- Research Article
- Open Access
- Published:
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
- 1.
Horn RA, Johnson CR: Matrix Analysis. Cambridge University Press, Cambridge, UK; 1985:xiii+561.
- 2.
Zhang F: Matrix Theory: Basic Results and Techniques. Springer, New York, NY, USA; 1999.
- 3.
Buzano ML: Generalizzazione della diseguaglianza di Cauchy-Schwarz. Rendiconti del Seminario Matematico Università e Politecnico di Torino 1974, 31 (1971/73): 405–409.
- 4.
Dragomir SS: Refinements of Buzano's and Kurepa's inequalities in inner product spaces. Facta Universitatis 2005, (20):65–73.
- 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.
- 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.
- 7.
Dannan FM: Matrix and operator inequalities. Journal of Inequalities in Pure and Applied Mathematics 2001., 2(3, article 34):
- 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.2008.06.006
- 9.
Tarazaga P: Eigenvalue estimates for symmetric matrices. Linear Algebra and its Applications 1990, 135: 171–179. 10.1016/0024-3795(90)90120-2
- 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-X
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.
Author information
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Keywords
- Condition Number
- Positive Real Number
- Product Space
- Diagonal Entry
- Positive Semidefinite
