Gohberg, I., Lancaster, P. and Rodman, L., Matrix Polynomials, Academic Press, New York, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....structures. In the interest of eciency and stability, any good numerical method for solving problems of this type should preserve and exploit these structures. 3 Linearization The classical approach to solving a k th degree polynomial eigenvalue problem of dimension m is to linearize it, [12], i.e. to transform it to an equivalent rstdegree equation Ax Bx = 0 of dimension km. There are many di erent such linearizations with very di erent numerical properties, see [26, 29] Since the problems that we have presented in Section 2 have a speci c symmetry structure we will demonstrate ....

I. Gohberg, P. Lancaster, and L. Rodman. Matrix Polynomials. Academic Press, New York, 1982.

....root ( 1 = 2 ) This is the smallest amount of damping for which no oscillation occurs in the free response q(t) The properties of systems of second order differential equations (2. 1) for n 1 have been analyzed in some detail by Lancaster [87] and more recently by Gohberg, Lancaster and Rodman [62]. For simplicity, we examine the case where all the eigenvalues i , i = 1: 2n of ( M K C)x = 0 are distinct. The case of non distinct eigenvalues is considered in section 3.7. We write = diag( 1 ; 2 ; 2n ) X = x 1 ; x 2 ; x 2n ] Y = y 1 ; y 2 ; y 2n ] ....

....= diag e 1 ( e 2 ( e n ( and e i ( is a monic polynomial such that e i ( divides e i 1 ( The diagonal matrix Gamma ( is called the Smith form or canonical form of Q( and it is unique, though E( and F ( are not. For a proof of the decomposition (3. 4) we refer to [58] [62], 149, pp. 19 20] The polynomials e 1 ( e n ( are called the invariant polynomials of Q( To gain some insight into the decomposition (3.4) we consider the case where Q( has 2n distinct eigenvalues. In this case, one can show (see [87, sec. 3.3] for instance) that e i ( 1, i = ....

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I. Gohberg, P. Lancaster, and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.

....eigenvalue and x the corresponding right eigenvector; y 0 is a left eigenvector if y P( 0. The set of eigenvalues of P is denoted by A(P) When Am is nonsingular P has mr, finite eigenvalues, while if Am is singular P has infinite eigenvalues. Good references for the theory of ) matrices are [8], 20] 21] 37] Throughout this paper we assume that P has only finite eigenvalues (and pseudoeigenvalues) how to deal with infinite eigenvalues is described in [16] 2.2) AP( XmAAm xm iAAm i AAo. We define the e pseudospectrum of P by A(P) C: P( AP( vc = 0 for ....

I. GOHBERG, P. LANCASTER, AND L. RODMAN, Matrix Polynomials, Academic Press, New York, 1982.

....0; 8t 2 R: Denote by K the set of all matrix valued polynomials from W which are nonnegative definite everywhere on R. The above calculation shows that K ae K: We wish to show that K = K: The crucial property here is that each w 2 K admits a factorization of the form (see e.g. [7]) w = Delta Delta Delta = A i t ; Delta A ; t 2 R: 11 Here A i 2 Matm (C) the set of all m by m matrices with complex entries ) and A is conjugate transpose of A i . If A i = B i Gamma1C i ; B i ; C i 2 Matn (R) B i B j C i C j )t Here we used ....

I. Gohberg, P. Lancaster and L. Rodman, Matrix polynomials, Academic Press, NY, 1982, pp 409.

....basis for the kernel of an analytic linear operator. In order to construct an analytic basis for the kernel of a linear operator, we will need the following two results due to Gohberg Rodman [14] The form of the first result that we need is Lemma S6.2 on page 389 in Gohberg, Lancaster Rodman [13]. Lemma 3.1 ( 14, 13] Let u 1 (#) u k (#) be n dimensional vector functions which are analytic for all # # . Suppose that for some # 0 # the vectors u 1 (# 0 ) u k (# 0 ) are linearly independent, and let # 0 = # : u 1 (#) u k (#) are linearly dependent . ....

....of an analytic linear operator. In order to construct an analytic basis for the kernel of a linear operator, we will need the following two results due to Gohberg Rodman [14] The form of the first result that we need is Lemma S6.2 on page 389 in Gohberg, Lancaster Rodman [13] Lemma 3. 1 ([14, 13]) Let u 1 (#) u k (#) be n dimensional vector functions which are analytic for all # # . Suppose that for some # 0 # the vectors u 1 (# 0 ) u k (# 0 ) are linearly independent, and let # 0 = # : u 1 (#) u k (#) are linearly dependent . Then there exist ....

[Article contains additional citation context not shown here]

I. Gohberg, P. Lancaster & L. Rodman. Matrix Polynomials, Academic Press: New York (1982).

....r 1 t I m where I m is the m m identity matrix. Thus P r (M) 2 IR if M 2 IR and P r (t) P r (tI m ) 2 IR . We will make use of the notation P r (M) v = v 0 M v r 1 M v 11 where M 2 IR and v 2 IR . So, P r (M) v) 2 IR . On matrix polynomials, see [17]. P r (A k ) v 0 = v 0 0 A k v 0 = v 0 0 v r 1 r 1 H k v r = 0: It follows that P r (A k ) v i = 0 for i = 0; r 1. Indeed, from what precedes, k (P r (A k ) v 0 ) 0 = A k v 0 0 A k v 0 (17) v i 0 A k v i r 1 ....

I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.

....P (x) 3. 1 Direct approach Since Popov forms appear at the denominator of irreducible fractions, the problem is to find a suitable transfer function H(x) associated to P (x) i.e. such that the Popov form of the denominator of an irreducible description of H(x) is T (x) But, as emphasized by [14], to study a nonsingular matrix polynomial P (x) it is natural to study systems for which the transfer function is H (x) Precisely we are going to see that H leads to a suitable choice H(x) to compute T (x) We denote by I(n; d) the problem of inverting a matrix of degree d in M n;n ....

I. Gohberg, P. Lancaster, and L. Rodman. Matrix polynomials. Academic Press, NewYork, 1982.

....h r 1 t I m where I m is the m m identity matrix. Thus P r (M) 2 IR if M 2 IR and P r (t) P r (tI m ) 2 IR . We will make use of the notation P r (M) v = vh 0 M vh r 1 M v where M 2 IR and v 2 IR . So, P r (M) v) 2 IR (on matrix polynomials, see [13]) Such polynomials were already used in the multiparameter Lanczos method [5] where they were related to formal vector orthogonality, and in the block Lanczos method [12] where they were related to formal matrix orthogonality. We have P r (A k ) v 0 = v 0 h 0 A k v 0 = v 0 h 0 ....

I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.

....of all i by i minors of A . Then the diagonal entries in the Smith normal form of A are s 1,1 = s 1 , and s i ,i = s i s i 1 , for i 1. 3. Two n by n matrices A and B have the same Smith normal form if and only if they are equivalent. For a proof see Gohberg, Lancaster, and Rodman [5] or Newman [17] Let C i denote all i element subsets of 1 , n and let A I ,J , for I , J C i , denote the minor of A restricted to the rows in I and columns in J . By the above theorem we could compute the Smith normal form of A by computing s = GCD I ,J C i n A I ,J . The ....

Gohberg, I., P. Lancaster, and L. Rodman, Matrix Polynomials, Academic Press, 1982.

..... H f Gamma1 7 7 7 7 7 7 7 7 7 ; V e = 6 6 6 6 6 6 6 6 6 V J . 7 7 7 7 7 7 7 7 7 : 41) By direct substitution one can show that EV e = V e J: Since V is nonsingular, the columns of V e are linearly independent. It is also shown in [22] that the eigenvalues of E are exactly the singularities of H(s) so E has m eigenvalues in C . We therefore conclude that Im V e is the C invariant subspace of E. Let us assume the existence of another C stable matrix G 1 satisfying H(G) 0. Let us also write G 1 in the form V 1 J 1 ....

I. C. Gohberg, P. Lancaster, and L. Rodman. Matrix Polynomials. Academic Press, New York, 1982.

....he was a visitor at CWI. The above equations give a realization (in the behavioral sense, see [14] of (1. 1) There is a straightforward generalization of this for vector equations of the form P ( w(t) 0 when P (s) 2 R p Thetap is monic, i.e. P (s) with P = I ; see for instance [3, 7, 8] where the term linearization is used rather than realization . The situation becomes more complicated if P is nonsingular or not even square. Indeed, assume that P (s) i=0 P i s is a p Theta (m p) polynomial matrix. One readily verifies that the system P ( dt )w = 0 is represented ....

I. Gohberg, P. Lancaster and L. Rodman (1982). Matrix Polynomials. Academic Press, New York.

..... 1 . F d d . A matrix C is in rational canonical form if C is block diagonal with companion matrices on its diagonal blocks, C = diag (C f 1 (x ) C f m (x ) and f i (x ) divides f i 1 (x ) for all 1 i m 1. We have the following lemma, cf [5] Chapter VI, or [8], Chapter S1. 4.1 Lemma. Let A , B F . 1) A is similar to B if and only if xI A and xI B are equivalent, they must have the same Smith normal forms. 2) Let diag (s 1 (x ) s n (x ) be the Smith normal form of xI A . Then C A = diag (C s 1 (x ) C s n (x ) is ....

....blocks may have the same eigenvalue and or the same size. In fact, each n i by n i block corresponds to an elementary divisor (x l i ) of A . The elementary divisors are simply the maximal powers of linear factors of the invariant factors of A . We refer, e.g. to [5] Chapter VI, 6, or [8], Chapter S1, for proofs of these facts. The only complication in formulating an algorithm for finding the Jordan normal form is that l i can lie in an algebraic extension of F and there is no unique way to represent l i . If we assume that F already contains the eigenvalues of A and that the ....

Gohberg, I., Lancaster, P., and Rodman, L., Matrix Polynomials, Academic Press, New York, NY, 1982.

.... Delta I Gamma H 2 Delta Delta Delta Gamma H f Gamma1 7 7 7 7 ; 23) V e = V (V J) Delta Delta Delta (V J : 24) By direct substitution one can show that EV e = V e J: Since V is nonsingular, the columns of V e are linearly independent. It is also shown in [10] that the eigenvalues of E are exactly the singularities of H(s) so E has m eigenvalues in C . We therefore conclude that Im V e is the C invariant subspace of E. Let us assume the existence of another C stable matrix G 1 satisfying H(G) 0. Let us also write G 1 in the form V 1 J 1 ....

I. C. Gohberg, P. Lancaster, and L. Rodman. Matrix Polynomials. Academic Press, New York, 1982.

....assumption In this section we brieAEy discuss the implementation of the algebraic reduction technique of [1] under the non degeneracy assumption. For more details on the inversion of an analytically perturbed operator the interested reader is also referred to the following papers and books: [7, 8, 13, 20, 23, 24]. Let (X; h: i) be a nite dimensional Hilbert space and consider a family of linear operators L (i) X X , i = 0; 1; Let L(z) X X be the linear operator L(z) L (0) zL (1) z 2 L (2) 11) that is, L(z) is an analytic perturbation of L (0) Consider the ....

I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press, Newyork, 1982.

....zero with multiplicity m 1 and that the perturbed matrices A( also have eigenvalue zero with multiplicity m for suOEciently small but dioeerent from zero. We emphasize that the dimension of the perturbed null space does not depend on in some small punctured neighbourhood around = 0 [11, 12]. When the perturbation parameter deviates from zero, the zero eigenvalues of the unperturbed matrix can split into zero and non zero eigenvalues [5, 17] This fact implies that m m. More detailed discussion on the stability properties of null spaces can be found in [4] and the references ....

....V = I m : 4) 2 Similarly, let v i ( i = 1; m be linearly independent eigenvectors of the perturbed matrix A( corresponding to the eigenvalue zero. Again one can form the matrix V ( v 1 ( v m ( which satises the equation A( V ( 0: 5) From Chapter S6 of [11] and the results of [12] we know that there exists a holomorphic family of vector valued functions v i ( which constitute a basis for the null space of A( for 6= 0. Therefore, V ( can be expressed as a power series in some neighbourhood of zero V ( V 0 V 1 2 V 2 Delta ....

I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.

....latter coeOEcient also coincides with the one obtained with the help of MATLAB symbolic toolbox. 6. 4 Asymptotics for P 0 (z) We know that the eigenprojection P 0 (z) of the perturbed operator corresponding to the identically zero eigenvalue is analytic in some (punctured) neighbourhood of z = 0 [4, 14], that is, P 0 (z) P 00 1 X k=1 z k P 0k ; for z 6= 0: 46) For regular perturbations, P 00 is just P 0 (0) and the group projection coincides with the eigenprojection. This is not the case for singular perturbations. Therefore, an interesting question is how P 00 in (46) relates to ....

I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, (Academic Press, New York, 1982).

....output. Fortunately, the latter can be eciently analysed by the techniques developed for the singular analytic perturbations. The rst comprehensive study of the analytic perturbation theory was given in the book of Kato [87] This book later became a classic in perturbation theory. Some authors [60, 108, 124] have also studied a particular case of (1.3) that is polynomial perturbation A( A 0 A 1 : p A p : 1.4) Now a sceptical reader has a right to ask the following question: If the perturbation theory has been greatly developed in several monographs, is it possible to nd enough ....

....on Hilbert space. In particular, he showed that the principal part of the Laurent series expansion for the inverse operator A 1 ( can be given in terms of generalized Jordan chains. The generalized Jordan chains were initially developed in the context of matrix and operator polynomials (see [60, 108, 124] and numerous references therein) However, this concept can be easily generalized for the case 5 Analytic Perturbation of Singular Linear Systems 6 of analytic perturbation A( A 0 A 1 2 A 2 : Following [61] and [62] we say that the vectors 0 ; r 1 form a generalized ....

[Article contains additional citation context not shown here]

I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.

....corresponding to fl 0 as S ( 1 b ) fl 0 (t; and setting S ( 1 b ) g (t; S ( 1 b ) t; we have [19] S ( 1 b ) fl 0 (t; S ( 1 b ) g (t; I p : 17) 5 Unimodularity of the Zibulski Zeevi matrices. A square polynomial matrix P(z) with z 2CI is unimodular [11, 24] if its determinant equals a nonzero constant, i.e. det[P(z) c with c 2 CI ; c 6= 0: The inverse of a unimodular matrix is again a unimodular polynomial matrix. Conversely, if P(z) is a polynomial matrix with polynomial inverse, P(z) is unimodular. In the following we use the terminology ....

....2 [0; 1) is given by j F 0 b k . Conversely, if g (Z 1 b g) t; has a finite maximum degree for 2 [0; 1) the function g is strictly band limited. 9 The proof of Lemma 2. 2 can be found in [4] In the following, we shall also need the Smith form decomposition of polynomial matrices [11, 24], which essentially allows to decompose polynomial matrices into simpler forms such as triangular and diagonal forms. This decomposition will prove useful in what follows. Since for a compactly supported g the q Theta p matrix G (a) t; is a polynomial matrix in e 2i for all t, it can be ....

[Article contains additional citation context not shown here]

I. Gohberg, P. Lancaster, and L. Rodman. Matrix Polynomials. Academic Press, 1982.

....= L since dim P = n. Now note that the embedding from Corollary 1 can be modified slightly into an embedding : R n , P n Gamma1 R Theta R. This last embedding can then be used to transform our system into the following randomized version of the n Thetan Thetad matrix polynomial problem [GLR82] (modulo a set of 0 measure) For all j 2 [0: d] let A j be an n Theta n matrix consisting of independent real Gaussian random variables with mean 0 and variance d j , and consider the values of such that the matrix A 0 A 1 Delta Delta Delta d A d is singular. Letting x : ....

Gohberg, I., Lancaster, P., and Rodman, L., Matrix Polynomials, Academic Press, 1982.

....operator pencil P(#) has discrete spectrum without finite accumulation points, every eigenvalue of P(#) is semisimple, and the system of eigenfunctions of P(#) is two fold complete. There is a large body of research that deals with the spectral theory of operators, in particular we mention [2, 4, 11, 12, 16, 17, 18, 19, 20, 21, 25]. 3.1 Orthogonality Relations In this section, we quote the new orthogonality relations between the eigenfunctions of an undamped gyroscopic quadratic operator pencil established in [26] which generalize the recent result established in [10] This result (the relation (12) plays a key role in ....

I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.

....of the matrix polynomial, M(u) have one to one correspondence with the eigendecomposition of C = 2 6 6 6 4 0 I n 0 : 0 . 0 0 0 : I n GammaM 0 GammaM 1 GammaM 2 : GammaM d Gamma1 3 7 7 7 5 (2) where M i = M Gamma1 d M i [GLR82]. In case M d is singular or ill conditioned, the intersection problem is reduced to a generalized eigenvalue problem [MD94] Algorithms to compute all the eigenvalues are based on QR orthogonal transformations [GL89] They compute all the real as well as complex eigenvalues. Algorithms to compute ....

I. Gohberg, P. Lancaster, and L. Rodman. Matrix Polynomials. Academic Press, New York, 1982.

.... A 1 ffl A 0 where r is the maximum degree in ffl of any matrix entry. If A r is non singular, we have A Gamma1 r A(ffl) ffl r A Gamma1 r A r Gamma1 ffl r Gamma1 Delta Delta Delta A Gamma1 r A 1 ffl A Gamma1 r A 0 9 and the determinant of the right hand side equals [25] the characteristic polynomial of C = 2 6 6 6 6 6 6 4 0 I t 0 Delta Delta Delta 0 0 0 I t Delta Delta Delta 0 . 0 0 0 Delta Delta Delta I t GammaA Gamma1 r A 0 GammaA Gamma1 r A 1 GammaA Gamma1 r A 2 Delta Delta Delta GammaA ....

I. Gohberg, P. Lancaster, and L. Rodman. Matrix Polynomials. Academic Press, New York, 1982.

....Furthermore, the roots of the matrix polynomial, M(u) are identical with the eigenvalues of C = 2 6 6 4 0 I2mn 0 : 0 . 0 0 0 : I2mn GammaM 0 GammaM 1 GammaM 2 : GammaM d Gamma1 3 7 7 5 (3) where M i = M Gamma1 d M i [GLR82] In case Md is singular or ill conditioned, the intersection problem is reduced to a generalized eigenvalue problem [MD94] Algorithms to compute all the eigenvalues are based on QR orthogonal transformations [GL89] 2.4 Power Iterations We use marching methods to trace the visibility curves ....

I. Gohberg, P. Lancaster, and L. Rodman. Matrix Polynomials. Academic Press, New York, 1982.

....assumption In this section we brieAEy discuss the implementation of the algebraic reduction technique of [1] under the non degeneracy assumption. For more details on the inversion of an analytically perturbed operator the interested reader is also referred to the following papers and books: [7,8,13,20,23,24]. Let (X# h:# :i) be a nite dimensional Hilbert space and consider a family of linear operators L (i) X X , i =0# 1#: Let L(z) X X be the linear operator L(z) L (0) zL (1) z 2 L (2) # (11) that is, L(z) is an analytic perturbation of L (0) Consider the linear ....

I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials,Academic Press, Newyork, 1982.

....root ( 1 = 2 ) This is the smallest amount of damping for which no oscillation occurs in the free response q(t) The properties of systems of second order differential equations (2. 1) for n 1 have been analyzed in some detail by Lancaster [87] and more recently by Gohberg, Lancaster and Rodman [62]. For simplicity, we examine the case where all the eigenvalues i , i = 1: 2n of ( 2 M K C)x = 0 are distinct. The case of non distinct eigenvalues is considered in section 3.7. We write = diag( 1 ; 2 ; 2n ) X = x 1 ; x 2 ; x 2n ] Y = y 1 ; y 2 ; y 2n ] ....

....e 1 ( e 2 ( e n ( Delta and e i ( is a monic polynomial such that e i ( divides e i 1 ( The diagonal matrix Gamma ( is called the Smith form or canonical form of Q( and it is unique, though E( and F ( are not. For a proof of the decomposition (3. 4) we refer to [58] [62], 149, pp. 19 20] The polynomials e 1 ( e n ( are called the invariant polynomials of Q( To gain some insight into the decomposition (3.4) we consider the case where Q( has 2n distinct eigenvalues. In this case, one can show (see [87, sec. 3.3] for instance) that e i ( 1, i = ....

[Article contains additional citation context not shown here]

I. Gohberg, P. Lancaster, and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.

....left eigenvector k , the sequence f k j k ; j = M;M 1; g ; 13) is a solution of equation (10) By combining multiple eigenvalues with each of their independent eigenvectors, we thus obtain a total of d linearly independent solutions. On the other hand, it is known (e.g. see [5]) that the dimensionality of the solution space of equation (10) is exactly d . Therefore, any solution of (10) can be expressed as a linear combination of the d solutions (13) v j = d X k=1 x k k j k ; j = M;M 1; 14) where x k (k = 1; 2; d) are arbitrary (complex) ....

I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press, 1982.

.... Gamma I) Z i U Z i 1 = 0; i = 1; Delta Delta Delta ; k Gamma 1; D Gamma E T ) Z k Gamma1 U Z k = 0; 1:55) 30 where the first and the last equations constitute the boundary conditions. The matrix difference equations in (1. 55) can be solved by the nonmonic matrix polynomial theory [GLR82] The nonmonic matrix polynomial Phi(i ) Ui 2 (D Gamma I)i L = 2 6 4 g 3 Gamma i g 2 i g 2 i g 3 i 2 Gamma i 3 7 5 (1:56) is the characteristic equation corresponding to the matrix difference equation (1.55) If (X; J) is the finite Jordan pair of (1.56) Theorem 8.3 in [GLR82] ....

....[GLR82] The nonmonic matrix polynomial Phi(i ) Ui 2 (D Gamma I)i L = 2 6 4 g 3 Gamma i g 2 i g 2 i g 3 i 2 Gamma i 3 7 5 (1:56) is the characteristic equation corresponding to the matrix difference equation (1.55) If (X; J) is the finite Jordan pair of (1.56) Theorem 8. 3 in [GLR82] gives the general solution of the homogeneous finite difference equations in (1.55) by the expressions Z i = XJ i C; i = 0; 1; 2; Delta Delta Delta ; 1:57) where C = c 0 ; c 1 ; c 2 ] T is a constant vector to be determined by the boundary conditions. The determinant of the matrix ....

I. Gohberg, P. Lancaster, and L. Rodman. Matrix Polynomials. Academic Press, New York, 1982.

....= a ijp z p : a ij0 . Let the characteristic polynomial of a polynomial matrix: d(z) det a(z) dm z m dm Gamma1 z m Gamma1 : d 0 be of degree m. The matrix a(z) has then typically m eigenvalues. The following facts relating p; n and m are relatively easy to establish [1, 2]: 1. a(z) is proper , m = n Delta p 2. If rank(a p ) n then m n Delta p 3. a 0 is nonsingular , d 0 6= 0 Now, the main point is that if q(z) is a nilpotent polynomial matrix with m q zero eigenvalues, that is, det q(z) z mq and if m m q = n Delta p then a(z) q(z) Delta a(z) is ....

I. Gohberg, P. Lancaster, and L. Rodman, Matrix Polynomials. London: Academic Press, 1982.

....We take the matrix equation, 15) and multiply it by A Gamma1 . Let Sigma 00 = Ix 2 3 A Gamma1 Bx 3 A Gamma1 C; where I is a 12 Theta 12 identity matrix. In practice A Gamma1 B and A Gamma1 C are computed by linear equation solvers. Given Sigma 00 , we use Theorem 1. 1 [GLR82] to construct a 24 Theta 24 matrix M of the form M = 0 I GammaA Gamma1 C GammaA Gamma1 B ; where 0; I are 12 Theta 12 null and identity matrices, respectively. It follows from the structure of M that the eigenvalues of M correspond exactly to the roots of determinant( Sigma ....

....M 1 and M 2 M 1 = I 0 0 A ; M 2 = 0 I GammaC GammaB ; where 0; I are 12 Theta 12 null and identity matrices, respectively. Furthermore, the roots of determinant( Sigma 00 ) 0, correspond exactly to the eigenvalues of the generalized eigenvalue problem M 1 Gamma x 3 M 2 [GLR82] The eigenvectors have the same structure as (16) Computing the eigendecomposition of a generalized eigenvalue problem is costlier than the eigenvalue problem by a factor of 2:5 to 3. In most cases, we can perform a linear transformation and reduce the problem to an eigenvalue problem. In ....

I. Gohberg, P. Lancaster, and L. Rodman. Matrix Polynomials. Academic Press, New York, 1982.

....The special structure of C simplifies the algorithm and reduces the number of parameter patches, as well as the number of parameters in each patch. Examples are provided at the end of x4 and x5. 2. The Companion Matrix The results presented in this section are well known and can be found in [1], 3] We present the full proofs, since our notation differs a bit from the references, and the proofs are quite short. The inverse of a matrix polynomial P is a matrix polynomial Q (of finite degree) with P (z)Q(z) I. Lemma 2.1. A matrix polynomial P has an inverse Q if and only if det P (z) ....

....I GammaP p GammaP p Gamma1 Delta Delta Delta Delta Delta Delta GammaP 1 1 C C C C C A : 2.1) Obviously, there is a one to one correspondence between P and C. Our subscript numbering is backwards compared to the usual definition, due to our nonstandard definition of monic . Following [1], we define E(z) 0 B B B B B P p Gamma1 j=0 P j z j P p Gamma2 j=0 P j z j 1 Delta Delta Delta P 1 j=0 P j z j p Gamma2 Iz p Gamma1 GammaI z Gamma1 0 Delta Delta Delta 0 0 0 GammaI z Gamma1 0 . 0 . 0 Delta Delta Delta 0 GammaI z ....

I. Gohberg, P. Lancaster, and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.

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Gohberg, I., Lancaster, P. and Rodman, L., Matrix Polynomials, Academic Press, New York, 1982.

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Gohberg, I., Lancaster, P. and Rodman, L., Matrix Polynomials, Academic Press, New York, 1982.

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I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials (Academic Press, 1982). 18 P. F. Curran & G. McDarby

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I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.

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I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press, Newyork, 1982.

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Gohberg I., Lancaster P., Rodman L., Matrix Polynomials, Academic Press Inc., 1982, New York. 21

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I. Gohberg, P. Lancaster, and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.

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Gohberg I., Lancaster P., Rodman L., 'Matrix Polynomials', Academic Press Inc, (1982), New York.

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Gohberg, I., Lancaster, P., and Rodman, L., Matrix Polynomials, Academic Press, New York, 1982.

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Gohberg, I. Langaster, P. and Rodman, L. 1982, Matrix Polynomials, Academic Press Inc., New York.

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I. Gohberg, P. Lancaster, and L. Rodman, Matrix polynomials. New York: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], 1982. Computer Science and Applied Mathematics. 18

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I. Gohberg, P. Lancaster, and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.

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I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Academic Press, New York, 1982, xiv+409 pp., ISBN 0-12-287160-X.

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I. Gohberg, P. Lancaster, and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.

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I. Gohberg, P. Lancaster, and L. Rodman. Matrix Polynomials. Academic Press, New York, 1982. xiv+409 pp. ISBN 0-12-287160-X.

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Gohberg, I.; Lancaster, P.; and Rodman, L.: Matrix Polynomials. Academic Press, 1982.

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Gohberg I., Lancaster P. and Rodman L. (1982) Matrix Polynomials. New York: Academic Press.

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I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.

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I. Gohberg, P. Lancaster, L. Rodman. Matrix Polynomials. Academic Press, New York, 1982.

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I. Gohberg, P. Lancaster, L. Rodman. Matrix Polynomials. Academic Press, New York, 1982.

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