- Research Article
- Open Access

# On The Frobenius Condition Number of Positive Definite Matrices

- Ramazan Türkmen
^{1}Email author and - Zübeyde Ulukök
^{1}

**2010**:897279

https://doi.org/10.1155/2010/897279

© R. Türkmen and Z. Ulukök 2010

**Received:**19 February 2010**Accepted:**15 June 2010**Published:**6 July 2010

## 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

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

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

The cosine of the angle between two real 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.

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.

where is the Frobenius condition number.

Proof.

where is the Frobenius condition number of .

Lemma 2.2.

Proof.

for positive real numbers and .

Theorem 2.3.

where is the Frobenius condition number.

Proof.

for positive real numbers and .

Theorem 2.4.

Proof.

and since and , we arrive at . Therefore, proof is completed.

Theorem 2.5.

Proof.

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

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 .

Example we have.

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

## Declarations

### 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.

## Authors’ Affiliations

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## Copyright

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.