## versão On-line ISSN 1807-0302

### Comput. Appl. Math. v.24 n.3 Petrópolis set./dez. 2005

Matrix polynomials with partially prescribed eigenstructure: eigenvalue sensitivity and condition estimation

Fermin S. Viloche Bazán*

Department of Mathematics, Federal University of Santa Catarina, Florianópolis, Santa Catarina, 88040-900, Brazil CERFACS Technical Report TR/PA/04/64. E-mail: fermin@mtm.ufsc.br

ABSTRACT

Let Pm(z) be a matrix polynomial of degree m whose coefficients At Î Cq×q satisfy a recurrence relation of the form: hkA0+ hk+1A1+...+ hk+m-1Am-1 = hk+m, k > 0, where hk = RZkL Î Cp×q, R Î Cp×n, Z = diag (z1,...,zn) with zi ¹ zj for i ¹ j, 0 < |zj| < 1, and L Î Cn×q. The coefficients are not uniquely determined from the recurrence relation but the polynomials are always guaranteed to have n fixed eigenpairs, {zj,lj}, where lj is the jth column of L*. In this paper, we show that the zj's are also the n eigenvalues of an n×n matrix CA; based on this result the sensitivity of the zj's is investigated and bounds for their condition numbers are provided. The main result is that the zj's become relatively insensitive to perturbations in CA provided that the polynomial degree is large enough, the number n is small, and the eigenvalues are close to the unit circle but not extremely close to each other. Numerical results corresponding to a matrix polynomial arising from an application in system theory show that low sensitivity is possible even if the spectrum presents clustered eigenvalues.
Mathematical subject classification: 65F20,65F15.

Key words: matrix polynomials, block companion matrices, departure from normality, eigenvalue sensitivity, controllability Gramians.

1 Introduction

We are concerned with matrix polynomials

whose coefficients At Î q×q (t = 0:m-1) satisfy a recurrence relation of the form

where hk Î p×q. The coefficients, known as predictor parameters, reflect intrinsic properties of the sequence {hk} such as frequencies, damping factors, plane waves, etc, whose estimation from a finite data set {hk}, is an important problem in science and engineering [1, 19, 20, 21, 22, 28]. In this work, we concentrate on polynomials arising in applications where the data are assumed to be modeled as

where Z = diag(z1,...,zn) with zi ¹ zj for i ¹ j,  |zi| < 1 , R Î p×n is of rank p and L Î n×q of rank q with rows scaled to unit length. Also, as usual in the applications of interest, we shall assume that n is a small number.

Model (3) covers, e.g., impulse response samples of dynamic linear systems [1, 4, 19, 28, 29], where the z's are system poles, time domain nuclear magnetic resonance (NMR) data [26, 27], and time series defined by

where fj,gj Î p×1, the z's are of the form zj = (i = ), n = 2d, and q = 1 (see [22] and references therein).

In these applications, one wants to estimate the parameters zj and the matrices R, L from a finite data set {hk}. The problem is difficult as n is not always known in advance and the available data are corrupted by noise. However, a relatively simple polynomial-based approach can be used. The approach relies on the fact that if the data are free of noise and the coefficients are estimated from a linear system constructed by stacking m' successive recurrence relations, where we assume that m' > m > n, and n is the rank of the coefficient matrix, then Pm(z) has zj (j = 1: n) as eigenvalue and lj (j = 1: n), the jth column of L*, as associated left eigenvector [1, 19, 28] (the star symbol denotes conjugate transpose). Details about eigenvalues of matrix polynomials can be foundin [12]. The remaining mq-n eigenvalues have no physical meaning and are commonly known as spurious eigenvalues. Once the eigenpairs {zj,lj} are available, the estimation of R is straightforward. The same approach can be used in the noisy data case but some criterion is needed to separate the eigenvalues of interest from the spurious ones.

Note that since {zj,lj} are eigenpairs of Pm(z), then there holds

This is an underdetermined linear system of the form

where and m is an n×mq full-rank Krylov matrix defined by

Thus, all polynomials whose coefficients satisfy (2) (and hence (5)) will have n fixed eigenpairs, {zj,lj}, but the remainder of their eigenstructure will depend on the solution chosen. In the sequel we refer to the zj's as prescribed eigenvalues of Pm(z) and to the polynomial itself as a polynomial with partially prescribed eigenstructure, or shortly, as a predictor matrix polynomial. For applications involving predictor polynomials, the reader is referred to [1, 19, 21, 25, 29, 22].

We observe also that associated with Pm(z) there is a block companion matrix CA defined by

This matrix has the same eigenvalues as Pm(z) [12], left eigenvectors of theform * = [l* zl*...zm-1 l*] with l a left eigenvector of Pm(z), and satisfies the matrix equation

In practice the coefficients At are never known exactly and one has to analyze the sensitivity of the zj's to perturbations in At. The problem has receivedthe attention of many researchers and many sensitivity analyses for the scalar case (i.e., for q = 1) are now available, see, e.g., [2, 6, 17, 21, 25]. Some results concerning sensitivity of eigenvalues of general matrix polynomials can be found in [14, 24]. However, to the best of our knowledge nothing has been done on sensitivity analysis of prescribed eigenvalues of predictor polynomials for q > 1. The goal of this work is to carry out a sensitivity analysis of prescribed eigenvalues only, focusing on the influence of the polynomial degree on such sensitivity. We show that this can be done by relating the zj's to a small n×n matrix obtained by projecting CA onto an appropriate subspace and then analyzing the projected eigenproblem. As a result, simple estimates of measures of sensitivity of the zj's in the form of informative upper bounds are given.

The following notation is used throughout the paper. For A Î m×n, ||A||2 and ||A||F  denote the 2-norm (or spectral) and Frobenius norm of  A,  respectively. A denotes the Moore-Penrose pseudo-inverse of A. The ith singular value of A is denoted by si(A). The 2-norm condition number of A, k(A), is defined by k(A) = ||A||2 ||A||2. The spectrum of A Î n×n is denoted by l(A). The identity matrix of order n is denoted by In and its jth column by ej.

The paper is organized as follows. In Section 2, we describe results concerning the singular values of projected companion matrices by extending the workin [5]. The results obtained are then exploited in Section 3, in which we analyze the departure of the projected companion matrix from normality. In Section 4, we analyze the condition numbers of the zj's introduced by Wilkinson [30],and the overall 2-norm condition number of the related eigenvalue problem. We show that these measures of sensitivity are governed by the 2-norm condition number of the Krylov matrix and conclude that eigenvalues near the unit circle become relatively insensitive to noise provided that the polynomial degree is large enough and the eigenvalues themselves are not extremely close to each other. In addition to this, we provide estimates for the 2-norm condition number of controllability Gramians of multi-input multi-output discrete dynamical systems in diagonal form. Numerical results corresponding to a matrix polynomial arising from an application in system theory show that low sensitivity is possible even if some eigenvalues are clustered.

2 Singular value analysis of the projected companion matrix

In order to start our analysis we introduce a new block companion matrix associated with the prescribed eigenvalues. Let CB be defined by

whose first column block, denoted by XB, is any solution of the underdetermined linear system mXB = Z-1L. This definition ensures that (j = 1: n) is an eigenvalue of CB and that there holds

Let the columns of form an orthonormal basis for (), the columnspace of . Notice that because of (8) and (10), () is a left invariant subspace of both CA and CB associated with the eigenvalues of interest. Let A(m,q) and B(m,q) be the matrices obtained by projecting CA and CB onto (), that is,

Then it is clear that

The goal of this section is to analyze the singular values of A(m,q), focusing on their behavior as function of m,q. Before proceeding we observe that when the dependence of A(m,q) and B(m,q) on m,q is not important for the understanding, these matrices will be denoted by A and B. Notice also that the projector orthogonal onto (), denoted by , satisfies

Two lemmas are needed.

Lemma 2.1. For m > n and q > 1 there holds A = .

Proof. Since m is positive definite Hermitian, it is clear that the columns of = (m)-1/2 form an orthonormal basis for (). Using this basis and the definitions of A and B we have

This reduces to identity on using (8), (10), and the fact that m = I.

Lemma 2.2. Let A = A1- B1 with A1 Î n×p and B1 Î n×q. Assume rank ([A1  B1]) = p+q < n. Then, the number of positive, negative, and zero eigenvalues of A, is p, q, and n-(p+q), respectively.

Proof. Let the nonzero eigenvalues of A be arranged so that l1(A) > l2(A) > ...> lp+q(A). Our proof relies on the minimax principle for eigenvalues [11]:

Let the matrix P = [B1|A1] have a QR factorization

where Q, R are partitioned such that Q1 Î n×q, Q2 Î n×p, R11 Î q×q, R12 Î p×p and R22 Î p×p. Clearly, both R11 and R22 are nonsingular.From (13) it follows that

Substituting B1 and A1 into A, it follows that the projection of A onto (Q2), the subspace spanned by the columns of Q2, is

Let x Î (Q2), x ¹ 0. Then, because AQ2 is positive definite by (14), putting x = Q2b Î p, b ¹ 0, we have

and so, by the minimax principle, we conclude that A has at least p positive eigenvalues. Considering matrix -A instead of A and proceeding as beforeit follows that A has at least q negative eigenvalues. Apart from this, it is clear that A has n-(p+q) zero eigenvalues. From these conclusions the assertions of the lemma follow.

In order to describe our results concerning the singular values of m,q, wefirst notice that the Krylov matrix m becomes a weighted Vandermondematrix when q = 1. When the weights are all ones this matrix will be denoted by Wm. Let the columns of form an orthonormal basis for (). Then the orthogonal projector onto (), , satisfies

Using this notation we set

where  e = [1,..., 1]T Î n.

We are now ready to describe the singular spectrum of matrix A(m,q).

Theorem 2.3.Let the singular values of A(m,q) be arranged so that s1(A) > ... > sn(A). Assume that rank ([ZmL  L]) = 2q. Then, for 1 < q < n/2, there holds

Furthermore, if q = 1 the singular values of A(m,1) do not depend onthe matrix L defined in (3), but rather on the Vandermonde matrix Wm. In this case they are given by

where x0 denotes the first component of x+.

Proof. We use the fact that the squared singular values of A are eigenvalues of A . In fact, using the definition of A,

The last equality comes from the fact that = because is a basis of the right invariant subspace of associated with prescribed eigenvalues. Now notice that if we write CA = [E2 E2 ... Em XA], where Ej denotes the block column vector having its jth entry equal to Iq and the remaining ones equal to the zero matrix, then

and this can be rewritten as

Hence, using the fact that XA solves the system (5), which implies that XA = + N, where N is a matrix whose columns belong to (m) = [()]^, we have

where

Now observe that [**P1] = (*)[ZmL L] and that * is nonsingular. From this and the assumption that \operatornamerank([ZmL L]) = 2q it follows that rank([**P1]) = 2q. Thus, if * is identified with A1 and *P1 with B1 in Lemma 2.2, it follows from (21) that A has n-2q zero eigenvalues, the remaining ones being of the form 1+gi (i = 1:2q) with gi the nonzero eigenvalues of -*P1im + **. As q of these gi are positive and the other q are negative, the inequalities in (17) follow, as desired.

To prove the statement of the theorem for q = 1, we observe that in this case L is a column vector and that the Krylov matrix can be rewritten as m = L(1)Wm, where L(1) = diag(L1,1,..., Ln,1) is nonsingular since, by assumption |Lj,1| = 1, j = 1: n. From this observation and pseudo-inverse properties, it is immediate to see that P1 reduces to p1, reduces to x+, and neither depend on L. Hence it follows that A(m,1) A(m,1)* does not depend on L and that

The equalities (18) follow on analyzing the eigenvalues of A(m,1) (m,1) from this equality; details can be found in [5].

Remark 1. The rank condition on [ZmL L] is no serious restriction in practice. This is because in practical problems L is dense, in which case one can prove, under mild conditions, that rank([ZmL L]) = 2q.

Remark 2. Theorem 2.3 generalizes one concerning the singular values of a particular projected companion matrix by Bazan (see, Thm. 4 in [5]), and shows also that the singular values of the projected block companion matrix in our context, inherits to some extent the singular value properties of general block companion matrices described in Lemma 2.7 in [15].

Since the singular values of A(m,1) do not depend on the matrix L, we can always compare the singular values of A(m,q) for the case where q > 1 with those corresponding to q = 1. This is given in the following theorem.

Theorem 2.4. Let A(m,q) as before. Then, for m > n and 1 < q < 2n, there holds

Proof. We shall prove the inequalities (23) for q = 2; the proof for the case q > 2 is similar. Notice that for q = 1, we have

while if q = 2, we have from (21)

where we have assumed that = [X1,X2], P1 = . The idea behind the proof is to rewrite (25) in terms of the matrix introduced in (24). For this we use the fact that

where i = *i , 1 = [e1 e3 ... e2m-1], 2 = [e2 e4 ... e2m], in which ei denotes the ith canonical vector in mq. This can be seen as follows. Let L = [L1,L2] and R1 = diag(L1,1,..., Ln,1). Since Z and R1 are diagonal, the definition of X1 implies (see (22))

But since R1 Wm = [R1e R1Ze ... R1Zm-1e] and = m, we have

Inserting this result in Eq. (27) yields

A similar work with X2, , and gives

The set of equations (26) follows on multiplying by * on both sides of equations (28), (29), (30), and (31). Here we have used the fact that *x+ = x+, *p1 = p1, since both x+ and p1 belong to ().

We turn now to the proof of the theorem. Using the Eq. (26) and (24), we have

Let u be a unit vector in p and define wi to be the unit vector with the same direction as , i = 1,2. Forming the Rayleigh-Ritz quotient in (32), we have

where w = wi such that w*w = max{}. Now using the definition of matrix 1, we have

where we have used the fact that * = *. A similar work gives

Summing up the two last inequalities it is not difficult to check that

Substituting this result in (33) gives

and the proof of the first inequality in (23) is concluded.

Finally, since sn(A(m,q)) = 1/s1(B(m,q)), by Lemma (2.1), proceeding as before it follows s1(B(m,q)) <s 1(B(m,1)). This proves the second inequality in (23) and the proof of the theorem is concluded.

A point that remains for discussion is the behavior of the singular values of A(m,q) for fixed q > 1 and varying m. This is a difficult problem; so we restrict ourselves to analyzing bounds for them.

Corollary 2.5 Let and P1 be as in (22). Then we have

Additionally, while the lower bound increases with m, the upper bound decreases.

Proof. First notice from (21) that the squared singular values of A that differ from 1 are the eigenvalues of W defined by

By comparing the eigenvalues of W with those of its Hermitian part, it follows

This proves (34). We shall now prove that both ||||2 and ||P1||2 are decreasing functions of m. Let

Then we shall prove that ||||2 < ||||2 and ||1||2 < ||P1||2. In fact, write m = [L | Z m] and notice that

where

Applying the Sherman-Morrison formula to the inverse above we obtain

where we have used the fact that m = In, and we set = Z-1L. Pre-multiplication by L*Zm+1* and post-multiplication by Zm+1 L on both sides of this equation yields

This shows that the singular values of of can not exceed those of , thus ensuring the statement of the theorem for . To prove that ||P1||2 decreases with m, it is sufficient to partition m as m = [m | Zm L], and then proceed as before.

The corollary is interesting because it provides a bound for the 2-norm condition number of A of the form

that decreases with m. Thus, reliable bounds for k(A) can be obtained provided both and are small enough. For the significant case where the prescribed eigenvalues lie inside the unit circle, the asymptotic of the bounds as m is going to infinite is readily determined. To do this the following technical result, the proof of which is straightforward, is needed.

Lemma 2.6. Suppose all zj fall inside the unit circle. Then ||||2 ® 0 as m ® ¥.

Corollary 2.7. Suppose all zj lie inside the unit circle. Then, as m ® ¥ we have

Proof. We first notice that for q = 1 we have s1(CA(m,1))sn(CA(m,1)) = . Using Corollary 2.5 and Lemma 2.6 it follows that

Now since sn(A(m,q)) >s n(A(m,1)) for all m > n and fixed q > 1, by Corollary 2.5 again, there holds

The assertion of the corollary follows on using this inequality and the definition of k(A).

3 Departure from normality of A(m,q)

The influence of nonnormality on several problems in scientific computinghas been known for long time and several measures of nonnormality either of theoretical or practical interest are now available [8, 10, 13]. An exhaustive discussion on the influence of nonnormality on many problems in scientific computing, using several measures of nonnormality, is given in Chaitin-Chatelin and Frayseé [8]. For A Î n×n the following measure has been introduced by Henrici (1962):

This measure plays an important role in our context because it can be related to the conditioning of the eigenbasis of A when A is diagonalizable. To clarify this recall that for general A Î n×n with simple eigenvalues lj and uj, vj as associated left and right eigenvectors, the condition number of lj, denoted by kj(lj), is defined by (see. e.g., Wilkinson [30, p. 314])

Smith [23] proved that

where dj measures the distance of lj to the rest of the spectrum. Thus the more the ill-conditioned lj, the larger the ratio D/dj, which means that D increases and/or dj is small. Another interpretation of the above result is possible. Of course, it says that for the eigenvalue lj to be well conditioned, it suffices that D/dj » 0 and n be a moderate number. We shall return to this point later.

The goal here is to analyze D(A(m,q)), concentrating on its behavior as a function of m,q for fixed q > 1 and increasing m. The following theorem shows that this can be made by comparing the singular values of A(m,q) with those of A(m,1). This is always possible, since by Theorem 2.3, the singular values of A(m,1) do not depend on the matrix L.

Theorem 3.1. Let a and b denote respectively the largest and the smallest singular values of A(m,1) and let the singular values sj of A(m,q) be ordered in the usual way, i.e., s1 > s2 > ... > sn. Let

Define

Then, for each m > n and 1 < q < n/2 it holds

Proof. We first notice that, because of Theorem (2.3), we have

Now since A(m,q) has the same spectrum as A(m,1) we have

If this is rewritten as

the geometric-arithmetic mean inequality leads to

Multiplying both sides of this inequality by the sum of the reciprocals of each term of the right hand side, we obtain

where

Kantorovic's inequality (see Horn and Johnson [16, Thm. 7.4.41]) leads then to

where is defined in (42). Hence it follows

The upper bound in (43) follows from this inequality on noting that

where r is defined in (42). To prove the lower bound, rewrite (45) as

The geometric-arithmetic mean inequality leads then to

The lower bound in (43) is a consequence of using (47) in this inequality.

The departure from normality of A(m,1) is analyzed in Bazan [5]. The conclusion drawn from that analysis is that this matrix becomes close to anormal matrix provided the eigenvalues zj fall near the unit circle and m is large enough. This is important in our context since if we take into accountthe inequalities (43), we can conclude that A(m,q) for the case q > 1 may become closer to normality than A(m,1). In terms of eigenvalue sensitivity, this means that prescribed eigenvalues of Pm(z) can be less sensitive to noise when regarded as eigenvalues of A(m,q) with q > 1 than when regarded as eigenvalues of A(m,1). This shall be theoretically demonstrated in the next section. Here we restrict ourselves to numerically illustrate the behavior of D(A(m,q)).

Example: departure from normality of A(m,q) arising from a dynamical system. The dynamical system under analysis is defined by the state space equations

and corresponds to a computer model of a flexible structure known as Mini-Mast [18]. Matrices A, B and C are of orders 10×10, 10×2 and 2×10, respectively; the entries of the matrices can be found in [18]. Impulse response samples are thus given as

hk = CeADtkB, k = 0,1,...

Matrices R and L of model (3) are thus of order 2×10 and 10×2, respectively, and can be found readily by computing an eigendecomposition of matrix A. According to our notation this implies that n = 10, p = q = 2; the eigenvalues are of the form zj = (j = 1: 10) where the sj's are eigenvalues of A. The time step is Dt = 0.03 s. The model comprises five modes (in complex conjugate pairs) and involves two closely spaced frequency pairs. Frequencies and damping expressed as the negative real part of the zj's as well as the eigenvalues in modulus and separations dj = min|zj-zi|, i ¹ j, are displayed in Table 1.

In order to illustrate the behavior of D2(A(m,q)) as a function of m,q the norms for increasing m and q = 1: 2 were computed from the relation (see (21))

All computations were carried out using MATLAB. The results displayed in Figure 1 are surprising: they not only show that D2(A(m,2)) really improves D2(A(m,1)) but also that this improvement can be dramatic when m is near n = 10. For illustration, while for q = 1 and m = 10,11 we obtain

which illustrate that A(10,1) and A(11,1) are highly nonnormal, for q = 2 and the same values of m we obtain

The influence of q on D2(A(m,q)) for q > 2 was also analyzed. For this, input matrices B with random numbers as entries of orders q×10 and q = 1: 4 were constructed. With these matrices at hand, the matrices L of corresponding orders were obtained in the same way as in the case for q = 2. Results corresponding to the seed value of the random generator equal to 10 (we use the MATLAB function randn), displayed in Figure 2, show once more that the departure from normality of matrix A(m,q) for q > 1 gets smaller than that corresponding to q = 1. However no conclusion can be drawn concerning the behavior of D2(A(m,q)) for values q > 2 in comparison with that corresponding to q = 2.

As in this example all eigenvalues lie inside the unit circle, the asymptotic value of D2(A(m,q)) as m is going to infinity can be readily computed: it suffices to use (49) taking into account that in this case

where

Asymptotic values of D2(A(m,q)) in this case are:

4 Condition numbers

We have seen that the prescribed eigenvalues zj of Pm(z) are eigenvalues of the projected companion matrix A(m,q). This fact is exploited here to carry out a sensitivity analysis of these eigenvalues. To this end , we choose as measures of sensitivity the Wilkinson condition numbers of the zj's (see (40)) viewed as eigenvalues of A(m,q), and the overall 2-norm condition number of the eigenvalue problem. In order to describe our results we recall that for m > n and fixed q, q > 1, m,q = m is positive definite Hermitian. In the sequel we shall always assume that the left eigenvector of Pm(z) (the rows of matrix L in (3)) are scaled using the 2-norm to unit length. The lemma below explains that the sensitivity of the eigenvalue problem associated with matrix A(m,q) is governed by the condition number of matrix m,q.

Lemma 4.1. Let A(m,q) be as before. Then there holds

Consequently, the sensitivity of the eigenvalue problem related to the prescribed eigenvalues is governed by .

Proof. Set = (m,q)-1/2. It is immediate to check that the columns of form an orthonormal basis of (). Using the definition of A(m,q) and this basis, we have

The proof concludes on using (8).

In the following, the condition number of zj related to A(m,q) for q > 1 (and hence to Pm(z)) is denoted by kq(zj), while the condition number of the same eigenvalue but related to A(m,1) is denoted by k1(zj).

Theorem 4.2. For m > n the following properties hold

(a) For 1 < q < n/2 we have

(b) The condition numbers k1(zj) do not depend on the matrix L but rather on the Vandermonde matrix Wm.

(c) For fixed m > n and q > 1 there holds kq(zj) < k 1(zj).

(d) Let dj = |zj-zk| Then, for 1 < q < n/2 there holds

where x+ denotes the minimum norm solution of the system (5) for the case q = 1.

Proof. To prove (a) notice from Lemma 4.1 that vj = and uj = are left and right eigenvectors of A(m,q), respectively, associated with the eigenvalue zj. These eigenvectors satisfy the condition = 1. Besides this

and

The last equality is because the rows of L in (3) are scaled to unit length byassumption. The equality (50) follows from these relations on using the definitions given in (40).

To prove property (b) notice that L becomes a column vector in n when q = 1. In this case we can write m,q = L(1)WmL(1)* where L(1) denotes a diagonal matrix with the components of L as entries and Wm the Vandermonde matrix introduced in the previous section. From this observation and the definition (40) it is immediate that

which proves (b).

The proof of (c) is based on the property that ||ej||2 <||ej||2, which can be seen as follows. Let fj = ej. This means that fj is the minimum 2-norm solution of the underdetermined linear system

Let = [L(1)Wm ... L(q)Wm], where L(i) = diag(L1,i,... Ln,i), i = 1... q. It is clear that = m with an appropriate permutation matrix. Introduce defined by

Then

and therefore is a right inverse of . Define now f = ej. It is not difficult to check that this vector is a solution of the system (52). Additionally

This equality proves property (c) as fj is the solution of minimum norm of (52).

Finally, property (d) is a consequence of estimate (41), property (c), and Lemma 7 in Bazán [5] where it is proved that

The main conclusion that can be drawn from the Theorem 4.2 is that the sensitivity of the zj's regarded as eigenvalues of the projected companion matrix essentially depends on intrinsic characteristics of the eigenvalues themselves and on the degree of the associated matrix polynomial. Concerning the estimates (51), since n is assumed to be small, the conclusion is that they can approach the optimum value 1 provided » 0 and the eigenvalues in modulus are reasonably close to the unit circle but not extremely close to each other. In spite of the fact that this conclusion seems to emerge under rather stringent conditions, namely, n small and zj's close to the unit circle, we emphasize that there are many applications in which these conditions appear frequently. In fact, in modal analysis of vibrating structures, the analysis of slow-decaying signals often involves eigenvalues very close to the unit circle and n small; in [4, 1, 19] examples are reported with n ranging from 15 to 20. Numerical examples showing that » 0 for moderate values of m are discussed in [7]. Another example involving the condition n small is encountered in NMR; genuine applications in this field point out n ranging from 2 to 16 [26, 27]. The condition » 0 in NMR is numerically verified in [3].

Apart from the conclusion above, a remark concerning the meaning of property (c) must be done: It predicts reduction in sensitivity of prescribed eigenvalues when extracted from projected companion matrices related to polynomials with q > 1. This will be illustrated numerically later.

The following theorem states that the conditioning of the eigenvalue problem associated with A(m,q) improves the conditioning of the eigenvalue problem associated with matrix A(m,1).

Theorem 4.3. Set = Wm. Then for each m > n, we have that

Proof. We shall prove that

In fact, let be as in the proof of the previous theorem. Then it is clear that = m,q. Using this result, for all unit vector u Î n, we have that

Let vj  (j = 1: q) be the unit vector with the same direction as L(j)*u. Substituting vj in the above equation and using the Rayleigh-Ritz characterization of eigenvalues of symmetric matrices, we get

The first inequality in (53) follows on noting that (||L(j)*u||2...+||L(q)*u||2) = 1 because by assumption all rows of L have 2-norm equal to one. The second inequality in (53) follows in the same way and the proof concludes.

Note that because of its definition, whenever all zj fall inside the unit circle, the limit of m,q as m ® ¥ is always guaranteed to exist, and the same result applies for .

Corollary 4.4. Let ¥,q denote the limiting value of m,q as m ® ¥. Suppose all prescribed eigenvalues zj of Pm(z) fall inside the unit circle. Define

Then

where

Proof. This corollary is a consequence of Theorem 4.3 and Corollary 9 in Bazán [5].

Example: conditioning of Mini-Mast eigenvalues. To confirm the theoretical predictions of Theorem 4.2 we have computed the condition numbers kq(zj) of the eigenvalues associated with the Mini-Mast model described in the previous section. The goal is to verify that severe reduction in sensitivity is possible when extracting the zj's from projected companion matrices related to polynomials with q > 1. Results corresponding to q ranging from 1 to 4 and some values of m are displayed in Table 2. Reduction in sensitivity is apparent from this table.

4.1 An application to linear system theory

We shall show that the Corollary 4.4 can be applied to estimating the 2-norm condition number of controllable Gramians in linear system theory. Consider a dynamical discrete linear system S described by the state equations

where A Î n×n, B Î n×q, and C Î q×n. Assume li(A) ¹ l j (A), for i ¹ j, and |li(A)|< 1  (i = 1:n) . Assume also that the system is controllable, i.e., the extended controllable matrix defined by

satisfies rank() = n. Then the controllable Gramian of the system S, defined as [9]

is guaranteed to be symmetric and positive definite, and its eigenvalues areknown to concentrate information that plays a crucial role when solving system identification and model order reduction problems. It turns out that if the system eigenvalues lj(A) are distinct, a change of basis of the state vector = T-1xk with T a matrix of right eigenvectors of A, will transform the state space representation (54) to another one in diagonal form. When this is done, reduces to a matrix like the block Krylov matrix m and the controllable Gramian reduces to one like ¥,q. This shows that the estimate for k(¥,q) of the Corollary 4.4 applies to estimating the 2-norm condition number of the Gramian .

5 Conclusions

Based on the fact that prescribed eigenvalues of predictor polynomials can be regarded as eigenvalues of projected block companion matrices, an eigenvalue sensitivity analysis was performed. As a result, simple estimates of measures of eigenvalue sensitivity in the form of informative upper bounds were derived. In particular, under the assumption that n is small, it was proved that prescribed eigenvalues near the unit circle can be relatively insensitive to noise provided the polynomial degree is large enough. The effect of the dimension of the coefficients on the sensitivity was also analyzed and it was concluded that prescribed eigenvalues of predictor polynomials can be less sensitive to noise when regarded as eigenvalues of projected companion matrices related to matrix polynomials with coefficients of order q > 1 than when regarded as eigenvalues of projected companion matrices related to scalar polynomials. The theory was numerically illustrated using a matrix polynomial with clustered eigenvalues arising from the modal analysis field. The results are of interest in system analysis where estimates for the 2-norm condition number of controllability Gramians of multi-input multi-output discrete dynamical systems play a crucial role.

The author is aware that further research is desirable for the case where the prescribed eigenvalues are almost defective: the bounds in property (d) of Thm. 4.2 can be pessimistic in this case as the ratio D/dj is no longer small, but as illustrated in Table 2, the conditioning itself remains excellent. Furthermore, an analysis for the case where the prescribed eigenvalues are defective is needed. This challenging development is the subject of future research.

Acknowledgments. The author wishes to thank I.S. Duff and members ofthe ALGO Team at CERFACS for providing a cordial environment. Special thanks go to S. Gratton for suggestions that have improved the presentationof the paper. Thanks also go to the referees for their suggestions and constructive criticism. The author is particularly grateful to one referee for an important observation concerning inequality (51).

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