# New Trace Bounds for the Product of Two Matrices and Their Applications in the Algebraic Riccati Equation

- Jianzhou Liu
^{1}Email author and - Juan Zhang
^{1}

**2009**:620758

**DOI: **10.1155/2009/620758

© J. Liu and J. Zhang. 2009

**Received: **25 September 2008

**Accepted: **19 February 2009

**Published: **8 March 2009

## Abstract

By using singular value decomposition and majorization inequalities, we propose new inequalities for the trace of the product of two arbitrary real square matrices. These bounds improve and extend the recent results. Further, we give their application in the algebraic Riccati equation. Finally, numerical examples have illustrated that our results are effective and superior.

## 1. Introduction

with and are symmetric positive definite matrices. To guarantee the existence of the positive definite solution to (1.4), we shall make the following assumptions: the pair ( ) is stabilizable, and the pair ( ) is observable.

In practice, it is hard to solve the (ARE), and there is no general method unless the system matrices are special and there are some methods and algorithms to solve (1.4), however, the solution can be time-consuming and computationally difficult, particularly as the dimensions of the system matrices increase. Thus, a number of works have been presented by researchers to evaluate the bounds and trace bounds for the solution of the (ARE) [6–12]. In addition, from [2, 6], we know that an interpretation of is that is the average value of the optimal cost as varies over the surface of a unit sphere. Therefore, consider its applications, it is important to discuss trace bounds for the product of two matrices. Most available results are based on the assumption that at least one matrix is symmetric [7, 8, 11, 12]. However, it is important and difficult to get an estimate of the trace bounds when any matrix in the product is nonsymmetric in theory and practice. There are some results in [13–15].

In this paper, we propose new trace bounds for the product of two general matrices. The new trace bounds improve the recent results. Then, for their application in the algebraic Riccati equation, we get some upper and lower bounds.

In the following, let denote the set of real matrices. Let be a real -element array which is reordered, and its elements are arranged in nonincreasing order. That is, . Let . For , let , , denote the diagonal elements, the eigenvalues, the singular values of , respectively, Let denote the trace, the transpose of , respectively. We define , The notation ( ) is used to denote that is a symmetric positive definite (semidefinite) matrix.

then it is said that is controlled weakly by , which is signed by .

then it is said that is controlled by , which is signed by .

Therefore, considering the application of the trace bounds, many scholars pay much attention to estimate the trace bounds for the product of two matrices.

where is orthogonal.

- F.Zhang and Q. Zhang in [15] have obtained the following: let be arbitrary matrices with the following singular value decomposition:(1.12)

where is orthogonal. They show that (1.13) has improved (1.9).

## 2. Main Results

The following lemmas are used to prove the main results.

Lemma 2.1 (see [16, page 92, H.2.c] ).

Lemma 2.2 (see [16, page 95, H.3.b] ).

Remark 2.3.

Lemma 2.4 (see [16, page 218, B.1] ).

Lemma 2.5 (see [16, page 240, F.4.a] ).

Lemma 2.6 (see [17] ).

Note that if or , obviously, (2.6) holds. If , choose , then (2.6) also holds.

Remark 2.7.

Remark 2.8.

Lemma 2.9.

- (1)Note that , or ,(2.13)

- (2)If , , for , choose , then and . Thus, is a convex function. As and , from the property of the convex function, we have(2.14)

- (3)If , without loss of generality, we may assume . Then from (2), we have(2.15)

This completes the proof.

Theorem 2.10.

Proof.

This completes the proof.

Since applying (2.18) with in lieu of we immediately have the following corollary.

Corollary 2.11.

Now using (2.18) and (2.30), one finally has the following theorem.

Theorem 2.12.

Remark 2.13.

This implies that (2.18) improves (1.11).

Remark 2.14.

This implies that (2.18) improves (1.13).

Remark 2.15.

Both (2.38) and (2.40) show that (1.13) is tighter than (1.7).

## 3. Applications of the Results

In this section, we obtain the application in the algebraic Riccati equation of our results including (3.1). Some of our results and (3.1) cannot contain each other.

Theorem 3.1.

If and is the positive semidefinite solution of the ARE (1.4), then

- (1)the trace of matrix has the lower and upper bounds given by(3.2)

- (2)If , then the trace of matrix has the lower and upper bounds given by(3.3)

- (3)If , then the trace of matrix has the lower and upper bounds given by(3.4)

- (1)Take the trace in both sides of the matrix ARE (1.4) to get(3.6)

- (2)Note that when , and by (2.34), (2.6) and (3.7), we have(3.15)

- (3)
Note that when , by (3.3), we obtain (3.4) immediately. This completes the proof.

Remark 3.2.

where

Remark 3.3.

From Remark 2.7 and Theorem 3.1, let in (3.2), then we obtain (3.1) immediately.

## 4. Numerical Examples

In this section, firstly, we will give two examples to illustrate that our new trace bounds are better than the recent results. Then, to illustrate the application in the algebraic Riccati equation of our results will have different superiority if we choose different and , we will give two examples when and .

Example 4.1 (see [13] ).

Neither nor is symmetric. In this case, the results of [6–12] are not valid.

where both lower and upper bounds are better than those of (4.2) and (4.3).

Example 4.2.

Neither nor is symmetric. In this case, the results of [6–12] are not valid.

Obviously, (4.10) is tighter than (4.6), (4.7), (4.8) and (4.9).

Example 4.3.

Moreover, the corresponding ARE (1.4) with , is stabilizable and is observable.

where both lower and upper bounds are better than those of (4.12).

Example 4.4.

Moreover, the corresponding ARE (1.4) with , is stabilizable and is observable.

where both lower and upper bounds are better than those of (4.15).

## 5. Conclusion

In this paper, we have proposed lower and upper bounds for the trace of the product of two arbitrary real matrices. We have showed that our bounds for the trace are the tightest among the parallel trace bounds in nonsymmetric case. Then, we have obtained the application in the algebraic Riccati equation of our results. Finally, numerical examples have illustrated that our bounds are better than the recent results.

## Declarations

### Acknowledgments

The author thanks the referee for the very helpful comments and suggestions. The work was supported in part by National Natural Science Foundation of China (10671164), Science and Research Fund of Hunan Provincial Education Department (06A070).

## Authors’ Affiliations

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