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A note on the Frobenius conditional number with positive definite matrices
Journal of Inequalities and Applications volume 2011, Article number: 120 (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
In this article, we use the following notations. Let ℂn×nand ℝn×nbe the space of n×n complex and real matrices, respectively. The identity matrix in ℂn×nis denoted by I = I n . Let AT, Ā, AH, and tr(A) denote the transpose, the conjugate, the conjugate transpose, and the trace of a matrix A, respectively. Re(a) stands for the real part of a number a. The Frobenius inner product < ·, · > F in ℂm×nis defined as < A,B > F = Re(tr(BHA)), for A,B ∈ ℂm×n, i.e., < A,B > is the real part of the trace of BHA. The induced matrix norm is , which is called the Frobenius (Euclidean) norm. The Frobenius inner product allows us to define the consine of the angle between two given real n × n matrices as
The cosine of the angle between two real n × n depends on the Frobenius inner product and the Frobenius norms of given matrices. A matrix A ∈ ℂn×nis Hermitian if AH= A. An Hermitian matrix A is said to be positive semidefinite or nonnegative definite, written as A ≥ 0, if (see, e.g., [, p. 159])
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 ∈ ℂnto 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, < ·, ·>).
Lemma 1.1 (). For any a, b, x ∈ H, there is
It is clear that for a = b, (1.4) becomes the standard Schwarz inequality
with equality if and only if there exists a scalar λ such that x = λa.
Also Dragomir  has stated the following inequality.
Lemma 1.2. For any a,b,x ∈ H, and x ≠ 0, there is the following
Dannan  showed the following inequality by using arithmetic-geometric inequality.
Lemma 1.3. For n-square positive definite matrices A and B,
where m is a positive integer.
By taking A = I, B = A-1, and m = 1 in (1.7), we obtain
In , Türkmen and Ulukök proposed the following,
Lemma 1.4. Let both A and B be n-square positive definite matrices, then
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
Theorem 2.1. Let A be a positive definite real matrix, α be any real number. Then,
where μ F (A) is the Frobenius conditional number of A.
Proof. Let X, A, B be positive definite real matrices. From Lemma 1.2, we have the following
Let B = A-1, then (2.4) turns into
Since both X and A are positive definite, we have
where μ F (A) is the Frobenius condition number of A.
By taking X = Aα(α is an arbitrary real number) into (2.6), there exists
Thus, it follows that
From (1.9), by replacing A with A1-α, we get
Taking (2.10) into (2.8), we can write
In particular, let α = 1, and by taking it into (2.7), we have the following
Note, when , (2.12) becomes
Taking α = 1 into (2.12), we obtain that
(2.15), (2.16) can be found in .
Here trA = 2.5, detA = 1 and n = 2. Then, from (2.15) and (2.16), we obtain two lower bounds of μ F (A):
Taking α = 1/4 into (2.1) and (2.2) from Theorem 2.1, another two lower bounds are obtained as follows:
In fact, μ F (A) = 4.25. Thus, Theorem 2.1 is indeed a generalization of (2.15) and (2.16) given in .
Lemma 2.3. Let a1, a2,..., a n be positive numbers, and
Then, f(x) is monotonously increasing for x ∈ [0,+ ∞).
Proof. It is obvious that
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, ∞).
Theorem 2.4. Let A be a positive definite real matrix. Then
Proof. Since A is positive definite, from Lemma 1.4, we have
Let all the positive real eigenvalues of A be λ1, λ2,..., λ n > 0. Then, the eigenvalues of A-1 are 1/λ1, 1/λ2,..., 1/λ n > 0, the eigenvalues of A2 are , and the eigenvalue of A-2 are . Next, we will prove the following by induction.
In case n = 1, it is obvious that (2.20) holds.
In case . From Lemma 2.3, 2 + (λ1/λ2)2 + (λ2/λ1)2 ≥ 2 + λ1/λ2 + λ2/λ1 = (λ1 + λ2)(1/λ1 + 1/λ2). Thus, (2.20) holds. Suppose that (2.20) holds, when n = k, i.e.,
In case n = k + 1,
By Lemma 2.3, we get
Thus, when n = k + 1, (2.20) holds. On the other hand, and . Therefore,
Taking (2.25) into (2.19), we obtain
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
Definition 3.1 (see ). Let A = (a ij )p xq∈ ℝp×q. A is a centrosymmetric matrix, if
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 ):
Lemma 3.2. Let A = (a ij )n×n(n = 2m) be centrosymmetric. Then, A has the form,
Where B,C ∈ ℂm×m, .
Lemma 3.3. Let Abeann × n(n = 2m) centrosymmetric positive definite matrix with the following form
where B,C ∈ ℝm×m. Then there exists an orthogonal matrix P such that
where M = B - J m C, N = B + J m C and M-1, N-1are positive definite matrices.
Proof. From Lemma 3.2, there exists an orthogonal matrix P such that
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.
Lemma 3.4. Let A, B are positive definite real matrices. Then,
Proof. Let X be positive definite. By Lemma 1.2, we have
By replacing X, B with I n and B-1, respectively, in inequality (3.4), we can obtain
From Schwarz inequality (1.5),
By taking (3.7) into (3.6), we have
Theorem 3.5. Let A be a centrosymmetric positive definite matrix with the form
Let M = B - J m C, N = B + J m C. Then,
Proof. By Lemma 3.2, there exists an orthogonal matrix P such that
From Lemma 3.4,
Here n = 4, m = 2, det (A) = 256, tr A = 20, tr M = 4, tr N = 16, det(M) = 4, det(N) = 64. From Theorem 3.5, a lower bound of μ F (A) is as follows:
On the other hand, the lower bounds of μ F (A) in (2.15) and (2.16) provided by  are
It can easily be seen that, in this example, the best lower bound is the first one given by Theorem 3.5.
Zhang F: Matrix Theory: Basic Results and Techniques. Springer, New York; 1999.
Horn RA, Johnson CR: Matrix Analysis. Cambridge University Press, Cambridge; 1985.
Buzano ML: Gemeralizzazione della diseguaglianza di Cauchy-Schwarz. Rendiconti del Seminario Matematico Universita e Politecnico di Torino 1971, 31: 405–409. (1974) (Italian)
Dragomir SS: Refinemens of Buzano's and Kurepa's inequalities in inner product spaces. Facta Universitatis 2005, 20: 65–73.
Danna FM: Matrix and operator inequalites. J Inequal Pure Appl Math 2001.,2(3): article 34
Ramazan T, Zübeyde U: On the Frobenius condition number of positive definite matrices. J Inequal Appl 2010. Article ID 897279 (2010)
Liu ZY: Some properties of centrosymmetric matrices. Appl Math Comput 2002, 141: 17–26.
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.
The authors declare that they have no competing interests.
Hongyi Li carried out studies on the linear algebra and matrix theory with applications, and drafted the manuscript. Zongsheng Gao read the manuscript carefully and gave valuable suggestions and comments. Di Zhao provided numerical examples, which demonstrated the main results of this article. All authors read and approved the final manuscript.
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Li, H., Gao, Z. & Zhao, D. A note on the Frobenius conditional number with positive definite matrices. J Inequal Appl 2011, 120 (2011). https://doi.org/10.1186/1029-242X-2011-120
- Lower Bound
- Condition Number
- Positive Real Number
- Orthogonal Matrix
- Matrix Norm