On The Frobenius Condition Number of Positive Definite Matrices

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 897279, 11 pages doi:10.1155/2010/897279 Research Article On The Frobenius Condition Number of Positive Definite Matrices Ramazan Türkmen and Zübeyde Ulukök Department of Mathematics, Science Faculty, Selçuk University, 42003 Konya, Turkey Correspondence should be addressed to Ramazan Türkmen, Received 19 February 2010; Revised 4 May 2010; Accepted 15 June 2010 Academic Editor: S. S. Dragomir Copyright q 2010 R. Türkmen and Z. Ulukök. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We present some lower bounds for the Frobenius condition number of a positive definite matrix depending on trace, determinant, and Frobenius norm of a positive definite matrix and compare these results with other results. Also, we give a relation for the cosine of the angle between two given real matrices. 1. Introduction and Preliminaries The quantity κA  ⎧   ⎨AA−1  if A is nonsingular, ⎩∞ if A is singular 1.1 is called the condition number for matrix inversion with respect to the matrix norm ·. Notice that κA  A−1 A ≥ A−1 A  I ≥ 1 for any matrix norm see, e.g., 1, page 336. The condition number κA  AA−1  of a nonsingular matrix A plays an important role in the numerical solution of linear systems since it measures the sensitivity of the solution of linear systems Ax  b to the perturbations on A and b. There are several methods that allow to find good approximations of the condition number of a general square matrix. Let Cn×n and Rn×n be the space of n × n complex and real matrices, respectively. The identity matrix in Cn×n is denoted by I  In . A matrix A ∈ Cn×n is Hermitian if A∗  A, 2 Journal of Inequalities and Applications where A∗ denotes the conjugate transpose of A. A Hermitian matrix A is said to be positive semidefinite or nonnegative definite, written as A ≥ 0, if see, e. g., 2, p.159 x∗ Ax ≥ 0, ∀x ∈ Cn , 1.2 A is further called positive definite, symbolized A > 0, if the strict inequality in 1.2 holds for all nonzero x ∈ Cn . An equivalent condition for A ∈ Cn×n to be positive definite is that A is Hermitian and all eigenvalues of A are positive real numbers. The trace of a square matrix A the sum of its main diagonal entries, or, equivalently, the sum of its eigenvalues is denoted by trA. Let A be any m × n matrix. The Frobenius Euclidean norm of matrix A is ⎛ AF  ⎝ m  n  ⎞1/2 2 aij ⎠ . 1.3 i1 j1 It is also equal to the square root of the matrix trace of AA∗ , that is, AF  √ tr AA∗ . 1.4 The Frobenius condition number is defined by κF A  AF A−1 F . In Rn×n the Frobenius inner product is defined by A, B F  tr AT B 1.5 for which we have the associated norm that satisfies A2F  A, A F . The Frobenius inner product allows us to define the cosine of the angle between two given real n × n matrices as cosA, B  A, B F . AF BF 1.6 The cosine of the angle between two real n × n matrices depends on the Frobenius inner product and the Frobenius norms of given matrices. Then, the inequalities in inner product spaces are expandable to matrices by using the inner product between two matrices. Buzano in 3 obtained the following extension of the celebrated Schwarz inequality in a real or complex inner product space H; ·, · : | a, x x, b | ≤ 1 ab  | a, b |x2 , 2 1.7 for any a, b, x ∈ H. It is clear that for a  b, the above inequality becomes the standard Schwarz inequality | a, x |2 ≤ a2 x2 , a, x ∈ H, 1.8 Journal of Inequalities and Applications 3 with equality if and only if there exists a scalar λ ∈ K K  R or C such that x  λa. Also Dragomir in 4 has stated the following inequality: a, x x, b − 2 x a, b 2 ≤ ab , 2 1.9 where a, b, x ∈ H, x /  0. Furthermore, Dragomir 4 has given the following inequality, which is mentioned by Precupanu in 5, has been showed independently of Buzano, by Richard in 6: 1 1  a, b − abx2 ≤ a, x x, b ≤  a, b  abx2 . 2 2 1.10 As a consequence, in next section, we give some bounds for the Frobenius condition numbers and the cosine of the angle between two positive definite matrices by considering inequalities given for inner product space in this section. 2. Main Results Theorem 2.1. Let A be positive definite real matrix. Then 2 tr A det A1/n − n ≤ κF A, 2.1 where κF A is the Frobenius condition number. Proof. We can extend inequality 1.9 given in the previous section to matrices by using the Frobenius inner product as follows: Let A, B, X ∈ Rn×n . Then we write A, X F X, B F − X2F A, B F AF BF ≤ , 2 2 2.2 where A, X F  tr AT X, and  · F denotes the Frobenius norm of matrix. Then we get     tr AT X tr X T B X2F   tr AT B AF BF ≤ . − 2 2 2.3 In particular, in inequality 2.3, if we take B  A−1 , then we have     tr AT X tr X T A−1 X2F     AF A−1 F tr AT A−1 ≤ . − 2 2 2.4 4 Journal of Inequalities and Applications Also, if X and A are positive definite real matrices, then we get   tr AXtr XA−1 X2F   AF A−1 F κF A n ≤  , − 2 2 2 2.5 where κF A is the Frobenius condition number of A. Note that Dannan in 7 has showed the following inequality by using the well known arithmetic-geometric inequality, for n-square positive definite matrices A and B: ndet A det Bm/n ≤ tr Am Bm , 2.6 where m is a positive integer. If we take A  X, B  A−1 , and m  1 in 2.6, then we get n det X det A−1 1/n ≤ tr XA−1 . 2.7 That is,   det X 1/n n ≤ tr XA−1 . det A 2.8 In particular, if we take X  I in 2.5 and 2.8, then we arrive at tr A tr A−1 n ≤ κF A, − n 2  n 1 det A 1/n 2.9 ≤ tr A−1 . Also, from the well-known Cauchy-Schwarz inequality, since n2 ≤ tr A tr A−1 , one can obtain 0<n≤2 tr A tr A−1 − n ≤ κF A. n 2.10 Furthermore, from arithmetic-geometric means inequality, we know that ndet A1/n ≤ tr A. 2.11 Since n ≤ tr A/det A1/n , we write 0 < n ≤ 2 tr A/det A1/n − n. Thus by combining 2.9 and 2.11 we arrive at 2 tr A det A1/n − n ≤ κF A. 2.12 Journal of Inequalities and Applications 5 Lemma 2.2. Let A be a positive definite matrix. Then tr A3/2 tr A−1/2 n − ≥ 0. tr A 2 2.13 Proof. Let λi be positive real numbers for i  1, 2, . . . , n. We will show that      k k k   k  −1/2 3/2 ≥ λi λi λi 2 i1 i1 i1 2.14 for all k  1, 2, . . . , n. The proof is by induction on k. If k  1, −1/2 λ3/2 1 · λ1  λ1 ≥ 1 λ1 . 2 2.15 Assume that inequality 2.14 holds for some k. that is,      k k k   k  −1/2 3/2 ≥ λi λi λi . 2 i1 i1 i1 2.16 Then       k1 k1 k k     −1/2 −1/2 −1/2 3/2 3/2 3/2  λi λi λi  λk1 λi  λ (...truncated)