On The Frobenius Condition Number of Positive Definite Matrices

The quantity

(1.1)

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)

(1.2)

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

(1.3)

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

(1.4)

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

(1.5)

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

(1.6)

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 :

(1.7)

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

(1.8)

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

(1.9)

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]:

(1.10)

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.

Theorem 2.1.

Let be positive definite real matrix. Then

(2.1)

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

(2.2)

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

(2.3)

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

(2.4)

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

(2.5)

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 :

(2.6)

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

(2.7)

That is,

(2.8)

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

(2.9)

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

(2.10)

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

(2.11)

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

(2.12)

Lemma 2.2.

Let be a positive definite matrix. Then

(2.13)

Proof.

Let be positive real numbers for . We will show that

(2.14)

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

(2.15)

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

(2.16)

Then

(2.17)

The first inequality follows from induction assumption and the inequality

(2.18)

for positive real numbers and .

Theorem 2.3.

Let be positive definite real matrix. Then

(2.19)

where is the Frobenius condition number.

Proof.

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

(2.20)

where and denotes the Frobenius norm. Then we get

(2.21)

Set . Then

(2.22)

Since by Lemma 2.2 and

(2.23)

Hence

(2.24)

Let be positive real numbers for . Now we will show that the left side of inequality (2.19) is positive, that is,

(2.25)

By the arithmetic-geometric mean inequality, we obtain the inequality

(2.26)

So, it is enough to show that

(2.27)

Equivalently,

(2.28)

We will prove by induction. If , then

(2.29)

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

(2.30)

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

(2.31)

for positive real numbers and .

Theorem 2.4.

Let and be positive definite real matrices. Then

(2.32)

In particular,

(2.33)

Proof.

We consider the right side of inequality (1.10):

(2.34)

We can extend this inequality to matrices as follows:

(2.35)

where . Since , it follows that

(2.36)

Let be identity matrix and and positive definite real matrices. According to inequality (2.36), it follows that

(2.37)

From the definition of the cosine of the angle between two given real matrices, we get

(2.38)

In particular, for we obtain that

(2.39)

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

(2.40)

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

(2.41)

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

Theorem 2.5.

Let be a positive definite real matrix. Then

(2.42)

Proof.

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

(2.43)

where are eigenvalues of . That is,

(2.44)

Also, from definition of the Frobenius norm, we get

(2.45)

Then, we obtain that

(2.46)

Likewise,

(2.47)

When inequalities (2.40) and (2.47) are combined, they produce the following inequality:

(2.48)

Therefore, finally we get

(2.49)

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:

(2.50)

where , , and .

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

(2.51)

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

(2.52)

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:

(2.53)

Example we have.

(2.54)

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.