A note on the Frobenius conditional number with positive definite matrices
© Li et al; licensee Springer. 2011
Received: 12 June 2011
Accepted: 24 November 2011
Published: 24 November 2011
In this article, we focus on the lower bounds of the Frobenius condition number. Using the generalized Schwarz inequality, we present some lower bounds for the Frobenius condition number of a positive definite matrix depending on its trace, determinant, and Frobenius norm. Also, we give some results on a kind of matrices with special structure, the positive definite matrices with centrosymmetric structure.
1 Introduction and preliminaries
A is further called positive definite, symbolized A > 0, if the strict inequality in (1.2) holds for all nonzero x ∈ ℂ n . An equivalent condition for A ∈ ℂ n to be positive definite is that A is Hermitian and all eigenvalues of A are positive real numbers.
is called the condition number of matrix μ(A) with respect to the matrix norm || · ||. Notice that μ(A) = ||A-1|| · ||A|| ≥ ||A-1A|| = ||I|| ≥ 1 for any matrix norm (see, e.g., [, p. 336]). The condition number μ(A) 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 system 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.
We first introduce some inequalities. Buzano  obtained the following extension of the celebrated Schwarz inequality in a real or complex inner product space (H, < ·, ·>).
with equality if and only if there exists a scalar λ such that x = λa.
Also Dragomir  has stated the following inequality.
Dannan  showed the following inequality by using arithmetic-geometric inequality.
where m is a positive integer.
In , Türkmen and Ulukök proposed the following,
As a consequence, in the following section, we give some bounds for the Frobenius condition numbers by considering inequalities given in this section.
2 Main results
where μ F (A) is the Frobenius conditional number of A.
where μ F (A) is the Frobenius condition number of A.
(2.15), (2.16) can be found in .
In fact, μ F (A) = 4.25. Thus, Theorem 2.1 is indeed a generalization of (2.15) and (2.16) given in .
Then, f(x) is monotonously increasing for x ∈ [0,+ ∞).
When a i ≥ 1, and x ∈ [0, + ∞), , and . Thus, .
When 0 < a i < 1, and x ∈ [0, + ∞), we have , and . Thus, . Therefore, f'(x) ≥ 0, and f(x) is increasing for x ∈ [0, ∞).
In case n = 1, it is obvious that (2.20) holds.
Remark 2.5. (2.17) can be extended to any 0 < α ≤ 1, i.e., .
3 The Frobenius condition number of a centrosymmetric positive definite matrix
where J n = (e n , en-1,..., e1), e i denotes the unit vector with the i-th entry 1.
Using the partition of matrix, the central symmetric character of a square centrosym-metric matrix can be described as follows (see ):
Where B,C ∈ ℂm×m, .
where M = B - J m C, N = B + J m C and M-1, N-1are positive definite matrices.
Since A is positive definite, then any eigenvalue of A is positive real number. Thus, the eigenvalues of H are all positive real numbers. That is to say, all eigenvalues of M and N are positive real numbers. Thus, M and N are both positive definite. It is obvious that M-1 and N-1 are positive definite matrices.
It can easily be seen that, in this example, the best lower bound is the first one given by Theorem 3.5.
The authors would like to thank two anonymous referees for reading this article carefully, providing valuable suggestions and comments which have been implemented in this revised version. This study was supported by the National Science Foundation of China No. 11171013 and the Nation Science Foundation of China No. 60831001.
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