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

Abstract

Let Pm(z) be a matrix polynomial of degree m whose coefficients At <FONT FACE=Symbol>Î</FONT> Cq×q satisfy a recurrence relation of the form: h kA0+ h k+1A1+...+ h k+m-1Am-1 = h k+m, k > 0, where h k = RZkL <FONT FACE=Symbol>Î</FONT> Cp×q, R <FONT FACE=Symbol>Î</FONT> Cp×n, Z = diag (z1,...,z n) with z i <FONT FACE=Symbol>¹</FONT> z j for i <FONT FACE=Symbol>¹</FONT> j, 0 < |z j| < 1, and L <FONT FACE=Symbol>Î</FONT> Cn×q. The coefficients are not uniquely determined from the recurrence relation but the polynomials are always guaranteed to have n fixed eigenpairs, {z j,l j}, where l j is the jth column of L*. In this paper, we show that the z j's are also the n eigenvalues of an n×n matrix C A; based on this result the sensitivity of the z j's is investigated and bounds for their condition numbers are provided. The main result is that the z j's become relatively insensitive to perturbations in C A 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.

matrix polynomials; block companion matrices; departure from normality; eigenvalue sensitivity; controllability Gramians

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: [e-mail]

ABSTRACT

Let P_m(z) be a matrix polynomial of degree m whose coefficients A_t Î C^q×q satisfy a recurrence relation of the form: h_kA₀+ h_k+1A₁+...+ h_k+m-1A_m-1 = h_k+m, k > 0, where h_k = RZ^kL Î C^p×q, R Î C^p×n, Z = diag (z₁,...,z_n) with z_i ¹ z_j for i ¹ j, 0 < |z_j| < 1, and L Î C^n×q. The coefficients are not uniquely determined from the recurrence relation but the polynomials are always guaranteed to have n fixed eigenpairs, {z_j,l_j}, where l_j is the jth column of L*. In this paper, we show that the z_j's are also the n eigenvalues of an n×n matrix C_A; based on this result the sensitivity of the z_j's is investigated and bounds for their condition numbers are provided. The main result is that the z_j's become relatively insensitive to perturbations in C_A 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 A_t Î ^q×q (t = 0:m-1) satisfy a recurrence relation of the form

where h_k Î ^p×q. The coefficients, known as predictor parameters, reflect intrinsic properties of the sequence {h_k} such as frequencies, damping factors, plane waves, etc, whose estimation from a finite data set {h_k}, 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,...,z_n) with z_i ¹ z_j for i ¹ j, |z_i| < 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 f_j,g_j Î ^p×1, the z's are of the form z_j = (i = ), n = 2d, and q = 1 (see [22] and references therein).

In these applications, one wants to estimate the parameters z_j and the matrices R, L from a finite data set {h_k}. 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 P_m(z) has z_j (j = 1: n) as eigenvalue and l_j(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 {z_j,l_j} 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 {z_j,l_j} are eigenpairs of P_m(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, {z_j,l_j}, but the remainder of their eigenstructure will depend on the solution chosen. In the sequel we refer to the z_j's as prescribed eigenvalues of P_m(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 P_m(z) there is a block companion matrix C_A defined by

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

In practice the coefficients A_t are never known exactly and one has to analyze the sensitivity of the z_j's to perturbations in A_t. 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 z_j's to a small n×n matrix obtained by projecting C_A onto an appropriate subspace and then analyzing the projected eigenproblem. As a result, simple estimates of measures of sensitivity of the z_j'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 s_i(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 I_n and its jth column by e_j.

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 z_j'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 C_B be defined by

whose first column block, denoted by X_B, is any solution of the underdetermined linear system _mX_B = Z^-1L. This definition ensures that (j = 1: n) is an eigenvalue of C_B 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 C_A and C_B associated with the eigenvalues of interest. Let _A(m,q) and _B(m,q) be the matrices obtained by projecting C_A and C_B 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 A₁Î ^n×p and B₁Î ^n×q. Assume rank ([A₁B₁]) = 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) > ...> l_p+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 Q₁Î ^n×q, Q₂Î ^n×p, R₁₁Î ^q×q, R₁₂Î ^p×p and R₂₂Î ^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 W_m. 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) > ... > s_n(_A). Assume that rank ([Z^mL 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 W_m. In this case they are given by

where x ₀ 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 C_A = [E₂E₂ ... E_m X_A], where E_j denotes the block column vector having its jth entry equal to I_q and the remaining ones equal to the zero matrix, then

and this can be rewritten as

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

where

Now observe that [**P₁] = (*)[Z^mL L] and that * is nonsingular. From this and the assumption that \operatornamerank([Z^mL 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+g_i (i = 1:2q) with g_i the nonzero eigenvalues of -*P₁im + **. As q of these g_i 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⁾W_m, where L⁽¹⁾ = diag(L_1,1,..., L_n,1) is nonsingular since, by assumption |L_j,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 [Z^mL 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([Z^mL 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 = [e₁e₃ ... e_2m-1], ₂ = [e₂e₄ ... e_2m], in which e_i denotes the ith canonical vector in ^mq. This can be seen as follows. Let L = [L₁,L₂] and R₁ = diag(L_1,1,..., L_n,1). Since Z and R₁ are diagonal, the definition of X₁ implies (see (22))

But since R₁W_m = [R₁e R₁Ze ... R₁Z^m-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 w_i to be the unit vector with the same direction as , i = 1,2. Forming the Rayleigh-Ritz quotient in (32), we have

where w = w_i 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 s_n(_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 = I_n, and we set = Z^-1L. Pre-multiplication by L*Z^m+1* and post-multiplication by Z^m+1L 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 | Z^m 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 z_j fall inside the unit circle. Then ||

||₂® 0 as m ® ¥.

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

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

Now since s_n(_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 l_j and u_j, v_jas associated left and right eigenvectors, the condition number of l_j, denoted by k_j(l_j), is defined by (see. e.g., Wilkinson [30, p. 314])

Smith [23] proved that

where d_j measures the distance of l_j to the rest of the spectrum. Thus the more the ill-conditioned l_j, the larger the ratio D/d_j, which means that D increases and/or d_j is small. Another interpretation of the above result is possible. Of course, it says that for the eigenvalue l_j to be well conditioned, it suffices that D/d_j » 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 s_j of

_A(m,q) be ordered in the usual way, i.e., s₁>s₂> ... > s_n. 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 z_j 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 P_m(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

h_k = Ce^ADtkB, 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 z_j = (j = 1: 10) where the s_j'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 z_j's as well as the eigenvalues in modulus and separations d_j = min|z_j-z_i|, 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 z_j of P_m(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 z_j'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 P_m(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 z_j related to _A(m,q) for q > 1 (and hence to P_m(z)) is denoted by k_q(z_j), while the condition number of the same eigenvalue but related to _A(m,1) is denoted by k₁(z_j).

Theorem 4.2. For m > n the following properties hold

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

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

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

(d) Let d_j =

|z_j-z_k| 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 v_j = and u_j = are left and right eigenvectors of _A(m,q), respectively, associated with the eigenvalue z_j. 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⁾W_m

L⁽¹⁾* where L⁽¹⁾ denotes a diagonal matrix with the components of L as entries and W_m 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 ||e_j||₂<||e_j||₂, which can be seen as follows. Let f_j = e_j. This means that f_j is the minimum 2-norm solution of the underdetermined linear system

Let = [L⁽¹⁾W_m ... L^(q)W_m], where L⁽ⁱ⁾ = diag(L_1,i,... L_n,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 =

e_j. It is not difficult to check that this vector is a solution of the system (52). Additionally

This equality proves property (c) as f_j 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 z_j'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 z_j'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 = W_m

. 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 v_j (j = 1: q) be the unit vector with the same direction as L^(j)*u. Substituting v_j 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 z_j 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 z_j of P_m(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 k_q(z_j) 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 z_j'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 l_i(A) ¹ l _j(A), for i ¹ j, and |l_i(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 l_j(A) are distinct, a change of basis of the state vector = T^-1x_k 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/d_j 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).

Received: 09/IX/04. Accepted: 12/XII/04.

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*

This research was performed while the author was a visitor at CERFACS, Toulouse, France, and was sponsored by CNPq, Brazil, grant 201407/03-5(NV).

Publication Dates

• Publication in this collection
  20 Apr 2006
• Date of issue
  Dec 2005