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# On The Frobenius Condition Number of Positive Definite Matrices

*Journal of Inequalities and Applications*
**volume 2010**, Article number: 897279 (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.

## 1. Introduction and Preliminaries

The quantity

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.

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

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 trace of a square matrix (the sum of its main diagonal entries, or, equivalently, the sum of its eigenvalues) is denoted by . Let be any matrix. The Frobenius (Euclidean) norm of matrix is

It is also equal to the square root of the matrix trace of , that is,

The Frobenius condition number is defined by . In the Frobenius inner product is defined by

for which we have the associated norm that satisfies . The Frobenius inner product allows us to define the cosine of the angle between two given real matrices as

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.

Buzano in [3] obtained the following extension of the celebrated Schwarz inequality in a real or complex inner product space :

for any . It is clear that for , the above inequality becomes the standard Schwarz inequality

with equality if and only if there exists a scalar ( or ) such that . Also Dragomir in [4] has stated the following inequality:

where . 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 be positive definite real matrix. Then

where 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 . Then we write

where and denotes the Frobenius norm of matrix. Then we get

In particular, in inequality (2.3), if we take , then we have

Also, if and are positive definite real matrices, then we get

where is the Frobenius condition number of .

Note that Dannan in [7] has showed the following inequality by using the well known arithmetic-geometric inequality, for -square positive definite matrices and :

where is a positive integer. If we take , , and in (2.6), then we get

That is,

In particular, if we take in (2.5) and (2.8), then we arrive at

Also, from the well-known Cauchy-Schwarz inequality, since , one can obtain

Furthermore, from arithmetic-geometric means inequality, we know that

Since , we write . Thus by combining (2.9) and (2.11) we arrive at

Lemma 2.2.

Let be a positive definite matrix. Then

Proof.

Let be positive real numbers for . We will show that

for all . The proof is by induction on . If ,

Assume that inequality (2.14) holds for some . that is,

Then

The first inequality follows from induction assumption and the inequality

for positive real numbers and .

Theorem 2.3.

Let be positive definite real matrix. Then

where is the Frobenius condition number.

Proof.

Let and . Then from inequality (1.9) we can write

where and denotes the Frobenius norm. Then we get

Set . Then

Since by Lemma 2.2 and

Hence

Let be positive real numbers for . 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 , then

Assume that the inequality (2.28) holds for some . Then

The first inequality follows from induction assumption and the second inequality follows from the inequality

for positive real numbers and .

Theorem 2.4.

Let and 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:

where . Since , it follows that

Let be identity matrix and and 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 matrices, we get

In particular, for we obtain that

Also, Chehab and Raydan in [8] have proved the following inequality for positive definite real matrix by using the well-known Cauchy-Schwarz inequality:

By combining inequalities (2.39) and (2.40), we arrive at

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

Theorem 2.5.

Let be a positive definite real matrix. Then

Proof.

According to the well-known Cauchy-Schwarz inequality, we write

where are eigenvalues of . That is,

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 is symmetric matrix, a necessary condition to be positive semidefinitematrix is that .

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

where , , and .

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

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 .

Since . Thus, we get

All these bounds can be combined with the results which are previously obtained to produce practical bounds for . 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 , , , 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.

## References

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*Matrix Theory: Basic Results and Techniques*. Springer, New York, NY, USA; 1999.Buzano ML: Generalizzazione della diseguaglianza di Cauchy-Schwarz.

*Rendiconti del Seminario Matematico Università e Politecnico di Torino*1974, 31 (1971/73): 405–409.Dragomir SS: Refinements of Buzano's and Kurepa's inequalities in inner product spaces.

*Facta Universitatis*2005, (20):65–73.Precupanu T: On a generalization of Cauchy-Buniakowski-Schwarz inequality.

*Annals of the " Alexandru Ioan Cuza" University of Iaşi*1976, 22(2):173–175.Richard U: Sur des inegalites du type Wirtinger et leurs application aux equations differentielles ordinaries.

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*Linear Algebra and its Applications*1990, 135: 171–179. 10.1016/0024-3795(90)90120-2Wolkowicz H, Styan GPH: Bounds for eigenvalues using traces.

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## 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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Türkmen, R., Ulukök, Z. On The Frobenius Condition Number of Positive Definite Matrices.
*J Inequal Appl* **2010**, 897279 (2010). https://doi.org/10.1155/2010/897279

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