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
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Quit e Computadores - Coimbra - Portugal
October of 2010 Introduction Sufficient Conditions . . . Block Circulants . . .
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Title Page 1. Introduction
JJ II 2. Sufficient Conditions for the Stability J I
Page2 of 24 3. Block Circulant Matrices with Com-
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Quit Introduction Sufficient Conditions . . . Block Circulants . . .
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J I 1. Introduction
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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) ∈ Rmn and the block matrix Title Page A11 A12 ··· A1m JJ II A21 A22 ··· A2m Ab = ∈ Mm(Pn(R)) . . ··· . J I Am1 Am2 ··· Amm
Page4 of 24 is partitioned into m × m commuting blocks of order n, that is
AijAlk = AlkAij, ∀i, j, l, k = 1, . . . , m. Go Back
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Definition 1. Let Ab ∈ 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 ∈ Mm(Pn(R)) is a block eigenvalue of Ab.
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Quit Introduction Sufficient Conditions . . . Block Circulants . . .
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m m−1 Title Page Definition 4. Let P (X) = X + A1X + ··· + Am−1X + Am, Ai ∈ Pn(R), be a matrix polynomial. The matrix Cb, of order mn, partitioned into blocks of order n, given by JJ II 0n In 0n ··· 0n . . . . . J I ...... . . . Cb = . .. .. 0 n Page6 of 24 0n ······ 0n In −Am −Am−1 · · · −A2 −A1 Go Back is said to be a block companion matrix associated to matrix polynomial P (X). Full Screen
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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 In JJ II Γ1 X1 = . . J I m−1 Γ1
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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 . . .
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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.
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Quit Introduction Sufficient Conditions . . . Block Circulants . . .
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J I 2. Sufficient Conditions for the Stabi- lity Page9 of 24
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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 ∈ 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.
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Quit Introduction Sufficient Conditions . . . Block Circulants . . .
Home Page Symmetrizable Matrices
Title Page A matrix N ∈ Rn×n is said to be symmetrizable if there exists a T Rn×n JJ II matrix R = R ∈ positive definite such that N T R = RN. J I
Page 11 of 24 n×n Two matrices N1,N2 ∈ R are said to be simultaneous symme- trizable if there exists a matrix R = RT ∈ Rn×n positive definite Go Back such that T N1 R = RN1 Full Screen and T N2 R = RN2.
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n×n JJ II Theorem 4. Let N1,N2 ∈ 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.
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Quit Introduction Sufficient Conditions . . . R Block Circulants . . . Proposition 2. Let Cb ∈ 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 ∈ Pn(R), with 0 ≤ α < β ≤ m, are diagonalizable; Title Page T mn×mn (iii) W = W = [Wij] ∈ R where 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 ∈ 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] ∈ R , where Full Screen 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 W 1/2 1/2 W Cb J I (iii) . is of full rank; . 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 ∈ Pn( ), be a matrix polynomial. The matrix Hb = Block Circulants . . . h i e R Hb(ij) ∈ Mm(Pn( )), where
Home Page 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
A0A1 0n A0A3 0n A0A5 Page 15 of 24 0n A1A2 − A0A3 0n A1A4 − A0A5 0n e H = A0A3 0n A0A5 − A1A4 + A2A3 0n A2A5 . Go Back b 0n A1A4 − A0A5 0n A3A4 − A2A5 0n
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 ∈ Pn( ), be a matrix polynomial. And let be the matrix Hb = Hb(ij) ∈ 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 A1 A3 A5 ········· A2m−1 J I A A A ········· A 0 2 4 2m−2 0n A1 A3 ········· A2m−3 . Page 16 of 24 0 A A .. ······ A Hb = n 0 2 2m−4 , .. . 0n 0n A1 ··· . ··· . . .. . Go Back . ············ . . 0n 0n 0n ········· Am Full Screen we call the block Hurwitz matrix.
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Quit Definition 9. If we perform the block elimination of the block Introduction Hurwitz matrix, Hb, we obtain the matrix Sufficient Conditions . . . Block Circulants . . . C21 C22 C23 ········· 0n C31 C32 C33 ······ Home Page 0n 0n C41 C42 ······ ∈ Mm(Pn(R)), (9) 0n 0n 0n C51 C52 ··· . . . . Title Page ...... 0n 0n 0n ··· 0n C(m+1)1 JJ II which we call the block Routh matrix. Definition 10. Let Sb = Sb(ij) ∈ Mm(Pn(R)), where J I In if j − i = 1 −Sk if i − j = 1 (k = 2, . . . , m) Page 17 of 24 Sb(ij) = . −S1 if j = i = m 0n otherwise Go Back The matrix Full Screen 0n In 0n ··· 0n −S 0 I ... . m n n S = ...... , (10) Close b 0n 0n . .. . . −S3 0n In
Quit 0n ··· 0n −S2 −S1 is named the block Schwarz matrix. Introduction Sufficient Conditions . . . Block Circulants . . .
Home Page A strong relationship between the block companion matrix and the −1 Title Page block Schwarz matrix is that they are similar, i. e., Sb = TCbT with
JJ II In . 0n 0n 0n 0n 0n 0n 0n ...... −1 C(m−1)1C(m−1)2 .In 0n 0n 0n 0n 0n 0n J I 0n . 0n In 0n 0n 0n 0n 0n −1 −1 T = C(m−3)1C(m−1)3 .C61 C62 0n In 0n 0n 0n 0n , (11) −1 0n . 0n C51 C52 0n In 0n 0n 0n Page 18 of 24 −1 −1 −1 C(m−5)1C(m−5)4 .C41 C43 0n C41 C42 0n In 0n 0n −1 −1 0n . 0n C31 C33 0n C31 C32 0n In 0n −1 −1 −1 ..C21 C24 0n C21 C23 0n C21 C22 0n In Go Back
where Cij are the elements of the block Routh matrix, with Ci1 ∈ R Full Screen Pn( ), i = 2, ··· , m, being nonsingular.
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Quit Introduction R Sufficient Conditions . . . Proposition 3. Let Sb ∈ Mm(Pn( )) be a block Schwarz matrix Block Circulants . . . and let the matrices S1,S1S2,S1S2S3,...,S1S2S3 ··· Sm−1Sm ∈ Pn(R) be diagonalizable and positive definite. Then, for a sym- Home Page metric and positive semi-definite matrix Q, there exists a unique solution M, symmetric and positive definite, of the Lyapunov ma- T Title Page trix equation Sb M + MSb = −Q.
Corollary 3. If Λ1, Λ2,..., Λm are a complete set of block eigen-
JJ II values Ab, then they are stable. Corollary 4. The equilibrium of the matrix differential equation
J I 0 y (t) = Aby(t)
Page 19 of 24 is asymptotically stable.
Go Back In the following result, we study the stability of a matrix differential equation, by using a block Hermite matrix. Full Screen e Proposition 4. Let Hb be a block Hermite matrix of P (X). If e R Hb is positive definite and if AαAβ ∈ Pn( ) are diagonalizable, Close 0 ≤ α ≤ β ≤ m, and if A0 is nonsingular, then the equilibrium 0 of the matrix differential equation y (t) = Aby(t) is asymptotically Quit stable. Introduction Sufficient Conditions . . . Block Circulants . . .
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JJ II
J I 3. Block Circulants Matrices with Commuting Blocks Page 20 of 24
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Quit Introduction Sufficient Conditions . . . Block Circulants . . .
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Title Page We consider now the equation
0 y (t) = Aby(t), (12) JJ II mn where y(t) ∈ R and the matrix Ab has a block circulant structure. Let consider a real mn × mn block circulant matrix J I A1 A2 ··· Am Page 21 of 24 Am A1 ··· Am−1 Ab = bcirc(A1,A2,...,Am) = , ············
Go Back A2 ··· Am A1
where A1,A2,...,Am ∈ Pn(R). Full Screen
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Quit Introduction Sufficient Conditions . . . ∗ 1 2 p−1 ( i2π ) Block Circulants . . . √ p Theorem 6. Let Fp = p V (1, w, w , . . . , w ), with w = e , | {z } Vandermonde Matrix ∗ Home Page and let Bi = FnAiFn , i = 1, 2, . . . , m. Then, we have: (i) Title Page M1 B1 M2 B2 √ ∗ M3 B3 = ( mFm ⊗ In) ; . . JJ II . . Mm Bm ∗ J I (ii) (Fm ⊗ Fn)Ab(Fm ⊗ Fn) = diag(M1,M2,...,Mm);
(iii) The eigenvalues of M1,M2,...,Mm are eigenvalues of Ab. Page 22 of 24
We notice that the matrices M1,M2,...,Mm are the Block Eigenva- Go Back lues of Ab.
Proposition 5. If M1,M2,...,Mm are stable then the equilibrium Full Screen of the matrix differential equation
0 Close y (t) = Aby(t) is asymptotically stable. Quit We approach the inertia of Ab, by considering two cases for m: Introduction Sufficient Conditions . . . (1) m is even. Block Circulants . . . m Let λk1, λk2, . . . , λkn be eigenvalues of Mk, k = 1,..., 2 +1, such that Home Page B1 B2 √ ∗ Title Page Mk = ( mF ⊗ In)k B3 , m . . Bm JJ II √ ∗ th where ( mFm ⊗ In)k is the k block row of type n by mn of the √ ∗ block matrix ( mF ⊗ In). J I m Thus,
Page 23 of 24 (i)if xk = Re(λk ), p = 1, . . . , n, N− (xk , xk , . . . , xk ) = lk, p p 1 2 n then the matrix Ab has l1 + 2 l2 + ··· + l m + l m eigen- [ 2 ] [ 2 ]+1 Go Back values with negative real parts;
(ii)if xkp = Re(λkp), p = 1, . . . , n, N+ (xk1, xk2, . . . , xkn) = rk, Full Screen then the matrix Ab has r1 + 2 r2 + ··· + r m + r m ei- [ 2 ] [ 2 ]+1 genvalues with positive real parts; Close (iii) if xk = Re(λk ), p = 1, . . . , n, N0 (xk , xk , . . . , xk ) = uk, p p 1 2 n Quit then the matrix Ab has u1 + 2 u2 + ··· + u m + u m ei- [ 2 ] [ 2 ]+1 genvalues with null real parts. Introduction (2) m is odd. Sufficient Conditions . . . m Let λk1, λk2, . . . , λkn be eigenvalues of Mk, k = 1,..., 2 +1, such Block Circulants . . . that B1 Home Page B2 √ ∗ Mk = ( mFm ⊗ In)k B3 , . Title Page . Bm √ ∗ th JJ II where ( mFm ⊗ In)k is the k block row of type n by mn of the √ ∗ block matrix ( mFm ⊗ In). J I Thus,
(i)if xkp = Re(λkp), p = 1, . . . , n, N− (xk1, xk2, . . . , xkn) = lk, Page 24 of 24 then the matrix Ab has l1 + 2 l2 + ··· + l m eigenvalues [ 2 ]+1 with negative real parts; Go Back (ii)if xk = Re(λk ), p = 1, . . . , n, N+ (xk , xk , . . . , xk ) = rk, p p 1 2 n Full Screen then the matrix Ab has r1 + 2 r2 + ··· + r m eigenvalues [ 2 ]+1 with positive real parts;
Close (iii) if xk = Re(λk ), p = 1, . . . , n, N0 (xk , xk , . . . , xk ) = uk, p p 1 2 n then the matrix Ab has u1 +2 u2 + ··· + u m eigenvalues [ 2 ]+1 Quit with null real parts.