Báo cáo hóa học: " Research Article New Trace Bounds for the Product of Two Matrices and Their Applications in the Algebraic Riccati Equation" potx

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 620758, 18 pages doi:10.1155/2009/620758 Research Article New Trace Bounds for the Product of Two Matrices and Their Applications in the Algebraic Riccati Equation Jianzhou Liu and Juan Zhang Department of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, China Correspondence should be addressed to Jianzhou Liu, liujz@xtu.edu.cn Received 25 September 2008; Accepted 19 February 2009 Recommended by Panayiotis Siafarikas 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. Copyright q 2009 J. Liu and J. Zhang. This is an open access article distributed under t he Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 1–5. Consider the following linear system see 4: ˙xtAxtBut,x0x 0 , 1.1 with the cost J   ∞ 0  x T Qx  u T u  dt. 1.2 Moreover, the optimal control rate u ∗ and the optimal cost J ∗ of 1.1 and 1.2 are u ∗  Px, P  B T K, J ∗  x T 0 Kx 0 , 1.3 2 Journal of Inequalities and Applications where x 0 ∈ R n is the initial state of the systems 1.1 and 1.2, K is the positive definite solution of the following algebraic Riccati equation ARE: A T K  KA − KRK  −Q, 1.4 with R  BB T and Q 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 A, R is stabilizable, and the pair Q, A 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 trK is that trK/n is the average value of the optimal cost J ∗ as x 0 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 R n×n denote the set of n × n real matrices. Let x x 1 ,x 2 , ,x n  be arealn-element array which is reordered, and its elements are arranged in nonincreasing order. That is, x 1 ≥ x 2 ≥ ··· ≥ x n .Let|x| |x 1 |, |x 2 |, ,|x n |. For A a ij  ∈ R n×n ,letdAd 1 A,d 2 A, ,d n A,λAλ 1 A,λ 2 A, ,λ n A,σA σ 1 A,σ 2 A, ,σ n A denote the diagonal elements, the eigenvalues, the singular values of A, respectively, Let trA,A T denote the trace, the transpose of A, respectively. We define A ii  a ii  d i A, A A  A T /2. The notation A>0 A ≥ 0 is used to denote that A is a symmetric positive definite semidefinite matrix. Let α, β be two real n-element arrays. If they satisfy k  i1 α i ≤ k  i1 β i ,k 1, 2, ,n, 1.5 then it is said that α is controlled weakly by β, which is signed by α≺ w β. If α≺ w β and n  i1 α i  n  i1 β i , 1.6 then it is said that α is controlled by β, which is signed by α ≺ β. Journal of Inequalities and Applications 3 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 A, B ∈ R n×n , then − n  i1 σ i Aσ i B ≤ trAB ≤ n  i1 σ i Aσ i B. 1.7 Xing et al. in 13 have observed another result. Let A, B ∈ R n×n be arbitrary matrices with the following singular value decomposition: B  Udiag  σ 1 B,σ 2 B, ,σ n B  V T , 1.8 where U, V ∈ R n×n are orthogonal. Then λ n AS n  i1 σ i B ≤ trAB ≤ λ 1 AS n  i1 σ i B, 1.9 where S  UV T is orthogonal. Liu and He in 14 have obtained the following: let A, B ∈ R n×n be arbitrary matrices with the following singular value decomposition: B  Udiag  σ 1 B,σ 2 B, ,σ n B  V T , 1.10 where U, V ∈ R n×n are orthogonal. Then min 1≤i≤n  V T AU  ii n  i1 σ i B ≤ trAB ≤ max 1≤i≤n  V T AU  ii n  i1 σ i B. 1.11 F. Zhang and Q. Zhang in 15 have obtained the following: let A, B ∈ R n×n be arbitrary matrices with the following singular value decomposition: B  Udiag  σ 1 B,σ 2 B, ,σ n B  V T , 1.12 where U, V ∈ R n×n are orthogonal. Then n  i1 σ i Bλ n−i1 AS ≤ trAB ≤ n  i1 σ i Bλ i AS, 1.13 where S  UV T is orthogonal. They show that 1.13 has improved 1.9. 4 Journal of Inequalities and Applications 2. Main Results The following lemmas are used to prove the main results. Lemma 2.1 see 16, page 92, H.2.c. If x 1 ≥ ··· ≥ x n ,y 1 ≥ ··· ≥ y n and x ≺ y, then for any real array u 1 ≥···≥u n , n  i1 x i u i ≤ n  i1 y i u i . 2.1 Lemma 2.2 see 16, page 95, H.3.b. If x 1 ≥ ··· ≥ x n ,y 1 ≥ ··· ≥ y n and x≺ w y, then for any real array u 1 ≥···≥ u n ≥ 0, n  i1 x i u i ≤ n  i1 y i u i . 2.2 Remark 2.3. Note that if x≺ w y, then for k  1, 2, ,n, x 1 , ,x k ≺ w y 1 , ,y k .Thus from Lemma 2.2 , we have k  i1 x i u i ≤ k  i1 y i u i ,k 1, 2, ,n. 2.3 Lemma 2.4 see 16, page 218, B.1. Let A  A T ∈ R n×n ,then dA ≺ λA. 2.4 Lemma 2.5 see 16, page 240, F.4.a. Let A ∈ R n×n ,then λ  A  A T 2  ≺ w      λ  A  A T 2       ≺ w σA. 2.5 Lemma 2.6 see 17. Let 0 <m 1 ≤ a k ≤ M 1 , 0 <m 2 ≤ b k ≤ M 2 ,k 1, 2, ,n, 1/p  1/q  1. Then n  k1 a k b k ≤  n  k1 a p k  1/p  n  k1 b q k  1/q ≤ c p,q n  k1 a k b k , 2.6 where c p,q  M p 1 M q 2 − m p 1 m q 2  p  M 1 m 2 M q 2 − m 1 M 2 m q 2  1/p  q  m 1 M 2 M p 1 − M 1 m 2 m p 1  1/q . 2.7 Journal of Inequalities and Applications 5 Note that if m 1  0,m 2 /  0 or m 2  0,m 1 /  0, obviously, 2.6 holds. If m 1  m 2  0, choose c p,q ∞,then2.6 also holds. Remark 2.7. If p  q  2, then we obtain Cauchy-Schwartz inequality n  k1 a k b k ≤  n  k1 a 2 k  1/2  n  k1 b 2 k  1/2 ≤ c 2 n  k1 a k b k , 2.8 where c 2    M 1 M 2 m 1 m 2   m 1 m 2 M 1 M 2  . 2.9 Remark 2.8. Note that lim p →∞  a p 1  a p 2  ··· a p n  1/p  max 1≤k≤n  a k  , lim p →∞ q → 1 c p,q  lim p →∞ q → 1 M p 1 M q 2 − m p 1 m q 2  p  M 1 m 2 M q 2 − m 1 M 2 m q 2  1/p  q  m 1 M 2 M p 1 − M 1 m 2 m p 1  1/q  lim p →∞ q → 1 M p 1  M q 2 − m 1 /M 1  p m q 2  M 1/p 1  p  m 2 M q 2 −m 1 /M 1 M 2 m q 2  1/p M q/p 1  q  m 1 M 2 −M 1 m 2 m 1 /M 1  p  1/q  lim p →∞ q → 1 M 2 M 1/pp/q−p 1 m 1 M 2  lim p →∞ q → 1 1 M 1/p−1 1 m 1  M 1 m 1 . 2.10 Let p →∞,q → 1in2.6, then we obtain m 1 n  k1 b k ≤ n  k1 a k b k ≤ M 1 n  k1 b k . 2.11 Lemma 2.9. If q ≥ 1, a i ≥ 0 i  1, 2, ,n,then  1 n n  i1 a i  q ≤ 1 n n  i1 a q i . 2.12 6 Journal of Inequalities and Applications Proof. 1 Note that q  1, or a i  0 i  1, 2, ,n,  1 n n  i1 a i  q  1 n n  i1 a q i . 2.13 2 If q>1, a i > 0, for x>0, choose fxx q , then f  xqx q−1 > 0andf  x qq − 1x q−2 > 0. Thus, fx is a convex function. As a i > 0and1/n  n i1 a i > 0, from the property of the convex function, we have  1 n n  i1 a i  q  f  1 n n  i1 a i  ≤ 1 n n  i1 fa i  1 n n  i1 a q i . 2.14 3 If q>1, without loss of generality, we may assume a i  0 i  1, ,r,a i > 0 i  r  1, ,n. Then from 2, we have  1 n − r  q  n  i1 a i  q   1 n − r n  i1 a i  q ≤ 1 n − r n  i1 a q i . 2.15 Since n − r/n q ≤ n − r/n,thus  1 n n  i1 a i  q   n − r n  q  1 n − r  q  n  i1 a i  q ≤ n − r n 1 n − r n  i1 a q i  1 n n  i1 a q i . 2.16 This completes the proof. Theorem 2.10. Let A, B ∈ R n×n be arbitrary matrices with the following singular value decomposition: B  Udiagσ 1 B,σ 2 B, ,σ n BV T , 2.17 where U, V ∈ R n×n are orthogonal. Then n  i1 σ i Bd n−i1  V T AU  ≤ trAB ≤ n  i1 σ i Bd i  V T AU  . 2.18 Proof. By the matrix theory we have trABtr  AUdiag  σ 1 B,σ 2 B, ,σ n B  V T   tr  V T AUdiag  σ 1 B,σ 2 B, ,σ n B   n  i1 σ i B  V T AU  ii . 2.19 Journal of Inequalities and Applications 7 Since σ 1 B ≥ σ 2 B ≥ ··· ≥ σ n B ≥ 0, without loss of generality, we may assume σB σ 1 B,σ 2 B, ,σ n B. Next, we will prove the left-hand side of 2.18: n  i1 σ i Bd n−i1  V T AU  ≤ n  i1 σ i Bd i  V T AU  . 2.20 If d  V T AU    d n  V T AU  ,d n−1  V T AU  , ,d 1  V T AU  , 2.21 we obtain the conclusion. Now assume that there exists j<ksuch that d j V T AU > d k V T AU, then σ j Bd k  V T AU   σ k Bd j  V T AU  − σ j Bd j  V T AU  − σ k Bd k  V T AU    σ j B − σ k B  d k  V T AU  − d j  V T AU  ≤ 0. 2.22 We use  dV T AU to denote the vector of dV T AU after changing d j V T AU and d k V T AU, then n  i1 σ i B  d i  V T AU  ≤ n  i1 σ i Bd i  V T AU  . 2.23 After limited steps, we obtain the the left-hand side of 2.18. For the right-hand side of 2.18, n  i1 σ i Bd i  V T AU  ≤ n  i1 σ i Bd i  V T AU  . 2.24 If d  V T AU    d 1 V T AU  ,d 2  V T AU  , ,d n  V T AU  , 2.25 we obtain the conclusion. Now assume that there exists j>ksuch that d j V T AU < d k V T AU, then σ j Bd k  V T AU   σ k Bd j  V T AU  − σ j Bd j  V T AU  − σ k Bd k  V T AU    σ j B − σ k B  d k  V T AU  − d j  V T AU  ≥ 0. 2.26 8 Journal of Inequalities and Applications We use  dV T AU to denote the vector of dV T AU after changing d j V T AU and d k V T AU, then n  i1 σ i Bd i  V T AU  ≤ n  i1 σ i B  d i  V T AU  . 2.27 After limited steps, we obtain the right-hand side of 2.18. Therefore, n  i1 σ i Bd n−i1  V T AU  ≤ trAB ≤ n  i1 σ i Bd i  V T AU  . 2.28 This completes the proof. Since trABtrBA, applying 2.18 with B in lieu of A, we immediately have the following corollary. Corollary 2.11. Let A,B ∈ R n×n be arbitrary matrices with the following singular value decomposition: A  Pdiagσ 1 A,σ 2 A, ,σ n AQ T , 2.29 where P, Q ∈ R n×n are orthogonal. Then n  i1 σ i Ad n−i1  Q T BP  ≤ trAB ≤ n  i1 σ i Ad i Q T BP. 2.30 Now using 2.18 and 2.30, one finally has the following theorem. Theorem 2.12. Let A, B ∈ R n×n be arbitrary matrices with the following singular value decompositions, respectively: A  Pdiag  σ 1 A,σ 2 A, ,σ n A  Q T , B  Udiag  σ 1 B,σ 2 B, ,σ n B  V T , 2.31 where P, Q, U, V ∈ R n×n are orthogonal. Then max  n  i1 σ i Ad n−i1  Q T BP  , n  i1 σ i Bd n−i1  V T AU   ≤ trAB ≤ min  n  i1 σ i Bd i  V T AU  , n  i1 σ i Ad i  Q T BP   . 2.32 Journal of Inequalities and Applications 9 Remark 2.13. We point out that 2.18 improves 1.11. In fact, it is obvious that min 1≤i≤n  V T AU  ii n  i1 σ i B ≤ n  i1 σ i Bd n−i1  V T AU  ≤ trAB ≤ n  i1 σ i Bd i  V T AU  ≤ max 1≤i≤n  V T AU  ii n  i1 σ i B. 2.33 This implies that 2.18 improves 1.11. Remark 2.14. We point out that 2.18 improves 1.13. Since for i  1, ,n, σ i B ≥ 0and d i V T AUd i V T AU V T AU T /2, from Lemmas 2.1 and 2.4, then 2.18 implies n  i1 σ i Bλ n−i1  V T AU   V T AU  T 2  ≤ n  i1 σ i Bd n−i1  V T AU   V T AU  T 2  ≤ trAB ≤ n  i1 σ i Bd i  V T AU   V T AU  T 2  ≤ n  i1 σ i Bλ i  V T AU   V T AU  T 2  . 2.34 In fact, for i  1, 2, ,n, we have λ i  V T AU V T AU T 2   λ i  V T AUV T AUV T  T 2 V   λ i  AUV T AUV T  T 2   λ i AS. 2.35 10 Journal of Inequalities and Applications Then 2.34 can be rewritten as n  i1 σ i Bλ n−i1 AS ≤ n  i1 σ i Bd n−i1  V T AU  ≤ trAB ≤ n  i1 σ i Bd i  V T AU  ≤ n  i1 σ i Bλ i AS. 2.36 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 λ AS≺ w σAS. 2.37 Since S is orthogonal, σASσA. Then 2.37 is rewritten as follows: λ AS≺ w σA. By using σ 1 B ≥ σ 2 B ≥··· ≥σ n B ≥ 0andLemma 2.2,weobtain n  i1 σ i Bλ i AS ≤ n  i1 σ i Bσ i A. 2.38 Note that λ i −AS−λ n−i1 AS,fromLemma 2.2 and 2.38, we have − n  i1 σ i Bλ n−i1 AS n  i1 σ i Bλ i −AS ≤ n  i1 σ i B   λ i AS   ≤ n  i1 σ i Bσ i A. 2.39 Thus, we obtain − n  i1 σ i Bσ i A ≤ n  i1 σ i Bλ n−i1 AS. 2.40 Both 2.38 and 2.40 show that 1.13 is tighter than 1.7. [...]...Journal of Inequalities and Applications 11 3 Applications of the Results Wang et al in 6 have obtained the following: let K be the positive semidefinite solution of the ARE 1.4 Then the trace of matrix K has the lower and upper bounds given by λn A λn A λ1 R 2 λ 1 R tr Q ≤ tr K ≤ 2 λ1 A λ n R /n tr Q λ1 A λ n R /n 3.1 In this section, we obtain the application in the algebraic Riccati equation of our... stabilizable and Q, A is 1.6039 ≤ tr K ≤ 5.6548 4.15 1.6771 ≤ tr K ≤ 5.5757, 4.16 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,... 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 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... , then 3.12 n σi K i 1 It is easy to see that tr AT K tr KA ≤ λ 1 AT tr K 2λ 1 AT A 2 λ 1 A tr K tr K 3.13 tr K 2λ 1 A tr K Combine 3.11 and 3.13 , we obtain 1 cp,q n2−1/q n i 1 1/p p λi R tr K 2 − 2tr K λ n A − tr Q ≤ 0 3.14 Solving 3.14 for tr K yields the right-hand side of the inequality 3.2 Similarly, we can obtain the left-hand side of the inequality 3.2 14 λi Journal of Inequalities and Applications. .. Lasserre, “Tight bounds for the trace of a matrix product, ” IEEE Transactions on Automatic Control, vol 42, no 4, pp 578–581, 1997 8 Y Fang, K A Loparo, and X Feng, “Inequalities for the trace of matrix product, ” IEEE Transactions on Automatic Control, vol 39, no 12, pp 2489–2490, 1994 9 J Saniuk and I Rhodes, “A matrix inequality associated with bounds on solutions of algebraic Riccati and Lyapunov equations,”... IEEE Transactions on Automatic Control, vol 51, no 9, pp 1506–1509, 2006 16 A W Marshall and I Olkin, Inequalities: Theory of Majorization and Its Applications, vol 143 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1979 17 C.-L Wang, “On development of inverses of the Cauchy and Holder inequalities,” SIAM Review, vol ¨ 21, no 4, pp 550–557, 1979 ... results including 3.1 Some of our results and 3.1 cannot contain each other Theorem 3.1 If 1/p 1 and K is the positive semidefinite solution of the ARE 1.4 , then 1/q 1 the trace of matrix K has the lower and upper bounds given by λn A λn A 2 p n i 1λ i p n i 1λ i ≤ tr K ≤ 2 If A 1/cp,q n1−1/q R λ1 A 1/p R tr Q 1/p 3.2 λ1 A 2 1/cp,q p n i 1λ i n2−1/q p n i 1λ i 1/cp,q n2−1/q R R 1/p tr Q 1/p AT /2 ≥ 0, then... “Existence condition on solutions to the algebraic Riccati equation,” Acta Automatica Sinica, vol 34, no 1, pp 85–87, 2008 5 K Ogata, Modern Control Engineering, Prentice-Hall, Upper Saddle River, NJ, USA, 3rd edition, 1997 18 Journal of Inequalities and Applications 6 S.-D Wang, T.-S Kuo, and C.-F Hsu, Trace bounds on the solution of the algebraic matrix Riccati and Lyapunov equation,” IEEE Transactions... 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 Journal of Inequalities and Applications 15 equation of our results will have different superiority if we choose different p and q, we will give two examples when p 2, q 2, and p → ∞, q → 1 Example 4.1 see 13 Now let ⎛ ⎞ 0.9140 0.6989... AT /2 ≥ 0, then the trace of matrix K has the lower and upper bounds given by p n i 1λ i A 1/p p n i 1λ i 1/cp,q n1−1/q p n i 1λ i R A 1/p 2 p n i 1λ i R 1/p tr Q 1/p ≤ tr K ≤ p n i 1λ i A 1/p p n i 1λ i A 2/p 1/cp,q n2−1/q 1/cp,q n2−1/q p n i 1λ i R 1/p p n i 1λ i R 1/p tr Q 3.3 12 Journal of Inequalities and Applications AT /2 ≤ 0, then the trace of matrix K has the lower and upper bounds given by . 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. 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,. Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 620758, 18 pages doi:10.1155/2009/620758 Research Article New Trace Bounds for the Product of Two