On the computation of the Jordan canonical form of regular matrix polynomials

Transcription

1 On the computation of the Jordan canonical form of regular matrix polynomials G Kalogeropoulos, P Psarrakos 2 and N Karcanias 3 Dedicated to Professor Peter Lancaster on the occasion of his 75th birthday Abstract In this paper, an algorithm for the computation of the Jordan canonical form of regular matrix polynomials is proposed The new method contains rank conditions of suitably defined block Toeplitz matrices and it does not require the computation of the Jordan chains or the Smith form The Segré and Weyr characteristics are also considered Keywords: matrix polynomial; companion linearization; Jordan canonical pair; Jordan chain; Segré characteristic; Weyr characteristic AMS Subject Classifications: 65F3; 5A2; 5A22; 34A3; 93C5 Introduction and preliminaries Consider a linear system of ordinary differential equations of the form A m q (m) (t) + A m q (m ) (t) + + A q () (t) + A q(t) = f(t), () where A j C n n (j =,,, m) and q(t), f(t) are continuous vector functions with values in C n (the indices on q(t) denote derivatives with respect to the independent variable t) Applying the Laplace transformation to () yields the associated matrix polynomial P (λ) = A m λ m + A m λ m + + A λ + A, (2) whose spectral analysis leads to the general solution of () The suggested references on matrix polynomials and their applications to differential equations Department of Mathematics, University of Athens, Panepistimioupolis 5784, Athens, GREECE ( gkaloger@mathuoagr) 2 Department of Mathematics, National Technical University, Zografou Campus 578, Athens, GREECE ( ppsarr@mathntuagr) 3 Control Engineering Research Centre, School of Engineering and Mathematical Sciences, City University, Northampton Square, London ECV HB, UK ( nkarcanias@cityacuk)

2 are [4,, 5] To facilitate the presentation, a brief description of the Jordan structure of the matrix polynomial P (λ) in (2) follows A system of vectors {x, x,, x k }, where x j C n (j =,,, k) and x, is called a Jordan chain of length k + of P (λ) corresponding to the eigenvalue λ and the eigenvector x if it satisfies the equations ξ j= j! P (j) (λ ) x ξ j = ; ξ =,,, k The vectors in a Jordan chain are not uniquely defined, and for m >, they need not be linearly independent [4] The set of all finite eigenvalues of P (λ), that is, σ F (P ) = {λ C : detp (λ) = }, is called the finite spectrum of P (λ) In this note, we assume that P (λ) in (2) is regular, ie, detp (λ) is not identically zero As a consequence, σ F (P ) contains no more than nm elements Let λ σ F (P ) and let {x,, x 2,,, x ρ, } be a basis of the null space Null P (λ ) Suppose that for every l =, 2,, ρ, a maximal Jordan chain of P (λ) corresponding to λ and x l, is of the form {x l,, x l,,, x l,sl } and has length s l Then define J P (λ ) = J J 2 J ρ, where J l (l =, 2,, ρ) is the s l s l (elementary) Jordan block having all its diagonal elements equal to λ, ones on the super diagonal and zeros elsewhere Choosing the order of the Jordan chains such that s s 2 s ρ, the set F P (λ ) = {s, s 2,, s ρ } is uniquely defined [4, ] and it is said to be the Segré characteristic of P (λ) at the eigenvalue λ (for linear pencils, see [8, 9, 4]) In this way, the matrix J P (λ ) is also uniquely defined Furthermore, if X P (λ ) = [ ] x, x, x,s x 2, x 2,s2 x ρ,sρ is the n (s + s s ρ ) matrix whose columns are the vectors of the Jordan chains above, then the pair (X P (λ ), J P (λ )) is known as a Jordan pair of P (λ) corresponding to λ [4, p 84] The value λ = is said to be an eigenvalue of P (λ) if ˆλ = is an eigenvalue of the algebraic dual matrix polynomial of P (λ), namely, ˆP (λ) = λ m P (/λ) = A λ m + A λ m + + A m λ + A m, (3) or equivalently, if the matrix A m is singular The spectrum of P (λ) is defined by σ(p ) = σ F (P ) when A m is nonsingular, and by σ(p ) = σ F (P ) { } when A m is singular Moreover, a Jordan pair of P (λ) corresponding to the infinity is defined by (X P,, J P, ) = (X ˆP (), J ˆP ()) [4, p 85] If the finite spectrum of P (λ) is σ F (P ) = {λ, λ 2,, λ k }, then denote X P,F = [ X P (λ ) X P (λ 2 ) X P (λ k ) ] and J P,F = k j= J P (λ j ) The nm nm matrix J P = J P,F J P, and the pair ([ X P,F X P, ], J P,F J P, ) are known as the Jordan canonical form (which is unique up to permutations of the diagonal Jordan blocks of J P,F ) and a Jordan canonical pair of P (λ), respectively, and provide a deep insight into the structure of the system () [3, 4, 6,, 3, 5] In particular, suppose that the size of J P,F is ζ ζ, ν is the smallest positive 2

3 integer satisfying JP, ν = and the vector function f(t) is (ν + m )-times continuously differentiable Then the general solution of () is given by the formula [4, Theorem 8] t ν q(t) = X P,F e t J P,F c + X P,F e (t s) J P,F Y P,F f(s) ds X P, J P, Y P, f (j) (t), t where the vector c C ζ is arbitrary, and the matrices Y P,F C ζ n and Y P, C (nm ζ) n are directly computable by ([ X P,F X P, ], J P,F J P, ) [4] In [, 5], one can find a second (algebraic) method for the construction of the Jordan matrix J P, which is based on the notions of the elementary divisors and the Smith form of P (λ) In general, this alternative procedure brings also some inconvenience since it requires the computation of the (monic) greatest common divisor of all j j minors of P (λ) for every j =, 2,, n In this paper, generalizing results of Karcanias and Kalogeropoulos [8] (see Theorem below), we obtain a new methodology for the construction of J P, which contains rank conditions of suitably defined block Toeplitz matrices (Theorem 2) Note that the calculation of the ranks can be done in an efficient and reliable way by using existing algorithms based on the well known singular value decomposition or the rank-revealing QR factorizations (see [, 2] and the references therein) The new method does not require the computation of the Jordan chains or the elementary divisors of P (λ), and it can also be formulated in terms of the notion of the Weyr characteristic [8, 9, 4] Moreover, in Section 3, we give an overall algorithm and two illustrative examples 2 A piecewise arithmetic progression property Let P (λ) = A m λ m + + A λ + A be an n n regular matrix polynomial as in (2) An nm nm linear pencil S λ + S is called a linearization of P (λ) if j= E(λ)(S λ + S )G(λ) = P (λ) I n(m ) for some nm nm unimodular matrix polynomials E(λ) and G(λ), ie, with constant nonzero determinants The companion linearization of P (λ) is defined by the nm nm linear pencil L P (λ) = (I n(m ) A m )λ C P I I I = λ I I A m A A A m 2 A m By [4, p 86] (see also [,, 3]), (4) E(λ)L P (λ)g(λ) = P (λ) I n(m ), (5) 3

4 where the nm nm matrix polynomials E (λ) E 2 (λ) E m (λ) I I E(λ) = I I with E m (λ) = A m and E j (λ) = A j + λe j+ (λ) (j = m, m 2,, ), and G(λ) = I Iλ I Iλ 2 Iλ Iλ m Iλ m 2 Iλ I are unimodular In particular, dete(λ) = ± and detg(λ) =, and the matrix functions E(λ) and G(λ) are also (unimodular) matrix polynomials By (5), it is obvious that σ(l P ) σ(p ) At this point, we remark that the notion of the Jordan canonical form of P (λ) defined in the previous section is equivalent to the Weierstrass canonical form T (λ) = (Iλ J P,F ) (J P, λ I) of the companion linearization L P (λ) The linear pencil T (λ) is also a linearization of P (λ) and satisfies T (λ) = ML P (λ)n for some nonsingular matrices M, N C nm nm [4, Theorems 73 and 76] Furthermore, the connection between J P, and the Smith form of P (λ) at infinity (see [5] for definitions and details) is given by [3, Theorem ] (see also [5, Proposition 44]) By the uniqueness of the Weierstrass canonical form of a linear pencil [3], one can obtain the following lemma, and for clarity, we give a simple proof Lemma The matrix polynomial P (λ) in (2) and its companion linearization L P (λ) in (4) have exactly the same Jordan canonical form Proof By (5) and [, Theorem 77] (see also [3, Proposition 2]), it is clear that J P,F = J LP,F For the infinite spectrum, consider the algebraic dual matrix polynomial ˆP (λ) in (3) and its companion linearization L ˆP (λ) For the nm nm unimodular matrix polynomial G(λ) in (6) and the algebraic dual pencil of L P (λ), that is, I Iλ I ˆL P (λ) = I Iλ A λ A λ A m 2 λ A m λ + A m, (6) 4

5 we have ˆL P (λ)g(λ) T = I I I A λ A λ 2 + A λ A λ m + + A m 2 λ ˆP (λ) Thus, ˆL P (λ) is equivalent to I n(m ) ˆP (λ) Moreover, by the discussion on the companion linearizations, it follows that there exist two nm nm unimodular matrix polynomials F (λ) and F 2 (λ) such that I n(m ) ˆP (λ) = F (λ)l ˆP (λ)f 2 (λ) Hence, the linear pencils L ˆP (λ) and ˆL P (λ) are equivalent, and by [, Theorem 77], J L ˆP, = J ˆLP, Consequently, J P, = J ˆP, = J L ˆP, = J ˆLP, = J L P, Keeping in mind the above lemma, we can compute J P = J P,F J P, applying to L P (λ) the following result [8] by Theorem (For linear pencils) Consider an n n regular linear pencil Aλ+B and an eigenvalue λ σ(aλ+ B) For k =, 2,, define the nk nk matrix Q k (λ ) = Aλ + B A Aλ + B A A Aλ + B when λ, and Q k ( ) = A B A B B A when λ = Then define the sequence and set ν (λ ) = ν k (λ ) = dim Null Q k (λ ) ; k =, 2, (i) For every k =, 2,, 2 ν k (λ ) ν k (λ ) + ν k+ (λ ) (ii) If 2 ν k (λ ) = ν k (λ ) + ν k+ (λ ) for some k =, 2,, then there is no k k Jordan block of Aλ + B corresponding to λ 5

6 (iii) If 2 ν k (λ ) > ν k (λ ) + ν k+ (λ ) for some k =, 2,, then the Jordan canonical form of Aλ + B has exactly 2 ν k (λ ) (ν k (λ ) + ν k+ (λ )) Jordan blocks of order k corresponding to λ Moreover, there exists a positive integer τ such that ν (λ ) < ν (λ ) < < ν τ (λ ) < ν τ (λ ) = ν τ +(λ ) = It is worth noting that for the companion linearization L P (λ) in (4), Q k (λ ) is an nmk nmk matrix Thus, the problem of obtaining a direct rank condition for P (λ) arises in a natural way Notice also that the derivative of the linear pencil Aλ+B in Theorem is (Aλ+B) () = A, and that for every j = 2, 3,, (Aλ + B) (j) = Theorem 2 (For matrix polynomials) Let P (λ) = A m λ m + + A λ + A be an n n regular matrix polynomial and let λ σ(p ) For k =, 2,, consider the nk nk matrix P (λ )! P () (λ ) P (λ ) R k (λ ) = 2! P (2) (λ )! P () (λ ) (k )! P (k ) (λ ) (k 2)! P (k 2) (λ )! P () (λ ) P (λ ) when λ, and R k ( ) = A m A m A m A m 2 A m A m k+ A m k+2 A m A m when λ =, where it is assumed that for j =, 2,, A j = Define the sequence ν k (λ ) = dim Null R k (λ ) ; k =, 2, and set ν (λ ) = (i) For every k =, 2,, 2 ν k (λ ) ν k (λ ) + ν k+ (λ ) (ii) If 2 ν k (λ ) = ν k (λ ) + ν k+ (λ ) for some k =, 2,, then there is no k k Jordan block of P (λ) corresponding to λ (iii) If 2 ν k (λ ) > ν k (λ ) + ν k+ (λ ) for some k =, 2,, then the Jordan canonical form of P (λ) has exactly 2 ν k (λ ) (ν k (λ )+ν k+ (λ )) Jordan blocks of order k corresponding to λ Moreover, there is a positive integer τ such that ν (λ ) < ν (λ ) < < ν τ (λ ) < ν τ (λ ) = ν τ +(λ ) = (7) 6

7 Proof Assume that λ σ F (P ) In the case λ =, the proof is similar (see also Remark below) For any k =, 2,, consider the nmk nmk matrix L P (λ ) I n(m ) A m L P (λ ) Q LP,k(λ ) = I n(m ) A m, I n(m ) A m L P (λ ) where L P (λ) is the companion linearization of P (λ) in (4) Recall (5) and observe that Q LP,k(λ ) has the same rank with the nmk nmk matrix E(λ ) G(λ ) E(λ ) M LP,k(λ ) = Q G(λ ) L P,k(λ ) E(λ ) G(λ ) = P (λ ) I n(m ) S(λ ) P (λ ) I n(m ) S(λ ) S(λ ) P (λ ) I n(m ) where the nm nm matrix polynomial S(λ) = E(λ)(I n(m ) A m )G(λ) is of the form B (λ) B 2 (λ) B m (λ) B m (λ) I S(λ) = Iλ I Iλ m 2 Iλ m 3 I, with B l (λ) = (m + l)a m λ m l + + 2A l+ λ + A l ; l =, 2,, m For j = k, k,, 2, by straightforward computations, we eliminate all the n n submatrices I, Iλ,, Iλ m 2 in the (j, j )-th block S(λ ) of the matrix M LP,k(λ ) by using the n n identity matrices on the main diagonal of the (j, j)-th block of M LP,k(λ ), that is, P (λ ) I P (λ ) I n(m ) = I I 7

8 This elimination process leads to an nmk nmk matrix of the form P (λ ) I I! P () (λ ) B 2 (λ ) B m (λ ) P (λ ) I I 2! P (2) (λ )! P () (λ ) P (λ ) I -st column (m+)-th column (2m+)-th column Consider this matrix as an mk mk block matrix with blocks of order n Then every diagonal block coincides with either the matrix P (λ ) or the identity matrix Moreover, for every ξ {, 2,, mk} \ {, m +,, (k )m + }, its ξ-th row is I ξ-th column and for every j =,,, k, its (jm + )-th column is (jm + )-th row P (λ ) (jm + m + )-th row! P () (λ ) (jm + 2m + )-th row 2! P (2) (λ ) 8

9 By changing the order of rows and columns, it follows that M LP,k(λ ) is equivalent to an nmk nmk matrix of the form [ ] Rk (λ ) I n(m )k As a consequence, rank Q LP,k(λ ) = rank M LP,k(λ ) = n(m )k + rank R k (λ ), or equivalently, dim Null Q LP,k(λ ) = nk rank R k (λ ) = dim Null R k (λ ) The proof is completed by Lemma and Theorem Remark For the algebraic dual matrix polynomial ˆP (λ) in (3) and for λ =, we have j! ˆP (j) () = A m j (j =,,, m) Hence, the motivation for the definition of the matrix R k ( ) (k =, 2, ) in the above theorem is clear Note also that for λ =, a simple proof of the second part of Theorem 2 (for the infinite spectrum) follows by applying the first part of the theorem (for the finite spectrum) to ˆP (λ) and its eigenvalue ˆλ = Remark 2 Theorem 2 admits an alternative direct proof, which is independent from the results in [8], based on the following considerations Let λ σ(p ) and consider the Segré characteristic F P (λ ) = {s, s 2,, s ρ } (s s 2 s ρ ) Let also x l,, x l,,, x l,sl ; l =, 2,, ρ be a set of corresponding (not uniquely defined) maximal Jordan chains If for any k =, 2,, we denote ζ l = min{k, s l } (l =, 2,, ρ), then by [5, Lemma 25], the vectors x l, x l, x l,j C nk ; l =, 2,, ρ, j = ζ l, ζ l 2,,, form a basis of the null space of R k (λ ) Thus, dim Null R k (λ ) = ρ l= ζ l, and the statements of Theorem 2 follow readily The positive integer τ that satisfies (7) is said to be the index of annihilation of J P (λ ) This index coincides with the largest length of Jordan chains of P (λ) 9

10 corresponding to λ, ie, with the size of the largest Jordan blocks of J P (λ ) (see [6] for meromorphic matrix functions) The set W P (λ ) = {w j (λ ) = ν j (λ ) ν j (λ ), j =, 2,, τ } is made up from positive integers and it is called the Weyr characteristic of J P (λ ) By [8, Remark 52] and Theorem 2, it follows that for every j =, 2,, τ, the difference w j (λ ) = ν j (λ ) ν j (λ ) is the number of the Jordan blocks of λ of order at least j An eigenvalue λ of the matrix polynomial P (λ) in (2) is called semisimple if all the corresponding Jordan blocks are of order, ie, the matrix J P (λ ) is diagonal Semisimple eigenvalues play an important role in the study of the stability of matrix polynomials under perturbations (see [7] and its references) Theorem 2 leads directly to a characterization of these eigenvalues Corollary An eigenvalue λ σ F (P ) is semisimple if and only if [ P (λ ) dim Null P (λ ) = dim Null P () (λ ) P (λ ) Proof Let λ σ F (P ) and consider the sequence ν (λ ), ν (λ ), ν 2 (λ ), defined in Theorem 2 Then λ is a semisimple eigenvalue of P (λ) if and only if the index of annihilation of J P (λ ) is τ =, or equivalently, if and only if ν (λ ) = ν 2 (λ ) 3 Numerical examples Let P (λ) = A m λ m + + A λ + A be an n n regular matrix polynomial, and let λ σ(p ) Then we can compute the Segré characteristic of P (λ) at the eigenvalue λ, F P (λ ) = {s, s 2,, s ρ } (s s 2 s ρ ), or equivalently, the Jordan matrix J P (λ ), by applying the following algorithm Step I Let ν =, ν = n rank P (λ ) and k = ] Step II While ν k ν k, repeat: (a) set k = k +, (b) if λ, then construct P (k ) (λ ), (c) construct the matrix R k (λ ) as in Theorem 2, (d) compute ν k = nk rank R k (λ ) Step III Let τ = k be the index of annihilation of J P (λ )

11 Step IV For j =, 2,, τ, compute the differences d j = 2 ν j (ν j + ν j+ ) Step V Let η = Step VI For j = τ, τ,, 2,, repeat: if d j >, then set s η+ = = s η+dj = j and η = η + d j Step VII Print the numbers s (= τ ) s 2 s ρ Our results are illustrated in the following two examples In the first example we construct the Jordan canonical form of a matrix polynomial with finite spectrum In the second one, we consider the infinite case Example [2, Example 6] Consider the 3 3 quadratic matrix polynomial P (λ) = I 3 λ λ Then it is easy to see that detp (λ) = (λ + 2) 6 Hence, P (λ) has exactly one eigenvalue, 2, of algebraic multiplicity 6 By [2], the Jordan canonical form of P (λ) is J P J P,F J P ( 2) = We can verify that ν ( 2) = [ 2 2 ν ( 2) = dim Null R ( 2) = dim Null P ( 2) = 2 [ ] P ( 2) ν 2 ( 2) = dim Null R 2 ( 2) = dim Null P () = 4 ( 2) P ( 2) ν 3 ( 2) = dim Null R 3 ( 2) P ( 2) = dim Null P () ( 2) P ( 2) = 5 2 P (2) ( 2) P () ( 2) P ( 2) ν 4 ( 2) = dim Null R 4 ( 2) P ( 2) = dim Null P () ( 2) P ( 2) 2 P (2) ( 2) P () ( 2) P ( 2) 2 P (2) ( 2) P () ( 2) P ( 2) ν 5 ( 2) = dim Null R 5 ( 2) = = 6 ] (8) = 6

12 Thus, our methodology implies that P (λ) has exactly Jordan block of order 4 and 2 ν 4 ( 2) (ν 3 ( 2) + ν 5 ( 2)) = 2 = 2 ν 2 ( 2) (ν ( 2) + ν 3 ( 2)) = 8 7 = Jordan block of order 2 corresponding to the eigenvalue λ = 2, and that the index of annihilation of J P ( 2) is τ = 4, confirming Theorem 2 Furthermore, by writing the Weyr characteristic W P ( 2) = {w j ( 2) = ν j ( 2) ν j ( 2), j =, 2, 3, 4} in the Ferrer diagram w ( 2) = ν ( 2) ν ( 2) = 2 = 2 w 2 ( 2) = ν 2 ( 2) ν ( 2) = 4 2 = 2 w 3 ( 2) = ν 3 ( 2) ν 2 ( 2) = 5 4 = w 4 ( 2) = ν 4 ( 2) ν 3 ( 2) = 6 5 = Jordan block Jordan block of order 4 of order 2 and observing that ν 5 ( 2) = ν 4 ( 2) = 6, we conclude once again that the Jordan canonical form of the matrix polynomial P (λ) is given by (8) Example 2 [4, Example 7] For the 2 2 matrix polynomial [ ] [ Q(λ) = λ ] [ λ ] [ λ + it is known [4] that a Jordan pair corresponding to the infinity is ([ ] [ ]) (X Q,, J Q, ) =, One can see that ν ( ) = [ ] ν ( ) = dim Null R ( ) = dim Null = ν 2 ( ) = dim Null R 2 ( ) = dim Null 3 = 2 ν 3 ( ) = dim Null R 3 ( ) ], 2

13 = dim Null = 2 Hence, according to Theorem 2, the matrix polynomial Q(λ) has exactly 2 ν 2 ( ) (ν ( ) + ν 3 ( )) = 4 3 = Jordan block of order 2 corresponding to the infinity Acknowledgments The authors acknowledge with thanks an anonymous referee for suggesting the direct proof described in Remark 2 References [] CH Bischof, G Quintana-Orti, Computing rank-revealing QR factorizations of dense matrices, ACM Transactions on Mathematical Software 24 (998) [2] P Freitas, P Lancaster, On the optimal value of the spectral abscissa for a system of linear oscillators, SIAM J Matrix Anal Appl 2 (999) [3] FR Gantmacher, The Theory of Matrices, 2nd edition, Chelsea Publishing Company, New York, 959 [4] I Gohberg, P Lancaster, L Rodman, Matrix Polynomials, Academic Press, New York, 982 [5] I Gohberg, L Rodman, Analytic matrix functions with prescribed local data, J Anal Math 4 (98) 9-28 [6] GE Hayton, AC Pugh, P Fretwell, Infinite elementary divisors of a matrix polynomial and its implications, Internat J Control 47 (988) [7] R Hryniv, P Lancaster, On the perturbations of analytic matrix functions, Integral Equations and Operator Theory 34 (999) [8] N Karcanias, G Kalogeropoulos, On the Segré characteristics of right (left) regular matrix pencils, Internat J Control 44 (986) 99-5 [9] N Karcanias, G Kalogeropoulos, The prime and generalized nullspaces of right regular pencils, Circuits Systems and Signal Process 4 (995) [] P Lancaster, M Tismenetsky, The Theory of Matrices, 2nd edition, Academic Press, Orlando, 985 [] AS Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Amer Math Society, Providence, Translations of Math Monographs, Vol 7, 988 3

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