76. LIO ND Z. I vector x 2 R n by a quasi-newton-type algorithm. llwright, Woodgate[2,3] and Liao[4] gave some necessary and sufficient conditions for

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1 Journal of Computational Mathematics, Vol.2, No.2, 23, LEST-SQURES SOLUTION OF X = D OVER SYMMETRIC POSITIVE SEMIDEFINITE MTRICES X Λ) nping Liao (Department of Mathematics, Hunan Normal University, Changsha 48) (Department of Mathematics and Information Science Changsha University, Changsha 43, China) Zhongzhi ai y (State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, cademy of Mathematics and System Sciences, Chinese cademy of Sciences, P.O. ox 279, eijing 8, China) bstract Least-squares solution of X = D with respect to symmetric positive semidefinite matrix X is considered. y making use of the generalized singular value decomposition, we derive general analytic formulas, and present necessary and sufficient conditions for guaranteeing the existence of the solution. y applying MTL 5.2, we give some numerical examples to show the feasibility and accuracy of this construction technique in the finite precision arithmetic. Key words: Least-squares solution, Matrix equation, Symmetric positive semidefinite matrix, Generalized singular value decomposition.. Introduction Denote by R n m the set of all real n m matrices, I k the identity matrix in R k k,or the set of all orthogonal matrices in R,SR the set of all symmetric matrices in R, and SR (SR ) the set of all symmetric positive semidefinite (definite) matrices in R. The notations O ( >O) represents that the matrix is a symmetric positive semidefinite (definite) matrix, represents the Moor-Penrose generalized inverse of a matrix, andk k denotes the Frobenius norm. We usebotho and to denote the zero matrix. The purpose of this paper is to study the least-squares problem of the matrix equation X = D with respect to X 2 SR, i.e., Problem I. For given matrices 2 R m n, 2 R n p, D 2 R m p, find a matrix ^X 2 SR such that k ^X Dk = min X2SR kx Dk: llwright[] first investigated a special case of Problem I, i.e., Problem. For given matrices ;D 2 R n m, find a matrix ^X 2 SR such that k ^X Dk = min X2SR kx Dk: The solution of Problem can be used as estimates of the inverse Hessian of a nonlinear differentiable function f : R n! R, which is to be minimized with respect to a parameter Λ Received ugust, 2. ) Subsidized by The Special Funds For Major State asic Research Project G y address: bzzlsec.cc.ac.cn.

2 76. LIO ND Z. I vector x 2 R n by a quasi-newton-type algorithm. llwright, Woodgate[2,3] and Liao[4] gave some necessary and sufficient conditions for the existence of solution of Problem as well as explicit formulas of the solution for some special cases. Dai and Lancaster[5] studied in detail another special case of Problem I, i.e., Problem. For given matrices 2 R n m, D 2 R m m, find a matrix ^X 2 SR such that k T ^X Dk = min X2SR k T X Dk: n inverse problem[6,7] arising in structural modification of the dynamic behaviour of a structure calls for solution of Problem. Dai and Lancaster[5] successfully solved Problem by using singular value decomposition(svd). Obviously, Problem I is a nontrivial generalization of Problems and, and the approachs adopted for solving Problems and in [-5] are not suitable for Problem I. In this paper, by applying the generalized singular value decomposition (GSVD) we will present necessary and sufficient conditions for the existence of the solution of Problem I, and give analytic expression of it, too. 2. Solutions of Problem I We first study the solution of Problem I when matrices ; ; D 2 R and ) = ) =n. Lemma 2. []. If ) =n, then Problem has a unique solution. Theorem 2.. If ; ; D 2 R and ) = ) = n, then Problem I has a unique solution. Proof. ecause ; 2 R and ) = ) =n,we easily know that )=n μ and kx Dk = k(x T )( T ) Dk = k X μ μ Dk; (2.) where μ = T, and μ X = X T. Now, it is obvious that μ X if and only if X. Therefore, by Lemma 2. and (2.) Problem I has a unique solution. To study the solvability of Problem I in general case, we decompose the given matrix pair [ T ;]by GSVD[8] as follows: where M is an n n nonsingular matrix, and I O O r O S ± = O C s O O O k r s O O O n k with r s m r s T = M± U T ; = M± V T ; (2.2) O O O ; O S ± = O C O O I O O O pk r s k r s r s k r s n k k = T ;); r = k ); s = T ) ) k; I and I are identity matrices, O and O are zero matrices, and S =diag(ff ;ff 2 ;:::;ff s ); S =diag(fi ;fi 2 ;:::;fi s ) >ff ff 2 ::: ff s > ; <fi» fi 2» :::» fi s < ; ff 2 i fi 2 i =(i =; 2;:::;s); U = (U ; U 2 ; U 3 ) 2 OR m m ; V = (V ; V 2 ; V 3 ) 2 OR p p : r s m r s pr k s k r s ;

3 Least-squares Solution of X = D Over Symmetric Positive Semidefinite Matrices X 77 Theorem 2.2. Let 2 R m n, 2 R n p, and D 2 R m p. ssume that the matrix pair [ T ;] has GSVD (2.2). Denote by D ij the matrix Ui T DV j(i; j = ; 2; 3). Then Problem I exists a solution if and onlyif ^X22 )=S DT 2; ^X22 ;S D 23); (2.3) where ^X22 is a unique minimizer of ks X 22 S D 22 k with respect to X 22 2 SR s s. Moreover, if Problem I exists a solution X, then this solution has the following expression: where 8Z 2 R k (n k) and X = M T Y YZ (YZ) T Z T YZ G 3 Y = X D 2 S ^X 22 S S DT 2 D 3 D 23 D T 3 D T 23S X 33 M ; (2.4) ; (2.5) X = D 2 S ^X 22 S DT 2 G ; (2.6) X 33 = D T ^X 23 S 22 S D 23 (D 3 D 2 S ^X 22 S D 23) T G (D 3 D 2 S ^X 22 S D 23)G 2 ; (2.7) 8G 2 SR r r ; 8G 2 2 SR (k r s) (k r s) ; 8G 3 2 SR (n k) (n k) ; and G satisfies the following condition: G )=G ;D 3 D 2 S ^X 22 S D 23): The following lemmas are necessary for proving Theorem 2.2. Lemma 2.2. Suppose that 2 R m n, 2 R n p and D 2 R m p, and there exists a minimizer ^X to Problem I. Then Proof. k ^X Dk = minx2sr It is obvious that k ^X Dk = minx2sr kx Dk = inf X2SR kx Dk: (2.8) kx Dk»inf X2SR kx Dk: On the other hand, if =, then ^X fi > forany f>, and k( ^X fi) Dk»k ^X Dk kfk < k ^X Dk f; if 6=, then by letting ffi =(2kk) f for any f>, we have ^X ffii >, and That is to say, k( ^X ffii) Dk»k ^X Dk kffik < k ^X Dk f: inf X2SR kx Dk»k ^X Dk = minx2sr kx Dk; and it follows straightforwardly that (2.8) holds. Lemma 2.3 (See [9], Theorem ). Let Q =(Q ij ) 2 2 be a 2 2 real block symmetric matrix, with Q and Q 22 square submatrices. Then Q is a symmetric positive semidefinite matrix if and onlyif Q ; Q 22 Q T 2Q Q 2 and Q )=Q ;Q 2 ): Lemma 2.3 directly results in the following Lemmas 2.4 and 2.5. Lemma 2.4. Let Q = (Q ij ) SR be a 2 2 real block symmetric matrix, with Q 2 SR r r the known submatrix, and Q 2 and Q 22 two unknown submatrices. Then there

4 78. LIO ND Z. I exist matrices Q 2 and Q 22 such that Q if and onlyif Q. Furthermore, all submatrices Q 2 and Q 22 such that Q can be expressed as Q 2 = Q Z; Q 22 = Z T Q Z G; where 8Z 2 R r (n r) and 8G 2 SR (n r) (n r). Lemma 2.5. Let Q = (Q ij ) R be a 3 3 real block symmetric matrix, with Q 2 R r r and Q 33 2 R s s two unknown submatrices, and the other blocks the known submatrices. Then there exist Q and Q 33 such that Q if and onlyif Q 22 and Q 22 )=Q 2 ;Q 22 ;Q 23 ): Furthermore, all submatrices Q and Q 33 such that Q can be expressed as ρ Q = Q 2 Q 22 Q 2 G ; Q 33 = Q 32 Q 22 Q 23 (Q 3 Q 32 Q 22 Q 2)G (Q 3 Q 2 Q 22 Q 23)G 2 ; where 8G 2 SR r r ; 8G 2 2 SR s s and G satisfies G )=G ;Q 3 Q 2 Q 22 Q 23): (2.9) Proof. It follows from Lemma 2.3 that Q Q 2 Q 3 Q 22 Q 2 Q 23 Q () Q 2 Q 22 Q 23 () Q 2 Q Q 3 Q 3 Q 32 Q 33 Q 32 Q 3 Q 33 Q Q () Q 22 ; 3 Q2 Q Q 3 Q 33 Q (Q 22 2;Q 23 ) and Q 22 ) = Q 22 ;Q 2 ;Q 23 ) 32 () Q 22 ; Q 22 ) = Q 2 ;Q 22 ;Q 23 ) and Q Q 2 Q Q 22 2 Q 3 Q 2 Q Q Q 3 Q 32 Q Q 22 2 Q 33 Q 32 Q Q : (2.) y Lemma 2.3 we know that there exist submatrices Q and Q 33 such that (2.) holds, and all submatrices Q and Q 33 such that (2.) holds can be expressed by (2.9) when Q 22. This completes the proof of this lemma. Proof of Theorem 2.2. For any X 2 SR, partition the matrix (M T XM)as It follows from (2.2) and (2.) that X X 2 X 3 X 4 M T X XM = T 2 X 22 X 23 X 24 C X T 3 X T 23 X 33 X 34 X T 4 X T 24 X T 34 X 44 r s k r s n k r s k r s n k : (2.) where kx Dk 2 = k± T M T XM± U T DV k 2 D X 2 S D 2 X 3 D 3 = D 2 S X 22 S D 22 S X 23 D 23 D 3 D 32 D 33 = ks X 22 S D 22 k 2 kx 2 S D 2 k 2 ks X 23 D 23 k 2 kx 3 D 3 k 2 C Λ ; 2 (2.2) C Λ = kd k 2 kd 2 k 2 kd 3 k 2 kd 32 k 2 kd 33 k 2 : (2.3)

5 Least-squares Solution of X = D Over Symmetric Positive Semidefinite Matrices X 79 From Theorem 2. we know that there exists a unique matrix ^X22 which minimizes ks X 22 S D 22 k with respect to X 22 2 SR s s. Obviously, if there exist submatrices X ;X 33 ;X 4 ;X 24 ;X 34 and X 44 such that the following matrix X: X D 2 S D 3 X 4 X = M T S DT 2 ^X 22 S D 23 X 24 C D T 3 D23S T X 33 X 34 M (2.4) X T 4 X T 24 X T 34 X 44 is a symmetric positive semidefinite matrix, then by (2.2) we know that the matrix X is certainly a solution of Problem I, where the sizes of submatrices X ;X 33 ;X 4 ;X 24 ;X 34 and X 44 are the same as (2.). Thus, if (2.3) holds, i.e., ^X22 ) = S DT 2; ^X22 ;S D 23); then by Lemma 2.5 we know that there exist submatrices X and X 33 such that X D 2 S D 3 S DT 2 ^X 22 S D 23 ; (2.5) D T 3 D23S T X 33 and that all submatrices X and X 33 such that(2.5) holds can be expressed as (2.5), (2.6) and (2.7). Therefore, by Lemma 2.4 we know that Problem I has a solution when ^X22 ) = S DT 2 ; ^X22 ;S D 23); and in this case the solution X of Problem I can be expressed by (2.4). Conversely, we only need to show that there is no minimum point to Problem I when Suppose that ^X22 ) 6= S DT 2; ^X22 ;S D 23): ^X22 ) 6= S DT 2 ; ^X22 ;S D 23); and that there is a minimum point, say X (M T X M)into μx X2 μ X3 μ μ X4 M T μx X M = T 2 μx 22 X23 μ X24 μ C 2 SR. Then by partitioning the matrix μx T 3 μx T 23 μx 33 X34 μ μx T 4 μx T 24 μx T 34 μx 44 r s k r s n k r s k r s n k ; (2.6) we obtain from Lemma 2.5 that X22 μ and X22 μ )= X μ T 2; X22 μ ; X23 μ ). If ( X μ T ; μ 2 X23 )=(S DT 2 ;S D 23); then μ X22 6= ^X22. Otherewise, we have ^X22 ) = μ X22 )= μ X T 2 ; μ X22 ; μ X23 ) = S DT 2 ; ^X22 ;S D 23); which obviously contradicts the initial assumption that and Hence, in both cases ^X22 ) 6= S DT 2; ^X22 ;S D 23): ( μ X T 2; μ X23 )=(S DT 2;S D 23) ( μ X T 2; μ X23 ) 6= (S DT 2;S D 23);

6 8. LIO ND Z. I it follows from (2.2), (2.6), (2.3)-(2.4) and Lemma 2.2 that kx Dk 2 = ks μ X22 S D 22 k 2 k μ X2 S D 2 k 2 ks μ X23 D 23 k 2 k μ X3 D 3 k 2 C Λ > ks ^X22 S D 22 k 2 C Λ = min X222SR s s = inf X222SR s s ks X 22 S D 22 k 2 C Λ ks X 22 S D 22 k 2 C Λ = inf X2SR kx Dk 2 = min X2SR kx Dk 2 : This contradicts the optimality ofx, and therefore, contradicts the existence of a minimum for Problem I when ^X22 ) 6= S DT 2; ^X22 ;S D 23): This completes the proof. Theorems 2. and 2.2 directly result in the following Corollary 2.. Corollary 2.. Let 2 R m n, 2 R n p and D 2 R m p. ssume that the matrix pair [ T ;] has GSVD (2.2). Denote by D ij the matrix (Ui T DV j)(i; j =; 2; 3). Then Problem I exists a unique solution if and only if ) =) =n: Moreover, if Problem I exists a unique solution ^X, then this solution has the expression ^X = M T ^X22 M ; where ^X22 is a unique minimizer of ks X 22 S D 22 k with respect to X 22 2 SR s s. 3. Numerical examples Without loss of generality, we take numerical examples only when ) = ) = n. ased on Corollary 2., we formulate the following algorithm to find the solution ^X of Problem I. lgorithm I. Let 2 R m n, 2 R n p and D 2 R m p satisfying ) = ) =n. Then Problem I can be solved in the following steps: Step. Make the GSVD of the matrix pair [ T ;] as (2.2), and determine the nonsingular matrix M, two orthogonal matrices U =(U 2 ; U 3 ); V =(V ; V 2 ); n m n and two diagonal matrices S and S. Step 2. Compute D 22 = U T DV 2 2. Step 3. Find ^X22 such thatks ^X22 S D 22 k = min X222SR ks X 22 S D 22 k by Steps : Step 3.. Let X 22 = R T R, where R is a real n n upper triangular matrix. Step 3.2. pply function leastsq in MTL 5.2 (in Optimization Toolbox) to find an n n upper triangular matrix ^R such that p n ks ^RT ^RS D 22 k = min R2UR ks R T RS D 22 k; where UR represents the set of all real n n upper triangular matrix. Step 3.3. Compute ^X22 = ^RT ^R. Step 4. Compute ^X = M T ^X22 M. Example. Let C C C = ; = ; D = : n

7 and the residual error of ^X as k ^X Dk =5: : and the residual error of ^X as k ^X Dk =5: : Least-squares Solution of X = D Over Symmetric Positive Semidefinite Matrices X 8 y lgorithm I we can obtain S = diag(: ; : ; : ; : ); S = diag(: ; : ; : ; : ); D 22 = 7: : : : : : : : C ; : : : : : : : : M = : : : : : : : : : : : : : : : : and ^X 22 = 6: : : : : : : : C : 6: : : : : : : : y Step 4 we obtain the solution ^X of Problem I as : : : : : : : : C ^X = : : : : : : : : C Example 2 []. Let 6 = ; = 4 3 ; D = 2 3 : :5 2 4 y lgorithm I we can obtain the solution of Problem I as : : : ^X = : : : ; : : : Example 3. Let ^X represent the solution of Problem I, min ( ^X) the smallest eigenvalue of ^X. Denote by hilb(n) the n-th order Hilbert matrix, pascal(n) the n-th order Pascal matrix, magic(n) the n-th order Magic matrix, toeplitz( : n) the n-th order Toeplitzmatrix whose first row is(; 2;:::;n), and hankel( : n) the n-th order Hankel matrix whose first row is(; 2;:::;n). For example, =2 =3 =4 =2 =3 =4 =5 C C hilb(4) = ; pascal(4) = ; =3 =4 =5 =6 3 6 =4 =5 =6 =7 4 2

8 82. LIO ND Z. I magic(4) = C C ; toeplitz( : 4) = ; C hankel(:4)= : We run a variety of numerical experiments about lgorithm I for the above classes of matrices of different dimensions. Some numerical results are listed in the following table. n D k ^X Dk min( ^X) 5 pascal(n) hilb(n) magic(n) toeplitz(:n) pascal(n) magic(n) e hankel(:n) pascal(n) hilb(n) pascal(n) hankel(:n) hilb(n) toeplitz(:n) hankel(:n) pascal(n) e pascal(n) toeplitz(:n) hilb(n) toeplitz(:n) hankel(:n) hilb(n) toeplitz(:n) hankel(:n) hilb(n) toeplitz(:n) hankel(:n) hilb(n) hankel(:n) toeplitz(:n) hilb(n) hankel(:n) toeplitz(:n) pascal(n) e hankel(:n) toeplitz(:n) hilb(n) From Examples -3, we clearly see that lgorithm I is feasible and accurate for solving Problem I. cknowledgements: Part of the first author's work was supported by State Key Laboratory of Scientific/Engineering Computing of Chinese cademy of Sciences. References [] J.C. llwright, Positive semidefinite matrices: Characterization via conical hulls and least-squares solution of a matrix equation, SIM J. Control Optim., 26(988), [2] J.C. llwright, K.G. Woodgate, Errata and ddendum to: Positive semidefinite matrices: Characterization via conical hulls and least-squares solution of a matrix equation, SIM J. Control Optim., 28(99), [3] K.G. Woodgate, Least-squares solution of F = P G over positive semidefinite matrices P, Linear lgebra ppl., 245(996), 7-9. [4].P. Liao, On the least squares problem of a matrix equation, J. Comput. Math., 7(999), [5] H. Dai, P. Lancaster, Linear matrix equations from an inverse problem of vibration theory, Linear lgebra ppl., 246(996), [6]. erman, Mass matrix correction using an incomplete set of measure model, I J., 7(979), [7]. erman and E.J. Nagy, Improvement of a large analytical model using test data, I J., 2(983), [8] C.C. Paige, M.. Saunders, Towards a generalized singular value decomposition, SIM J. Numer. nal., 8(98), [9]. lbert, Conditions for positive and nonnegative in terms of pseudoinverses, SIM J. ppl. Math., 7(969),