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New Trace Bounds for the Product of Two Matrices and Their Applications in the Algebraic Riccati Equation
Journal of Inequalities and Applications volumeÂ 2009, ArticleÂ number:Â 620758 (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
In the analysis and design of controllers and filters for linear dynamical systems, the Riccati equation is of great importance in both theory and practice (see [1â€“5]). Consider the following linear system (see [4]):
with the cost
Moreover, the optimal control rate and the optimal cost of (1.1) and (1.2) are
where is the initial state of the systems (1.1) and (1.2), is the positive definite solution of the following algebraic Riccati equation (ARE):
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 timeconsuming 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) [6â€“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 [13â€“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.
Let be two real element arrays. If they satisfy
then it is said that is controlled weakly by , which is signed by .
If and
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.
Marshall and Olkin in [16] have showed that for any then
Xing et al. in [13] have observed another result. Let be arbitrary matrices with the following singular value decomposition:
where are orthogonal. Then
where is orthogonal.
Liu and He in [14] have obtained the following: let be arbitrary matrices with the following singular value decomposition:
where are orthogonal. Then

F.
Zhang and Q. Zhang in [15] have obtained the following: let be arbitrary matrices with the following singular value decomposition:
(1.12)
where are orthogonal. Then
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] ).
If and , then for any real array ,
Lemma 2.2 (see [16, page 95, H.3.b] ).
If and , then for any real array ,
Remark 2.3.
Note that if , then for , . Thus from Lemma 2.2, we have
Lemma 2.4 (see [16, page 218, B.1] ).
Let , then
Lemma 2.5 (see [16, page 240, F.4.a] ).
Let , then
Lemma 2.6 (see [17] ).
Let . Then
where
Note that if or , obviously, (2.6) holds. If , choose , then (2.6) also holds.
Remark 2.7.
If , then we obtain CauchySchwartz inequality
where
Remark 2.8.
Note that
Let in (2.6), then we obtain
Lemma 2.9.
If , , then
Proof.

(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)
Since , thus
This completes the proof.
Theorem 2.10.
Let be arbitrary matrices with the following singular value decomposition:
where are orthogonal. Then
Proof.
By the matrix theory we have
Since , without loss of generality, we may assume . Next, we will prove the lefthand side of (2.18):
If
we obtain the conclusion. Now assume that there exists such that then
We use to denote the vector of after changing and , then
After limited steps, we obtain the the lefthand side of (2.18). For the righthand side of (2.18),
If
we obtain the conclusion. Now assume that there exists such that then
We use to denote the vector of after changing and , then
After limited steps, we obtain the righthand side of (2.18). Therefore,
This completes the proof.
Since applying (2.18) with in lieu of we immediately have the following corollary.
Corollary 2.11.
Let be arbitrary matrices with the following singular value decomposition:
where are orthogonal. Then
Now using (2.18) and (2.30), one finally has the following theorem.
Theorem 2.12.
Let be arbitrary matrices with the following singular value decompositions, respectively:
where are orthogonal. Then
Remark 2.13.
We point out that (2.18) improves (1.11). In fact, it is obvious that
This implies that (2.18) improves (1.11).
Remark 2.14.
We point out that (2.18) improves (1.13). Since for , and , from Lemmas 2.1 and 2.4, then (2.18) implies
In fact, for , we have
Then (2.34) can be rewritten as
This implies that (2.18) improves (1.13).
Remark 2.15.
We point out that (1.13) improves (1.7). In fact, from Lemma 2.5, we have
Since is orthogonal, . Then (2.37) is rewritten as follows: By using and Lemma 2.2, we obtain
Note that , from Lemma 2.2 and (2.38), we have
Thus, we obtain
Both (2.38) and (2.40) show that (1.13) is tighter than (1.7).
3. Applications of the Results
Wang et al. in [6] have obtained the following: let be the positive semidefinite solution of the ARE (1.4). Then the trace of matrix has the lower and upper bounds given by
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)
where
Proof.

(1)
Take the trace in both sides of the matrix ARE (1.4) to get
(3.6)
Since is symmetric positive definite matrix, , and from Lemma 2.9, we have
By the CauchySchwartz inequality (2.8), it can be shown that
Note that
, then by (2.34), use (2.6), considering (3.7) and (3.9), we have
From (2.34), note that and then we obtain
It is easy to see that
Combine (3.11) and (3.13), we obtain
Solving (3.14) for yields the righthand side of the inequality (3.2). Similarly, we can obtain the lefthand side of the inequality (3.2).

(2)
Note that when , and by (2.34), (2.6) and (3.7), we have
(3.15)
Thus,
From (3.11) and (3.16), with similar argument to (1), we can obtain (3.3) easily.

(3)
Note that when , by (3.3), we obtain (3.4) immediately. This completes the proof.
Remark 3.2.
From Remark 2.7 and Theorem 3.1, let in (3.2), then we obtain
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] ).
Now let
Neither nor is symmetric. In this case, the results of [6â€“12] are not valid.
Using (1.9) we obtain
Using (1.11) yields
By (2.18), we obtain
where both lower and upper bounds are better than those of (4.2) and (4.3).
Example 4.2.
Let
Neither nor is symmetric. In this case, the results of [6â€“12] are not valid.
Using (1.7) yields
From (1.9) we have
Using (1.11) yields
By (1.13), we obtain
The bound in (2.18) yields
Obviously, (4.10) is tighter than (4.6), (4.7), (4.8) and (4.9).
Example 4.3.
Consider the systems (1.1), (1.2) with
Moreover, the corresponding ARE (1.4) with , is stabilizable and is observable.
Using (3.17) yields
Using (3.1) we obtain
where both lower and upper bounds are better than those of (4.12).
Example 4.4.
Consider the systems (1.1), (1.2) with
Moreover, the corresponding ARE (1.4) with , is stabilizable and is observable.
Using (3.1) we obtain
Using (3.17) yields
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.
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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).
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Liu, J., Zhang, J. New Trace Bounds for the Product of Two Matrices and Their Applications in the Algebraic Riccati Equation. J Inequal Appl 2009, 620758 (2009). https://doi.org/10.1155/2009/620758
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DOI: https://doi.org/10.1155/2009/620758
Keywords
 Convex Function
 System Matrice
 Optimal Cost
 Positive Definite Matrix
 Linear Dynamical System