# On The Frobenius Condition Number of Positive Definite Matrices

## 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 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.

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 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.

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 Open image in new window is the Frobenius condition number.

Proof.

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

Lemma 2.2.

Proof.

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

Theorem 2.3.

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

Proof.

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

Theorem 2.4.

Proof.

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.

Proof.

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 .

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

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.

## References

- 1.Horn RA, Johnson CR:
*Matrix Analysis*. Cambridge University Press, Cambridge, UK; 1985:xiii+561.CrossRefMATHGoogle Scholar - 2.Zhang F:
*Matrix Theory: Basic Results and Techniques*. Springer, New York, NY, USA; 1999.CrossRefMATHGoogle Scholar - 3.Buzano ML: Generalizzazione della diseguaglianza di Cauchy-Schwarz.
*Rendiconti del Seminario Matematico Università e Politecnico di Torino*1974, 31 (1971/73): 405–409.MathSciNetMATHGoogle Scholar - 4.Dragomir SS: Refinements of Buzano's and Kurepa's inequalities in inner product spaces.
*Facta Universitatis*2005, (20):65–73.Google Scholar - 5.Precupanu T: On a generalization of Cauchy-Buniakowski-Schwarz inequality.
*Annals of the " Alexandru Ioan Cuza" University of Iaşi*1976, 22(2):173–175.MathSciNetMATHGoogle Scholar - 6.Richard U: Sur des inegalites du type Wirtinger et leurs application aux equations differentielles ordinaries.
*Proceedings of the Colloquium of Analysis, August 1972, Rio de Janeiro, Brazil*233–244.Google Scholar - 7.Dannan FM: Matrix and operator inequalities.
*Journal of Inequalities in Pure and Applied Mathematics*2001., 2(3, article 34):Google Scholar - 8.Chehab J-P, Raydan M: Geometrical properties of the Frobenius condition number for positive definite matrices.
*Linear Algebra and its Applications*2008, 429(8–9):2089–2097. 10.1016/j.laa.2008.06.006MathSciNetCrossRefMATHGoogle Scholar - 9.Tarazaga P: Eigenvalue estimates for symmetric matrices.
*Linear Algebra and its Applications*1990, 135: 171–179. 10.1016/0024-3795(90)90120-2MathSciNetCrossRefMATHGoogle Scholar - 10.Wolkowicz H, Styan GPH: Bounds for eigenvalues using traces.
*Linear Algebra and its Applications*1980, 29: 471–506. 10.1016/0024-3795(80)90258-XMathSciNetCrossRefMATHGoogle Scholar

## Copyright information

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.