On The Frobenius Condition Number of Positive Definite Matrices

Open Access
Research Article

Abstract

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.

Keywords

Condition Number Positive Real Number Product Space Diagonal Entry Positive Semidefinite 

1. Introduction and Preliminaries

The quantity

is called the condition number for matrix inversion with respect to the matrix norm Open image in new window . Notice that Open image in new window for any matrix norm (see, e.g., [1, page 336]). The condition number Open image in new window of a nonsingular matrix Open image in new window plays an important role in the numerical solution of linear systems since it measures the sensitivity of the solution of linear systems Open image in new window to the perturbations on Open image in new window and Open image in new window . There are several methods that allow to find good approximations of the condition number of a general square matrix.

Let Open image in new window and Open image in new window be the space of Open image in new window complex and real matrices, respectively. The identity matrix in Open image in new window is denoted by Open image in new window . A matrix Open image in new window is Hermitian if Open image in new window , where Open image in new window denotes the conjugate transpose of Open image in new window . A Hermitian matrix Open image in new window is said to be positive semidefinite or nonnegative definite, written as Open image in new window , if (see, e. g., [2], p.159)

Open image in new window is further called positive definite, symbolized Open image in new window , if the strict inequality in (1.2) holds for all nonzero Open image in new window . An equivalent condition for Open image in new window to be positive definite is that Open image in new window is Hermitian and all eigenvalues of Open image in new window are positive real numbers.

The trace of a square matrix Open image in new window (the sum of its main diagonal entries, or, equivalently, the sum of its eigenvalues) is denoted by Open image in new window . Let Open image in new window be any Open image in new window matrix. The Frobenius (Euclidean) norm of matrix Open image in new window is
It is also equal to the square root of the matrix trace of Open image in new window , that is,
The Frobenius condition number is defined by Open image in new window . In Open image in new window the Frobenius inner product is defined by
for which we have the associated norm that satisfies Open image in new window . The Frobenius inner product allows us to define the cosine of the angle between two given real Open image in new window matrices as

The cosine of the angle between two real Open image in new window 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 Open image in new window :
for any Open image in new window . It is clear that for Open image in new window , the above inequality becomes the standard Schwarz inequality
with equality if and only if there exists a scalar Open image in new window ( Open image in new window or Open image in new window ) such that Open image in new window . Also Dragomir in [4] has stated the following inequality:
where Open image in new window . 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]:

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 Open image in new window be positive definite real matrix. Then

where Open image in new window 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 Open image in new window . Then we write
where Open image in new window and Open image in new window denotes the Frobenius norm of matrix. Then we get
In particular, in inequality (2.3), if we take Open image in new window , then we have
Also, if Open image in new window and Open image in new window are positive definite real matrices, then we get

where Open image in new window is the Frobenius condition number of Open image in new window .

Note that Dannan in [7] has showed the following inequality by using the well known arithmetic-geometric inequality, for Open image in new window -square positive definite matrices Open image in new window and Open image in new window :
where Open image in new window is a positive integer. If we take Open image in new window , Open image in new window , and Open image in new window in (2.6), then we get
In particular, if we take Open image in new window in (2.5) and (2.8), then we arrive at
Also, from the well-known Cauchy-Schwarz inequality, since Open image in new window , one can obtain
Furthermore, from arithmetic-geometric means inequality, we know that
Since Open image in new window , we write Open image in new window . Thus by combining (2.9) and (2.11) we arrive at

Lemma 2.2.

Let Open image in new window be a positive definite matrix. Then

Proof.

Let Open image in new window be positive real numbers for Open image in new window . We will show that
Assume that inequality (2.14) holds for some Open image in new window . that is,
The first inequality follows from induction assumption and the inequality

for positive real numbers Open image in new window and Open image in new window .

Theorem 2.3.

Let Open image in new window be positive definite real matrix. Then

where Open image in new window is the Frobenius condition number.

Proof.

Let Open image in new window and Open image in new window . Then from inequality (1.9) we can write
where Open image in new window and Open image in new window denotes the Frobenius norm. Then we get
Let Open image in new window be positive real numbers for Open image in new window . Now we will show that the left side of inequality (2.19) is positive, that is,
By the arithmetic-geometric mean inequality, we obtain the inequality
So, it is enough to show that
Equivalently,
We will prove by induction. If Open image in new window , then
Assume that the inequality (2.28) holds for some Open image in new window . Then
The first inequality follows from induction assumption and the second inequality follows from the inequality

for positive real numbers Open image in new window and Open image in new window .

Theorem 2.4.

Let Open image in new window and Open image in new window be positive definite real matrices. Then
In particular,

Proof.

We consider the right side of inequality (1.10):
We can extend this inequality to matrices as follows:
Let Open image in new window be identity matrix and Open image in new window and Open image in new window positive definite real matrices. According to inequality (2.36), it follows that
From the definition of the cosine of the angle between two given real Open image in new window matrices, we get
In particular, for Open image in new window we obtain that
Also, Chehab and Raydan in [8] have proved the following inequality for positive definite real matrix Open image in new window by using the well-known Cauchy-Schwarz inequality:
By combining inequalities (2.39) and (2.40), we arrive at

and since Open image in new window and Open image in new window , we arrive at Open image in new window . Therefore, proof is completed.

Theorem 2.5.

Let Open image in new window be a positive definite real matrix. Then

Proof.

According to the well-known Cauchy-Schwarz inequality, we write
Also, from definition of the Frobenius norm, we get
Then, we obtain that
Likewise,
When inequalities (2.40) and (2.47) are combined, they produce the following inequality:
Therefore, finally we get

Note that Tarazaga in [9] has given that if Open image in new window is symmetric matrix, a necessary condition to be positive semidefinitematrix is that Open image in new window .

Wolkowicz and Styan in [10] have established an inequality for the spectral condition numbers of symetric and positive definite matrices:

where Open image in new window , Open image in new window , and Open image in new window .

Also, Chehab and Raydan in [8] have given the following practical lower bound for the Frobenius condition number Open image in new window :

Now let us compare the bound in (2.49) and the lower bound obtained by the authors in [8] for the Frobenius condition number of positive definite matrix Open image in new window .

All these bounds can be combined with the results which are previously obtained to produce practical bounds for Open image in new window . In particular, combining the results given by Theorems 2.1, 2.3, and 2.5 and other results, we present the following practical new bound:

Example we have.

Here Open image in new window , Open image in new window , Open image in new window , and have Open image in new window . Then, we obtain that Open image in new window , Open image in new window , Open image in new window , and Open image in new window . Since Open image in new window , in this example, the best lower bound is the second lower bound given by Theorem 2.3.

Notes

Acknowledgments

The authors thank very much the associate editors and reviewers for their insightful comments and kind suggestions that led to improving the presentation. This study was supported by the Coordinatorship of Selçuk University's Scientific Research Projects.

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Copyright information

© R. Türkmen and Z. Ulukök 2010

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.Department of MathematicsScience Faculty, Selçuk UniversityKonyaTurkey

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