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Stability of Matrix Differential Equations with Commuting Matrix Introduction Stability of Matrix Differential Sufficient Conditions . Block Circulants . Equations with Commuting Matrix Home Page Constant Coefficients Title Page Fernando Martins Edgar Pereira JJ II M. A. Facas Vicente Jos´eVit´oria J I Coimbra College of Education - Polytechnic Institute of Coimbra Page1 of 24 Department of Informatics - University of Beira Interior Department of Mathematics - University of Coimbra Go Back Instituto de Telecomunica¸c~oes- Coimbra Full Screen Delega¸c~aoda Covilh~a- Portugal Close Instituto de Engenharia de Sistemas Quit e Computadores - Coimbra - Portugal October of 2010 Introduction Sufficient Conditions . Block Circulants . Home Page Title Page 1. Introduction JJ II 2. Sufficient Conditions for the Stability J I Page2 of 24 3. Block Circulant Matrices with Com- Go Back muting Blocks Full Screen Close Quit Introduction Sufficient Conditions . Block Circulants . Home Page Title Page JJ II J I 1. Introduction Page3 of 24 Go Back Full Screen Close Quit Introduction Sufficient Conditions . We consider systems of first order linear differential equations with Block Circulants . matrix constant coefficients that can be written in a matrix form 0 Home Page y (t) = Aby(t) (1) where y(t) 2 Rmn and the block matrix Title Page 2 3 A11 A12 ··· A1m 6 7 JJ II 6 A21 A22 ··· A2m 7 Ab = 6 7 2 Mm(Pn(R)) 6 . ··· . 7 4 5 J I Am1 Am2 ··· Amm Page4 of 24 is partitioned into m × m commuting blocks of order n, that is AijAlk = AlkAij; 8i; j; l; k = 1; : : : ; m: Go Back Full Screen Definition 1. Let Ab 2 Mm(Pn(R)) be a block matrix. The char- Close acteristic matrix polynomial of Ab is P (X) = det (I ⊗ X − A ) : Quit b m b Introduction Sufficient Conditions . Block Circulants . Home Page Definition 2. A matrix Γ of order n is a (right) solvent of the matrix polynomial P (X) if P (Γ) = 0n: Title Page Definition 3. Let Ab be a block matrix of order mn. If JJ II AbX1 = X1Λ; (2) J I where Λ is a block (a matrix of order n) and the block vector X1 (a matrix of dimension mn × n) is full rank, then Λ is called a Page5 of 24 (right) block eigenvalue of Ab and X1 is the corresponding (right) block eigenvector. Go Back Theorem 1. Any solvent of the characteristic matrix polynomial Full Screen of the matrix Ab 2 Mm(Pn(R)) is a block eigenvalue of Ab. Close Quit Introduction Sufficient Conditions . Block Circulants . Home Page m m−1 Title Page Definition 4. Let P (X) = X + A1X + ··· + Am−1X + Am, Ai 2 Pn(R), be a matrix polynomial. The matrix Cb, of order mn; partitioned into blocks of order n; given by JJ II 2 3 0n In 0n ··· 0n . J I 6 . .. .. .. 7 6 . 7 Cb = 6 . .. .. 0 7 6 n 7 Page6 of 24 4 0n ······ 0n In 5 −Am −Am−1 · · · −A2 −A1 Go Back is said to be a block companion matrix associated to matrix polynomial P (X). Full Screen Close Quit Introduction Sufficient Conditions . Theorem 2. If the matrix Γ1; of order n; is a solvent of the matrix Block Circulants . polynomial P (X), then Home Page CbX1 = X1Γ1; where Cb is the block companion matrix of P (X) and Title Page 2 3 In JJ II 6 7 6 Γ1 7 X1 = 6 7 : 6 . 7 4 5 J I m−1 Γ1 Page7 of 24 Go Back Theorem 3. Let Cb be a block companion matrix associated with the matrix polynomial P (X) and let Λ1 be a block eigenvalue of Cb associated with the full rank block eigenvector X1, i. e., Full Screen CbX1 = X1Λ1: Close Under these conditions, if the first block X11; of X1, is nonsingular, −1 then Γ1 = X11Λ1X11 is a solvent of P (X). Quit Introduction Sufficient Conditions . Block Circulants . Home Page Title Page JJ II Definition 5. Let Ab be a block matrix of order mn and let Λ1; Λ2;:::; Λm be a set of block eigenvalues of Ab. This set is said J I to be a complete set of block eigenvalues when all the eigen- values, and respective partial multiplicities, of these block eigen- Page8 of 24 values are the eigenvalues, with the same partial multiplicities, of the matrix Ab. Go Back Full Screen Close Quit Introduction Sufficient Conditions . Block Circulants . Home Page Title Page JJ II J I 2. Sufficient Conditions for the Stabi- lity Page9 of 24 Go Back Full Screen Close Quit Introduction Sufficient Conditions . Block Circulants . Home Page Definition 6. Let Λ1 be a block eigenvalue of the block matrix Ab. If Title Page Re(λi) < 0; (3) for all λi 2 σ(Λ1); i = 1; : : : ; n, then Λ1 is said to be stable. JJ II J I The stability criterion in terms of blocks for the equilibrium of the matrix differential equation (1) is stated next. Page 10 of 24 Go Back Proposition 1. Let Λ1; Λ2;:::; Λm be a complete set of block ei- genvalues of the block matrix Ab. If all block eigenvalues, Λj, are stable, then the equilibrium of the matrix differential equation (1) Full Screen is asymptotically stable. Close Quit Introduction Sufficient Conditions . Block Circulants . Home Page Symmetrizable Matrices Title Page A matrix N 2 Rn×n is said to be symmetrizable if there exists a T Rn×n JJ II matrix R = R 2 positive definite such that N T R = RN: J I Page 11 of 24 n×n Two matrices N1;N2 2 R are said to be simultaneous symme- trizable if there exists a matrix R = RT 2 Rn×n positive definite Go Back such that T N1 R = RN1 Full Screen and T N2 R = RN2: Close Quit Introduction Sufficient Conditions . Block Circulants . Home Page Title Page n×n JJ II Theorem 4. Let N1;N2 2 C be diagonalizable. Then, N1 and N2 comute if and only if they are simultaneously diagonalizable. J I Page 12 of 24 Theorem 5. A set of matrices simultaneously diagonalizable is also a set of matrices simultaneously symmetrizable. Go Back Full Screen Close Quit Introduction Sufficient Conditions . R Block Circulants . Proposition 2. Let Cb 2 Mm(Pn( )) be the block companion ma- trix associated with the matrix polynomial P (X). If Home Page (i) σ(Cb) \ σ(−Cb) = Ø; (ii) AαAβ 2 Pn(R); with 0 ≤ α < β ≤ m; are diagonalizable; Title Page T mn×mn (iii) W = W = [Wij] 2 R where 8 JJ II < 2RAm+1−jAm+1−i if m + i and m + j are even, i ≤ j Wij = Wji = (4) : 0n if m + i or m + j is odd J I (i; j = 1; 2; : : : ; m) and R = RT 2 Rn×n is positive definite, Page 13 of 24 then the Lyapunov matrix equation T Go Back Cb V + VCb = −W (5) T mn×mn has a unique solution V = V = [Vij] 2 R , where Full Screen 8 i−1 > X k+i−1 <> (−1) RAm−i−j+k+1Am−k if i + j is even, i ≤ j Close Vij = Vji = k=0 : (6) > > : 0n if i + j is odd Quit Introduction Sufficient Conditions . Corollary 1. If the Lyapunov matrix equation Block Circulants . T Cb V + VCb = −W; (7) Home Page has a unique symmetric solution V , such that: Title Page (i) V is positive definite; (ii) W is positive semi-definite; JJ II 2 3 W 1=2 6 1=2 7 6 W Cb 7 J I (iii) 6 . 7 is of full rank; 4 . 5 W 1=2Cm−1 Page 14 of 24 b (iv) Λ1; Λ2;:::; Λm are a complete set of block eigenvalues Ab, Go Back then all block eigenvalues, Λj, are stable. Full Screen Corollary 2. The equilibrium of the matrix differential equation 0 Close y (t) = Aby(t) is asymptotically stable. Quit Definition 7. Let P (X) = A Xm + A Xm−1 + ··· + A X + Introduction 0 1 m−1 R e Sufficient Conditions . Am;Ai 2 Pn( ); be a matrix polynomial. The matrix Hb = Block Circulants . h i e R Hb(ij) 2 Mm(Pn( )), where Home Page 8 i−1 > X k+i−1 e e < (−1) AkAi+j−k−1 if i + j is even, i ≤ j Hb(ij) = Hb(ji) = (8) k=0 :> Title Page 0n if i + j is odd (i; j = 1; 2; : : : ; m) is said to be the block Hermite matrix of JJ II P (X). J I Example 1. For m = 5, we have 2 A0A1 0n A0A3 0n A0A5 3 Page 15 of 24 6 0n A1A2 − A0A3 0n A1A4 − A0A5 0n 7 e 6 7 H = 6 A0A3 0n A0A5 − A1A4 + A2A3 0n A2A5 7 : Go Back b 6 7 6 7 4 0n A1A4 − A0A5 0n A3A4 − A2A5 0n 5 Full Screen A0A5 0n A2A5 0n A4A5 Close e Remark 1. The matrix Hb is block symmetric, that is Quit e e Hb(ij) = Hb(ji): Introduction Sufficient Conditions . Block Circulants . Home Page m m−1 Definition 8. Let P (X) = A0X + A1X + ··· + Am−1X + R Title Page Am;Ai 2 Pn( ); be a matrix polynomial. And let be the matrix Hb = Hb(ij) 2 Mm(Pn(R)); where Hb(ij) = A2j−i with Ar = 0n if r < 0 or r > m (i; j = 1; 2; : : : ; m). To this matrix Hb, given by JJ II 2 3 A1 A3 A5 ········· A2m−1 J I 6 A A A ········· A 7 6 0 2 4 2m−2 7 6 0n A1 A3 ········· A2m−3 7 6 . 7 Page 16 of 24 6 0 A A .
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