 p ã ApV : 21 p ã 21 p à 22 p p B and zp R n 1 is the real analytic solution of fz z p in B where fz w p is defined in 6 z and deti n 1 + zpzp for all p B. In addition the associated unit singular vectors v 1 p and u 1 p are given by v 1 p V 1 zp 1 + zp zp 1 2 u 1 p Apv 1 p/σ 1 p p B. 42 Finally the following formulas for the simple nonzero singular value σ 1 p and the associated singular vectors v 1 p and u 1 p defined by hold: σ1 p Ap Re u 1 v 1 43 v1 p Ap Ap V 2 Φ 1 V2 Φ T 2 U2 u1 44 v 1 u1 p U 2 Φ 2 V 2 + i Im u 1 σ 1 Ap Ap Φ 3 U2 Ap u1 v 1 u 1 45 v 1 2 σ 1 p Re +Re u 1 2 Ap u1 v 1 v 1 V2 Φ 1 V2 V 2 Φ T 2 U 2 U 2 Φ 2 V2 U 2 Φ 3 U2 + 1 Ap Im u 1 σ 1 Ap Ap Ap v 1 Im u 1 Ap Ap u1 v 1 v 1 46 where j k 1... N u 1 v 1 U 2 and V 2 are defined by 38 and Φ 1 σ 1 σ 2 1I n 1 Λ T 2 Λ 2 1 Φ 2 Λ 2 σ 2 1I n 1 Λ T 2 Λ 2 1 Φ 3 σ 1 σ 2 1I m 1 Λ 2 Λ T 2 1 in which Λ 2 is defined by

13 4 Applications In this section we give some examples to show that our results are useful for computing the sensitivity and the second order perturbation expansions of simple nonzero finite generalized singular values of a complex matrix pair analytically dependent on several parameters. Based on Theorems 3.1 and 3.2 we first define the sensitivity of simple nonzero finite generalized singular values as follows 7. Definition 4.1 Let p p 1... p N T R N Ap C m n and Bp C l n. Suppose that Ap Bp has full row rank and ReAp ReBp ImAp and ImAp are real analytic matrix-valued functions of p in some neighborhood B of the origin. If {A B} has the GSVD 3 then the quantity σ1 p s pj σ 1 47 is called the sensitivity of the simple nonzero finite generalized singular value σ 1 1 with respect to the parameter p j the quantity s pi1 p i2...p im σ 1 m s 2 p ik σ 1 48 is called the sensitivity of the simple nonzero finite generalized singular value σ 1 1 with respect to the parameter p i1 p i2... p im the quantity s p σ 1 N s 2 p j σ 1 49 is called the sensitivity of the simple nonzero finite generalized singular value σ 1 1. j1 k1 Example 4.2 Let Ap 3 + 2p 1 + 4p 2 + ip 1 p 1 + ip 2 p 2 2 p 2 p p 1 5 and Bp where p p 1 p 2 T R 2 and i p 1 + ip 2 p 1 p p 1 p p 2 51 We observe that A B

14 where σ is a simple nonzero finite generalized singular value of {A B} and Using 5 and 51 we get Ap p 1 Bp p 1 By and 49 u 1 v 1 1 T. 2 + i Ap p 2 Bp p 2 s p1 3 2 s p2 3 1 s p i i 1 1 Next we present an example to show how the second order perturbation expansions of simple nonzero finite generalized singular values work.. Example 4.3 Let Ap p 1 +ip and Bp 5 3 p 1 +ip where p p 1 p 2 T R 2. Obviously the matrix pair {A B} is given by 6 2 A and B 2 6 which has the following GSVD: where Thus one has U AQ U V V 1 BQ 2 Q σ 1 2 Λ 2 Σ 2 2 Φ 1 Φ Φ Φ u T U T v T V T T Q T 14.

15 Ap p 1 Bp p 1 2 Ap p Ap p 1 p 2 2 Ap p i 1 Using 21 and 25 a simple calculation yields σ1 p p 1.5 Ap p 2 Bp p 2 2 Bp p Bp p 1 p 2 2 Bp p 2 2 σ1 p p 2 i 3i 3 3i 3. and 2 σ 1 p p Therefore σ 1 p has the expansion 2 σ 1 p p 1 p 2 2 σ 1 p p σ 1 p 2.5p p p O p 3 in a neighborhood of the origin where denotes the Euclidean vector norm. These examples show that our results are effective for evaluating the sensitivity and the second order Taylor expansions of simple nonzero finite generalized singular values. 5 Future work In this paper we give the first order and second order partial derivatives of simple nonzero finite generalized singular values of a complex matrix pair analytically dependent on several parameters. These results may be used to investigate the effectiveness of the GSVD-based methods for practical applications. An interesting problem is to discuss the sensitivity analysis and the second order perturbation expansions of zero generalized singular values infinite generalized singular values and multiple generalized singular values analytically dependent on several parameters. This needs further study. Acknowledgments We would like to thank the referees for their useful suggestions. 15

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