Sensitivity analysis for the multivariate eigenvalue problem

Transcription

1 Electronic Journal of Linear Algebra Volume 6 Volume 6 (03) Article Sensitivity analysis for the multivariate eigenvalue problem Xin-guo Liu wgwang@ouc.edu.cn Wei-guo Wang Lian-hua Xu Follow this and additional works at: Recommended Citation Liu, Xin-guo; Wang, Wei-guo; and Xu, Lian-hua. (03), "Sensitivity analysis for the multivariate eigenvalue problem", Electronic Journal of Linear Algebra, Volume 6. DOI: This Article is brought to you for free and open access by Wyoming Scholars Repository. It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Repository. For more information, please contact scholcom@uwyo.edu.

2 SENSITIVITY ANALYSIS FOR THE MULTIVARIATE EIGENVALUE PROBLEM XIN-GUO LIU, WEI-GUO WANG, AND LIAN-HUA XU Abstract. This paper concerns with the sensitivity analysis for the multivariate eigenvalue problem (MEP). The concept of a simple multivariate eigenvalue of a matrix is generalized to the MEP and the first-order perturbation expansions of a simple multivariate eigenvalue and the corresponding multivariate eigenvector are presented. The explicit expressions of condition numbers, perturbation upper bounds and backward errors for multivariate eigenpairs are derived. Key words. Multivariate eigenvalue problem, Simple eigenvalue, Condition number, Perturbation bound, Backward error. AMS subject classifications. 65H0, 5A, 65F5.. Introduction. Given a symmetric positive definite matrix A R n n, and a set of positive integers P m = {n,n,...,n m } with m n i = n, i= partition A into the block form A A A m A A A m A =......, A m A m A mm where A ij R ni nj. Let λ,λ,...,λ m be m real scalars and Λ := diag{λ I n,λ I n,...,λ m I nm }, Received by theeditors onapril7, 0. Accepted forpublication onaugust5, 03. Handling Editor: Panayiotis Psarrakos. School of Mathematical Sciences, Ocean University of China, Qingdao, 6600, P.R. China (liuxinguo@ouc.edu.cn). Supported in part by the National Natural Science Foundation of China Grant and 078, and Shandong Province Natural Science Foundation Grant Y008A07. School of Mathematical Sciences, Ocean University of China, Qingdao, 6600, P.R. China (wgwang@ouc.edu.cn). Supported in part by the Fundamental Research Funds for the Central Universities Grant School of Mathematical Sciences, Ocean University of China, Qingdao, 6600, P.R. China (lotus xu006@6.com). 636

3 Sensitivity Analysis for the Multivariate Eigenvalue Problem 637 where I ni R ni ni is the identity matrix. Denote S := {x = [ x T x T... x T m] T R n xj R nj, x j =, j =,,...,m}. The multivariate eigenvalue problem (MEP) is to find a pair (Λ,x) such that (.) { Ax = Λx, x S, where Λ is called a multivariate eigenvalue, x a multivariate eigenvector, and (Λ, x) a multivariate eigenpair or a solution to MEP (.). MEP arises as a necessary condition to the solution of the maximal correlation problem (MCP): (.) max x T Ax, s.t. x S, and finds applications in pattern analysis [0]. For more information about background materials, see [, 6, 3, ]. Several special cases of MEP have been well understood. It is a symmetric eigenvalue problem for a positive definite matrix when m = (see e.g., [3]). When m = n, the MEP has exactly n solutions which can be explicitly formulated. Chu and Watterson [] also discovered that there are precisely m i= (n i) solutions for a generic matrix A whose n eigenvalues are distinct. Horst [8] proposed an iterative method (named the power method in [] and the Horst method in [9, 3]) for computing a solution to the MEP. Detailed convergence analysis for the Horst iteration method are presented in [] and [9]. Following the innate iterative structure, a SOR-style generalization (P-SOR) has been proposed by Sun [4] as a natural improvement upon the Horst method. The convergence of the P-SOR method has been established in [4]. Recently, several results on the global maximizer of the maximal correlation problem are presented by [5, 8, 9, 0, ]. Although, as mentioned above, MEP is a symmetric eigenvalue problem for a positive definite matrix when m =, it must be pointed out that the generic case < m < n has essential differences from the classical eigenvalue problem. For example, neither characteristic polynomial nor Schur-type decomposition can be readily employed. It is known [3, 7] that both condition number and backward error play important roles in matrix computation. However, to our knowledge, little attention has been paid to the sensitivity analysis of the MEP. Golub and Zha [4] present a perturbation analysis of the canonical correlations of matrix pair.

4 638 X.-G. Liu, W.-G. Wang, and L.-H. Xu This paper deals with the sensitivity analysis for MEP. The rest of this paper is organized as follows. Section establishes the concept of simple multivariate eigenpair and presents first-order expansions. A sufficient and necessary condition for the simple multivariate eigenpairs is examined. In Section 3, following Rice s idea [], we introduce the condition numbers for multivariate eigenvalues and eigenvectors and present explicit formulae for them. In Section 4, we derive perturbation bounds for multivariate eigenpair by applying the Brouwer fixed-point theorem. Section 5 deals with backward error of approximate multivariate eigenpair. In Section 6, some simple numerical examples are shown. Finally, we give our concluding remarks in Section 7.. First order expansions of simple multivariate eigenpair. The MEP (.) can be equivalently written as the following nonlinear system (.) Ax = Λx, xt i x i =, x i R ni, i =,,...,m. Then the Jacobi matrix at a multivariate eigenpair (Λ,x) is the (n+m) (n+m) matrix [ ] A Λ B(x) J(Λ,x) := B(x) T, 0 where x x B(x) :=... Rn m. x m First we generalize the concept of simple root of a function (see, e.g., []) to the nonlinear system (.) as follows. Definition.. A multivariate eigenpair (Λ, x) of the MEP (.) with nonsingular J(Λ, x) is called a simple multivariate eigenpair. Remark.. It is easy to see that if m =, then (λ,x) is a simple eigenpair of A if and only if λ is a simple eigenvalue of A. After a careful look at the proof of [, Lemma 3.6], we know that the following result holds. Theorem.3. Let A R n n be a symmetric positive definite matrix whose n eigenvalues are distinct. Then any multivariate eigenpair of the MEP (.) is simple.

5 Sensitivity Analysis for the Multivariate Eigenvalue Problem 639 To simplify the notation, without loss of generality, we assume that n j >, j =,,...,m. For x j R nj, x j =, there exists an orthogonal matrix Q j = [ xj X j ] such that (If n j =, we just take Q j = x j.) Set Q T j x j = [ ] T, j =,,...,m. (.) (.3) A λ = G(x) T AG(x), G(x) = diag(x,...,x m ), Λ = diag(λ I n,...,λ m I nm ). Theorem.4. Let (Λ,x) be a multivariate eigenpair of MEP (.). Then (Λ,x) is simple if and only if the matrix A λ Λ is nonsingular. Proof. Let Q = diag(q,...,q m ), and let P R n n be a permutation matrix such that [ ] P diag(e () Im,...,e(m) ) =, 0 where e (j) = [ ] T R n j, j =,,...,m. Then (.4) [ PQ T 0 0 I m ] [ ] QP T 0 J(Λ, x) = J pq, 0 I m where (.5) M M I m J pq = M T M 0, I m 0 0 [ ] M M M T = PQ T (A Λ)QP T, M = A λ M Λ. From (.4) and (.5), we see that and our conclusion holds. det(j(λ,x)) = det(a λ Λ), In the rest of this paper, we assume that (Λ,x) is a simple multivariate eigenpair of the MEP (.).

6 640 X.-G. Liu, W.-G. Wang, and L.-H. Xu Consider the perturbation of A A(t) = A+tE, where t is a real parameter and E R n n is a symmetric matrix. Partition A(t) as where A ij (t) R ni nj with m n j = n. A (t) A (t) A m (t) A (t) A (t) A m (t) A(t) =......, A m (t) A m (t) A mm (t) j= We consider the associated MEP A(t)x(t) = Λ(t)x(t), where Λ(t) = diag(λ (t)i n,λ (t)i n,...,λ m (t)i nm ) and x(t) S. Set λ(t) = [ λ (t) λ (t)... λ m (t) ] T [ ] T., and λ = λ λ... λ m First applying the implicit function theory, we prove the following result. Theorem.5. Let E be a real symmetric matrix. Then there exist unique functions x(t) and λ(t) which are analytic in some neighborhood of t 0 = 0 such that (.6) { A(t)x(t) = Λ(t)x(t), x(t) S, with x(0) = x and Λ(0) = Λ. Proof. Let F(y,µ,t) := A(t)y Ωy, G(y,t) := [ (yt y )... (yt m y m ) ] T. where µ = [ µ... µ m ] T, Ω = diag(µ I n,...,µ m I nm ), and y = [ y T... y T m] T, y j R nj. Then we have. F(y,µ,0) = 0, and G(y,0) = 0.. F(y,µ,t) and G(y,t) are real analytic functions of the real variables y, µ, and t. [ ] F F 3. det y G y µ G µ (y,µ,t)=(x,λ,0) = det(j(λ,x)) 0.

7 Sensitivity Analysis for the Multivariate Eigenvalue Problem 64 Applying the implicit function theory establishes our conclusion. Theorem.5 implies that x(t) and λ(t) in (.6) have expansions of the form (.7) x(t) = x+ẋ(0)t+o(t ), λ(t) = λ+ λ(0)t+o(t ), where ẋ(0) = dx(t) dt and λ(0) = dλ(t) dt. t=0 t=0 The following theorem gives the expressions for the first order derivatives ẋ(0) and λ(0). Theorem.6. Assume that (Λ, x) is a simple multivariate eigenpair of MEP (.). Let P be an orthogonal matrix and Q a permutation matrix such that (.4) and (.5) hold. Then (.8) (.9) [ ] 0 0 ẋ(0) = Q T P 0 M PQ T Ex, λ(0) = [ ] I m Z PQ T Ex, where Z = M M, and the matrices M and M are defined in (.5). Proof. Differentiating the first equation in (.6) with respect to t at t = 0 gives λ (0)x I n (.0) Ex+Aẋ(0) =. + λ... λ m (0)x m Also, differentiating x i (t) T x i (t) = at t = 0 gives (.) x T i ẋ i (0) = 0, i =,,...,m. Combining (.0) with (.), we get [ẋ(0) ] [ ] Ex J(Λ, x) =. λ(0) 0 λ m I nm Since (Λ,x) is simple, then J(Λ,x) is nonsingular. Thus, we get (.) [ẋ(0) ] λ(0) [ ] Ex = J(Λ,x). 0 ẋ(0). From (.4), we get [ ] [ ] Q J(Λ,x) = T P 0 PQ Jpq T 0, 0 I m 0 I m

8 64 X.-G. Liu, W.-G. Wang, and L.-H. Xu where (.3) J pq = 0 0 I m 0 M Z T, I m Z Z Z = M M, Z = M M MT M. Substituting this into (.) yields (.8) and (.9). 3. Condition numbers. Although there were earlier references dealing with conditioning, e.g. [7], the general framework of the conditioning theory was introduced by Rice []. Note that the multivariate eigenvalue problem (.) is equivalent to the nonlinear system (.). Naturally, the condition numbers of multivariate eigenvalue Λ and the corresponding eigenvector x of the MEP (.) are defined as follows. Definition 3.. Let (λ(t), x(t)) satisfy (.6). Then the condition number of the multivariate eigenvalue Λ is defined by (3.) κ(λ) := lim t 0 sup E ξ { λ(t) λ α t and the condition number of the corresponding eigenvector x is defined by (3.) κ(x) := lim t 0 sup E ξ { x(t) x α t where α,α and ξ are all positive parameters which allow us some flexibility. For example: Taking α = α = ξ =, the condition numbers defined by (3.) and (3.) are the absolute condition numbers. Taking α = λ, α = x, ξ = A, the condition numbers defined by (3.) and (3.) are the relative condition numbers. The following results give explicit expressions for κ(λ) and κ(x). Theorem 3.. }, }, (3.3) (3.4) κ(λ) = κ(x) = m ξ [ ] I m Z, α m ξ M α, where the matrices Z and M are defined in (.3) and (.5), respectively.

9 Sensitivity Analysis for the Multivariate Eigenvalue Problem 643 Proof. From (.7), (.8) and (.9), we see that { } λ(0) κ(λ) = sup Similarly, we get E ξ { = sup E ξ = max = α y R n, y mξ α [ I m Z ] PQ T Ex m α ξ [ I m Z ]. κ(x) = sup E ξ } { [ } ] I m Z PQ T y α { } ẋ(0) α { [ ] } = sup 0 0 E ξ α 0 M PQ T Ex m = ξ M α. It is worthy pointing out that both κ(λ) and κ(x) depend on M. The following corollary indicates a particular case when the eigenvalue Λ is wellconditioned. Corollary 3.3. m (3.5) κ(λ) = ξ Xi T α A ijx j = 0, i j and Xi T (A ii λ i I ni )x i = 0. Proof. From (3.3), we have m κ(λ) = ξ Z = 0 α M = 0 (see the second equality in (.3)) X T i A ij x j = 0. i j and X T i (A ii λ i I ni )x i = 0. Although the matrix X j is not unique, both κ(λ) and κ(x) are independent of choices of X j. Several remarks on (3.3) (3.5) are in order. Remark 3.4. Corollary 3.3 involves the condition X T i (A ii λ i I ni )x i = 0. This means that (A ii λ i I ni )x i is orthogonal to all columns of X i implying it must point in the direction of x i. Therefore, A ii x i = µ i x i for some µ i follows.

10 644 X.-G. Liu, W.-G. Wang, and L.-H. Xu Proposition 3.5. If there exist orthogonal matrices X i R ni ni, (i =,...,m) such that Q T i A ijq j are diagonal for i,j =,...,m, then for any eigenvalue Λ of MEP (.), κ(λ) = m α ξ. Proof. The hypotheses that Q T i A ijq j are diagonal imply X T i A ijx j = 0 for i j and X T i (A ii λ i I ni )x i = 0. Thus, the desired result follows from Corollary 3.3. Below we outline several cases where Proposition 3.5 is applicable.. m =, A = I n, A = I n. For this case, we can take the matrices whose columns are comprised of the left and right singular vectors of A as Q and Q.. m = n. For this case, we can take Q i = (). 3. A ij = 0, (i j), i.e., A = diag(a,...,a mm ) is a block-diagonal matrix. For this case, we can take the eigenvector matrix of A ii as Q i. Remark 3.6. From (3.3)-(3.5), we see that both κ(λ) and κ(x) depend on M for general m and A. If M is large, then x is ill-conditioned, but λ may be wellconditioned (if, for example M = 0) or ill-conditioned. It is known [5, Theorem 3.6] that all eigenvalues are well-conditioned for the standard symmetric eigenvalue problem. [ ] A A Example 3.7. Consider the MEP (.) with A = A T, where A = A [ ] [ ] a 0 α 0 0, A =, A = diag(a 33,a 44,a 55 ). Assume that A has n 0 a 0 α 0 distinct eigenvalues. From [], A has precisely 4 eigenpairs, one of them is Λ = diag((a +α )I, (a 33 +α )I 3 ), x = [ ] T. Accordingly, a a α α 0 A λ Λ = α a 44 a 33 α a 55 a 33 α If a 55 a 33 +α, then x is ill-conditioned. On the other hand, from Proposition 3.5, all multivariate eigenvalues of A are

11 well-conditioned. Sensitivity Analysis for the Multivariate Eigenvalue Problem 645 Example 3.8. Let 0 A = , n = n =, 0 < ε. 0 4+ε The matrix A is symmetric positive definite and has four distinct eigenvalues. Let Λ = diag(,,4,4), x = [ 0 0 ] T. Then (Λ,x) is a multivariate eigenpair of A. Note that 0 A Λ = ε From (.4), we get A λ Λ = M = Z = M M = ε [ ] [ 0 0 0, M T 0 ε = [ ] 0, M 0 = ε. Thus, the absolute condition numbersareκ(x) = ε andκ(λ) = + 4 ε.thismeans that Λ and x both are ill-conditioned. Consider the perturbation of A A(t) = A+tE. Let E R 4 4 be a Householder transformation such that [ ] 0 Ex = QP T y (), y () R, y () = Then from (.8), we have ẋ(0) = Q T P [ ] 0. [ ][ ] M y () = Q T Ph, where h = [ ε] T. Thus, ẋ(0) = ε, and x(t) = x+qt Pht+O(t ). ],

12 646 X.-G. Liu, W.-G. Wang, and L.-H. Xu From (.9), we have λ(0) = [ I m Z ] PQ T QP T [ 0 y () = Z y () = [ ][ ] 0 0 = ε 0 ε ] [ ]. Thus, λ(t) = λ+ ε [ ] t+o(t ). 4. Perturbation bounds. Let (Λ, x) be a simple multivariate eigenpair of the MEP (.). Suppose that A is perturbed to A+ A. Consider the following nonlinear system (4.) { (A+ A)(x+ x) = (Λ+ Λ)(x+ x), x+ x S, where Λ = diag( λ I n,..., λ m I nm ) and λ = [ λ... λ m ] T. Set (4.) τ := J(Λ,x). Assume that A satisfies { } (4.3) A < min (4 mτ +)τ, λ min(a), where λ min (A) denotes the smallest eigenvalue of A. Applying the technique of [4], we prove the following result. Theorem 4.. Assume A satisfies (4.3). Then A + A has a multivariate eigenvalue Λ+ Λ = diag(µ I n,...,µ m I nm ) and a corresponding eigenvector y S such that (4.4) [ y x µ λ] mτ τ A A, where λ = [ λ... λ m ] T and µ = [ µ... µ m ] T.

13 Sensitivity Analysis for the Multivariate Eigenvalue Problem 647 Proof. Reformulating the nonlinear system (4.) as A x+ Λ x [ ] [ ] x A x (4.5) J(Λ, x) = + x λ 0.. x m By (4.5), we get [ ] { x λ τ m A [ ] x + A λ + Λ x + x }. x m { m A [ ] x τ + A λ + [ ] x λ + } x. Note that the last inequality follows from x + λ λ x. Hence, [ ] { [ ] x τ m A } x (4.6) +. λ τ A λ Consider the following quadratic equation about ξ τ ξ = ( m A +ξ ), τ A which has two positive solutions under the condition (4.3). The smaller one, denoted by ξ, satisfies mτ A ξ = τ A + ( τ A ) 4 mτ A (4.7). mτ A τ A We now define a mapping M by M and a set D by (4.8) [ ] x λ A x+ Λ x [ ] A x = (J(Λ,x)) + x, 0. x m D := {[ ] x [ ] } R n+m x λ λ ξ.

14 648 X.-G. Liu, W.-G. Wang, and L.-H. Xu Obviously, D is a bounded closed convex subset of R n+m. From (4.6) and (4.7), we see that M is a continuous mapping which maps D into ] D. By the Brouwer fixedpoint theorem, the mapping M has a fixed-point D such that + [ ] ] [ x x [ x λ λ λ is a solution of the nonlinear system (4.). Note that A < λ min (A) implies the perturbed matrix A + A is positive definite. From (4.5), (Λ + λ,x + x) is a multivariate eigenpair of A + A, and the desired conclusion is established. The inequality (4.4) reveals the relationship between the sensitivity of the multivariate eigenpair (Λ,x) and τ. Let A 0 = diag(a,a,...,a mm ) and the spectral decomposition of A jj be A jj = U j Λ j U T j, Λ j = diag(ˆλ (j) [ ] (j) (j),ˆλ,...,ˆλ n j ), U j = u (j) u (j)... u (j) n j. Suppose A 0 has n distinct eigenvalues. By Chu and Watterson [], the MEP A 0 y = Ωy, y S, has exactly m j= (n j) eigenpairs (Ω,y): Ω = diag(µ I n,µ I n,...,µ m I nm ), Ω = diag(ˆλ () k I n,ˆλ () k I n,...,ˆλ (m) k m I nm ), k j n j, j =,,...,m, [ ] T. y = ±(u () k ) T ±(u () k ) T... ±(u (m) k m ) T [ Let Λ = diag(λ,λ,...,λ m ), e = e (n) [ ] T R n i. Some calculation gives that T e (n ) T... e (n m) T ] T, and e (n i) [ ] [ ] τ (0) = J(Ω,y) A0 Ω B(y) Λ Ω B(e) = B(y) T = 0 B(e) T = 0 α, where α = min i m min j n i j k i ˆλ(i) k i (i) ˆλ j. = Let E = A A 0. As a consequence of Theorem 4., we get the following result. Theorem 4.. Suppose A 0 = diag(a,a,...,a mm ) has n distinct eigenvalues. Assume that E = A A 0 satisfies { } E < min (4 mτ (0) +)τ (0), λ min(a 0 ).

15 Sensitivity Analysis for the Multivariate Eigenvalue Problem 649 Then the MEP (.) has exactly m j= (n j) solutions. Furthermore, for any multivariate eigenpair (Ω,y) of A 0, there exists a multivariate eigenpair (Λ,x) of A such that [ y x µ λ] mτ (0) τ (0) E E, where λ = [ λ λ... λ m ] T and µ = [ µ µ... µ m ] T. We can also prove the following inclusion results concerning multivariate eigenvalues. Theorem 4.3. Let (Λ, x) be a multivariate eigenpair of (.). Then (4.9) σ min (λ i I A ii ) min{ν (i),ν(i) }, where ν (i) = A ij, j i ν (i) = m [ A i... A i,i A i,i+... A im ], i =,,...,m. In particular, if x is a global maximizer of the MCP, then (4.0) 0 λ i λ max (A ii ) min{ν (i),ν(i) }, i =,,...,m. Proof. Since Ax = Λx, we have that (λ i I A ii )x i = j i A ij x j, i =,,...,m. The inequalities (4.9) are obtained straightforward. If x is a global maximizer of the MCP (.), then the first inequality in (4.0) is already proved by Zhang and Chu [9]. Note that A ii is symmetric, the second inequality in (4.0) follows from (4.9) due to λ i λ max (A ii ) = σ min (λ i I ni A ii ). 5. Backward errors. In this section, we deal with the normwise backward errors of approximate multivariate eigenpairs of the MEP (.). Let ( Λ, x) be an approximate multivariate eigenpair of (.), where Λ = diag( λ I n, λ I n,..., λ m I nm ), x = [ x T x T... x T m] T S. Consider the following matrix equation about E 0 (5.) (A+E 0 ) x = Λ x,

16 650 X.-G. Liu, W.-G. Wang, and L.-H. Xu where E 0 R n n is a symmetric matrix. The normwise backward error of ( Λ, x) is defined by Note that (5.) can be rewritten as η( Λ, x) := min{ E 0 F E 0 satisfies (5.)}. E 0 x = Λ x A x. Applying a result of Sun ([4, Lemma.4]), we see that the following result holds. Theorem 5.. The normwise backward error η( Λ, x) is given by r η( Λ, x) = m ( xt r) m, where r = A x Λ x. Theorem 5. may be used to analyze the accuracy of the computed results of iterative methods for MEP (.). For example, we consider the Horst algorithm summarized below. Algorithm 5.. (Horst algorithm []) Given x (0) S; For k =,,..., until convergence; 3 For i =,,...,m; 4 Compute y (k) i = m A ij x (k) j ; 5 Compute µ (k) i 6 Compute x (k+) i 7 End 8 End j= = y (k) i ; = y(k) i µ (k) i ; [ ] T Let x (k) = (x (k) )T... (x (k) m ) T and Ω (k) = diag(µ (k) I n,...,µ (k) m I nm ) 0. Then the algorithm can be formulated as Ω (k) x (k+) = Ax (k), k =,,... The next result shows that if x (k+) x (k) ε, then (Ω (k),x (k+) ) is an approximate multivariate eigenpair of A. Theorem 5.3. There exists a symmetric matrix E R n n such that (A+E)x (k+) = Ω (k) x (k+),

17 Sensitivity Analysis for the Multivariate Eigenvalue Problem 65 where E F m A(x(k+) x (k) ). Proof. Let (5.) r = Ax (k+) Ω (k) x (k+). By Theorem 5., there exists a symmetric matrix E R n n such that where Recall that, by definition, Ex (k+) = r, E F m r. Ax (k) = Ω (k) x (k+). Substituting it into (5.), the desired conclusion is established. From the definition of M in (.5), we see that M = /σ min (A λ Λ). Theorem 3. shows that the sensitivity of the eigenvector x depends on σ min (A λ Λ). From (.3) and (3.3), loosely speaking, the sensitivity of the eigenvalue Λ is also dependent on σ min (A λ Λ). Let E(n) := {H R n n H T = H}. Suppose that (Λ,x) is a simple multivariate eigenpair of A and define E := {H H E(n),(Λ,x) is a multivariate eigenpair ofa+h, but not simple}. Below we prove an interesting property of σ min (A λ Λ) in connection with E. Theorem 5.4. If σ min (A λ Λ) < λ min (A), then min H E H F = σ min (A λ Λ), where A λ and Λ are defined respectively by (.), so (.3). Proof. Eckhart-Young theorem [] and Theorem.6 together imply that min H E H F min H E G(x) T HG(x) F = σ min (A λ Λ), where G(x) is defined in (.).

18 65 X.-G. Liu, W.-G. Wang, and L.-H. Xu Let (A λ Λ)υ = λ min υ, υ R n m, υ =, where λ min denotes the eigenvalue of A λ Λ satisfying λ min = σ min (A λ Λ). Now consider the matrix equation (5.3) G(x) T NG(x) = λ min υυ T with the unknown N E(n). Let u j = X j υ j, u = [ u T u T... u T m] T, uj R nj. Then, a simple calculation shows that N = λ min uu T = λ min G(x)υυ T G(x) T solves (5.3), and N F = σ min (A λ Λ). The assumption σ min (A λ Λ) < λ min (A) implies that N E and the desired result is obtained. Finally, a relationship between τ 0 := /σ min (A λ Λ) and τ := J(Λ,x) is given by the next theorem. Theorem 5.5. τ 0 τ max{τ 0 (+ M ),+ M +τ 0 ( M + M )}. Proof. From (.4) and (4.), τ = (J(Λ,x)) = J pq. Note that M = [ 0 I n m 0 ] 0 Jpq I n m and M = A λ Λ, thus τ 0 τ. 0 On the other hand, from (.3), τ = J pq λ max 0 τ 0 Z 0 τ 0 Z Z Z Z Z = max{τ 0 + Z,+ Z + Z } max{τ 0 (+ M ),+ M +τ 0 ( M + M )}. The first inequality is obtained from [6, Lemma ]. In summary, Theorems 3., 4. and 5.5 show that the sensitivity of the multivariate eigenpair (Λ,x) of A is dependent on τ 0.

19 Sensitivity Analysis for the Multivariate Eigenvalue Problem Numerical examples. In this section, we present test results with the condition numbers and the backward errors of multivariate eigenvalue of A. All computations are carried out using MATLAB with precision ǫ In our experiments, the computed solutions are given by using Horst method. Example 6.. ([])The matrix A is given by A = with m = and P = {,3}. The computed multivariate eigenvalue and eigenvector of A are λ = [ ] T, x = [ ] T. Example 6.. ([]) The matrix A is given by A = with m = and P = {,3}. The computed multivariate eigenvalue and eigenvector of A are λ = [ ] T, x = [ ] T. Example 6.3. ([5]) The matrix A is given by A = with m = 3 and P = {,,}. The computed multivariate eigenvalue and eigenvector of A are λ = [ ] T,

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