Distance bounds for prescribed multiple eigenvalues of matrix polynomials

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1 Distance bounds for prescribed multiple eigenvalues of matrix polynomials Panayiotis J Psarrakos February 5, Abstract In this paper, motivated by a problem posed by Wilkinson, we study the coefficient perturbations of a (square) matrix polynomial to a matrix polynomial that has a prescribed eigenvalue of specified algebraic multiplicity and index of annihilation For an n n matrix polynomial P(λ) and a given scalar µ C, we introduce two weighted spectral norm distances, E r (µ) and E r,k (µ), from P(λ) to the n n matrix polynomials that have µ as an eigenvalue of algebraic multiplicity at least r and to those that have µ as an eigenvalue of algebraic multiplicity at least r and maximum Jordan chain length (exactly) k, respectively Then we obtain a lower bound for E r,k (µ), and derive an upper bound for E r (µ) by constructing an associated perturbation of P(λ) Keywords: matrix polynomial, eigenvalue, algebraic multiplicity, index of annihilation, perturbation, singular value, singular vector AMS Subject Classifications: 5A8, 5A, 65F5, 65F35 Introduction and preliminaries Wilkinson s problem [4, 5] concerns computing the spectral norm distance (known as Wilkinson s distance) from an n n matrix with n distinct eigenvalues to the set of n n matrices having multiple eigenvalues, and has a strong connection to ill-conditioning of eigenvalue problems Malyshev [9] provided a solution to Wilkinson s problem by obtaining a singular value characterization of Wilkinson s distance Recently, Mengi [] (extending Malyshev s methodology) derived a singular value optimization characterization for the smallest perturbation to a matrix that has an eigenvalue of specified algebraic multiplicity Papathanasiou and Psarrakos [3] studied the case of polynomial eigenvalue problems, and applied Malyshev s technique to derive lower and upper bounds for a weighted distance from a given n n matrix polynomial to the n n matrix polynomials that have a prescribed multiple eigenvalue Motivated by the above, we consider an n n matrix polynomial P(λ) = A m λ m + A m λ m + + A λ + A, () where λ is a complex variable and A j C n n (j =,,,m) with A m We also assume that P(λ) is regular, that is, the determinant det P(λ) is not identically zero Department of Mathematics, National Technical University of Athens, Zografou Campus, 578 Athens, Greece (ppsarr@mathntuagr)

2 The study of matrix polynomials, especially with regard to their spectral analysis, has a long history and important applications; see [4, 8, ] and the references therein A scalar λ C is called an eigenvalue of P(λ) if the system P(λ )x = has a nonzero solution x C n This solution x is known as a (right) eigenvector of P(λ) corresponding to λ The set of all eigenvalues of P(λ), σ(p) = {λ C : detp(λ) = }, is the spectrum of P(λ), and since P(λ) is regular, it contains no more than nm finite elements The algebraic multiplicity of a λ σ(p) is the multiplicity of λ as a zero of the (scalar) polynomial detp(λ), and it is always greater than or equal to the geometric multiplicity of λ, that is, the dimension of the null space of matrix P(λ ) An eigenvalue of P(λ) is called semisimple if its algebraic and geometric multiplicities are equal, and it is called defective otherwise Suppose that for an eigenvalue λ σ(p), there exist x, x,,x k C n with x, such that ξ j= j! P (j) (λ )x ξ j = ; ξ =,,,k, () where P (j) (λ) denotes the jth derivative of P(λ) and k+ cannot exceed the algebraic multiplicity of λ Then the vector x is clearly an eigenvector of λ, and the vectors x, x,,x k are known as generalized eigenvectors The set {x, x,,x k } is called a Jordan chain of length k + of P(λ) corresponding to the eigenvalue λ Moreover, it is apparent that any set {x, x,,x ξ }, ξ =,,,k, is also a Jordan chain of P(λ) corresponding to λ Any eigenvalue of P(λ) of geometric multiplicity p has p maximal Jordan chains associated with p linearly independent eigenvectors, with total number of eigenvectors and generalized eigenvectors equal to the algebraic multiplicity of this eigenvalue The largest length of Jordan chains of P(λ) corresponding to λ σ(p) is known as the index of annihilation of λ [7] This index coincides with the size of the largest Jordan blocks of the Jordan canonical form of P(λ) corresponding to λ, and it is equal to if and only if the eigenvalue λ is semisimple; for details on the Jordan structure of matrix polynomials, see [4, 8] We are interested in (additive) perturbations of the matrix polynomial P(λ) of the form m Q(λ) = P(λ) + (λ) = (A j + j )λ j, (3) where the matrices j C n n (j =,,,m) are arbitrary For a given parameter ε > and a given set of nonnegative weights w = {w, w,,w m } with w >, we define the class of admissible perturbed matrix polynomials j= B(P, ε, w) = {Q(λ) as in (3) : j εw j, j =,,,m}, where denotes the spectral matrix norm (ie, the norm subordinate to the Euclidean vector norm), and the polynomial w(λ) = w m λ m + + w λ + w The weights w, w,,w m allow freedom in how perturbations are measured Moreover, B(P, ε, w) is convex and compact, with respect to the max norm P(λ) = max j m A j []

3 Now we can introduce weighted distances from P(λ) to the matrix polynomials that have a prescribed eigenvalue of algebraic multiplicity at least r, and to those that have a prescribed eigenvalue of algebraic multiplicity at least r and index of annihilation (exactly) k Definition For the matrix polynomial P(λ) in () and a given µ C, we define the distance from P(λ) to µ as an eigenvalue of algebraic multiplicity at least r by E r (µ) = inf {ε : Q(λ) B(P, ε, w) with µ as an eigenvalue of algebraic multiplicity at least r}, and the distance from P(λ) to µ as an eigenvalue of algebraic multiplicity at least r and index of annihilation k by E r,k (µ) = inf {ε : Q(λ) B(P, ε, w) with µ as an eigenvalue of algebraic multiplicity at least r and index of annihilation k} Notice that we have the identity E r (µ) = min E r,k(µ) Here, we allow values k=,,,mn of k greater than r because the optimal perturbed matrix polynomial Q(λ) may have µ as an eigenvalue of (both) algebraic multiplicity and index of annihilation greater than r The singular values of an n n complex matrix A, ie, the nonnegative square roots of the eigenvalues of A A, are denoted by s (A) s (A) s n (A) It is apparent that for the matrix polynomial P(λ), σ(p) = {λ C : s n (P(λ)) = }, and a scalar λ C is an eigenvalue of P(λ) of geometric multiplicity exactly p if and only if s (P(λ )) s n p (P(λ )) > s n p+ (P(λ )) = = s n (P(λ )) = If P(λ) = Iλ A for some A C n n, then σ(p) coincides with the standard spectrum of A, σ(a), and if in addition w = {w, w } = {, }, then B(P, ε, w) = {Iλ (A + E) : E ε} In this case, Malyshev [9] (inspired by []) has proved that E (µ) = sup γ> ([ Iµ A s n γi Iµ A ]) Extending Malyshev s methodology, Ikramov and Nazari [6] (for r = 3), and Mengi [] have obtained (always for P(λ) = Iλ A and w = {w, w } = {, }, ie, for constant matrices) E r (µ) = sup s rn r+ γ i,j C\{} Iµ A γ, I Iµ A γ 3, I γ 3, I Iµ A γ r, I γ r, I γ r,3 I Iµ A (4) In this work, our goal is to derive a primer estimation of the distances E r,k (µ) and E r (µ) for matrix polynomials Unfortunately, in the case of matrix polynomials, Malyshev s technique leads to lower and upper bounds for the distance E (µ) and not to its exact value [3] Hence, to be able to exploit the definition of Jordan chains of 3

4 matrix polynomials in () and avoid multivariable optimization problems, we consider block-toeplitz matrices of higher order with only one real parameter γ > and its powers instead of the r(r )/ complex parameters γ,, γ 3,, γ 3,,,γ r,r in (4) (see Definition below and the discussion before Theorem 4) Generalizing results of [3], in Section, we obtain a lower bound for E r,k (µ), and in Section 3, we construct an upper bound for E r (µ) and an associated perturbation of P(λ) Simple numerical examples are given in Section 4 to illustrate our results A lower bound for the distance E r,k (µ) By Theorem 4 of [3], we know that if both the required algebraic and geometric multiplicities of µ are equal to r (ie, the index of annihilation is equal to ), then E r, (µ) s n r+ (P(µ))/ Hence, it follows that E (µ) = s n(p(µ)) E r (µ) s n r+(p(µ)) E r, (µ) So, in the special case s n (P(µ)) = s n (P(µ)) = = s n r+ (P(µ)), it is clear that E r (µ) = s n r+ (P(µ))/, and an optimal perturbation of P(λ) is given by [3, formula (3)] Thus, in what follows, we assume that s n (P(µ)) s n r+ (P(µ)) and consider perturbations of P(λ) such that the perturbed matrix polynomial has µ as a defective eigenvalue (ie, k ) of algebraic multiplicity at least r The next definition is necessary for the remainder Definition For the matrix polynomial P(λ) in (), a positive integer k and a scalar γ C, we define the kn kn matrix polynomial P(λ) γ P () (λ) P(λ) γ F k [P(λ); γ] =! P () (λ) γ P () (λ) γ k (k )! P (k ) γ (λ) k (k )! P (k ) (λ) P(λ) We observe that a scalar λ C is an eigenvalue of P(λ) if and only if it is an eigenvalue of F k [P(λ); γ] Furthermore, if λ is an eigenvalue of P(λ) of algebraic multiplicity r and index of annihilation k, then for every γ, the null space of matrix F k [P(λ ); γ] has dimension at least r (for γ =, see [5, Lemma 5] and [7]) Lemma Suppose λ C is an eigenvalue of P(λ) of algebraic multiplicity at least r and index of annihilation k Then for any nonzero γ C, we have s kn r+ (F k [P(λ ); γ]) = Proof By hypothesis, we may assume that P(λ) has p Jordan chains corresponding to λ, namely, {x,, x,,,x,k }, {x,, x,,,x,k },, {x p,, x p,,,x p,kp }, with x,, x,,,x p, linearly independent eigenvectors, k = k k k p and (k + ) + (k + ) + + (k p + ) = r Notice that the first Jordan chain, 4

5 {x,, x,,,x,k }, is necessarily maximal Then, recalling (), we see that the r vectors x i, γx i,, x i,,, γ x i, γx i, C kn ; i =,,,p (5) γ ki x i,ki γ ki x i,ki γ ki x i,ki γ ki x i,ki x i, satisfy F k [P(λ ); γ] x i, γx i, γ x i, γ ξ x i,ξ γ ξ x i,ξ = P(λ )x i, γ j! P (j) (λ )x i, j j= = γ j! P (j) (λ )x i, j j= ξ γ ξ j! P (j) (λ )x i,ξ j for all i =,,,p and ξ =,,,k i Hence, they lie in the null space of matrix F k [P(λ ); γ] Moreover, one can verify that the vectors in (5) are linearly independent, keeping in mind their block form and the linear independence of the eigenvectors x,, x,,,x p, C n As a consequence, the dimension of the null space of F k [P(λ ); γ] is greater than or equal to r (ie, is an eigenvalue of matrix F k [P(λ ); γ] of geometric multiplicity at least r), and thus, s kn r+ (F k [P(λ ); γ]) = By this lemma, if a scalar µ C is a multiple eigenvalue of a perturbed matrix polynomial Q(λ) = P(λ) + (λ) (as in (3)) of algebraic multiplicity at least r and index of annihilation k, then for any nonzero γ C, µ is an eigenvalue of the kn kn matrix polynomial F k [Q(λ); γ] of geometric multiplicity at least r This observation and the discussion in Section 3 of [3] yield readily the following lemma Lemma 3 If µ C is an eigenvalue of a matrix polynomial Q(λ) = P(λ) + (λ) of algebraic multiplicity at least r and index of annihilation k, then for every γ, s kn r+ (F k [P(µ); γ]) F k [ (µ); γ] u v u It is straightforward to verify that, v Ckn (u j, v j C n, j =,,,k) is a pair of left and right singular vectors corresponding to a singular 5 u k j= v k

6 u (γ/ γ )u v (γ/ γ )v value of F k [P(µ); γ] (γ ) if and only if, is (γ/ γ ) k u k (γ/ γ ) k v k a pair of left and right singular vectors of F k [P(µ); γ ] corresponding to the same singular value Hence, for convenience, and without loss of generality, from this point and in the remainder of the paper, we assume that the parameter γ is real positive Theorem 4 Suppose µ C is an eigenvalue of a perturbed matrix polynomial Q(λ) = P(λ) + (λ) B(P, ε, w) of algebraic multiplicity at least r and index of annihilation k Then for every γ >, ε F k[ (µ); γ] F k [; γ ] s kn r+(f k [P(µ); γ]) F k [; γ ] Proof For the matrix polynomial (λ) and its derivatives, we have and (i) (µ) (µ) m j µ j ε j= m j (j ) (j i + ) j µ j i εw (i) ( µ ); i =,,,r j=i Thus, for any γ >, there is a unit vector ŷ = y y y k Ckn with y, y,,y k C n such that (µ)y γ () (µ)y + (µ)y F k [ (µ); γ] = F k [ (µ); γ] ŷ = k γ i i! (i) (µ)y k i i= = (µ)y + γ () k γ i (µ)y + (µ)y + + i! (i) (µ)y k i i= (µ) y + γ () (µ) y + γ (µ) () (µ) y y + (µ) y + + (µ) y k 6

7 (ε) y + γ (εw () ( µ )) y = + γ(ε)(εw () ( µ )) y y + (ε) y + + (ε) y k ε y γεw () ( µ ) y + ε y k γ i i! εw(i) ( µ ) y k i i= ε F k [; γ] The proof is completed by Lemma 3 As mentioned before, for k =, E r, (µ) s n r+ (P(µ))/ [3, Theorem 4] Corollary 5 For any γ >, the inequalities E r,k (µ) s kn r+(f k [P(µ); γ]) F k [; γ] ; k =,,,r and hold E r (µ) s kn r+ (F k [P(µ); γ]) min k=,,,nm F k [; γ] Let us denote by η(r, k) the smallest integer that is greater than or equal to n r k One can see that as γ + s kn r+ (F k [P(µ); γ]) F k [; γ] s η(r,k)(p(µ)) 3 An upper bound for the distance E r (µ) E n η(r,k)+, (µ) The technique applied in the proof of Theorem of [3] can be extended for the derivation of an upper bound for the distance E r (µ) from P(λ) in () to the n n matrix polynomials that have µ C as an eigenvalue of algebraic multiplicity at least r The following definition is necessary Definition 3 For any γ > and r {, 3,,n}, let u (γ) u (γ), v (γ) v (γ) Crn u r (γ) v r (γ) (u j (γ), v j (γ) C n, j =,,,r) be a pair of left and right singular vectors of s rn r+ (F r [P(µ); γ]), respectively Then we define the n r matrices U(γ) = [u (γ) u (γ) u r (γ)] and V (γ) = [v (γ) v (γ) v r (γ)] 7

8 For any γ > with rank(v (γ)) = r {, 3,n}, we will construct a perturbation γ (λ) such that the perturbed matrix polynomial Q γ (λ) = P(λ) + γ (λ) has µ as a defective eigenvalue with an associated (not necessarily maximal) Jordan chain of length r First we define the quantities φ i = w(i) ( µ ) (i!) ( ) µ i ; i =,,,r, µ setting µ/ µ = whenever µ =, and recalling that w > We also consider the r r upper triangular Toeplitz matrix γφ γ (φ φ ) γ 3 (φ φ φ 3 φ 3 ) γφ γ (φ φ ) Θ γ = [θ i,j ] = γφ, whose entries above the main diagonal are given by the recursive formulae θ i,j = θ i,i γ j i φ j i θ i,i+ γ j (i+) φ j (i+) θ i,j γφ ; i < j r (6) Denoting by V (γ) the Moore-Penrose pseudoinverse of V (γ), we consider the n n matrix γ = s rn r+ (F r [P(µ); γ])u(γ)θ γ V (γ), and define the matrices γ,j = and the matrix polynomial w ( ) j µ j γ ; j =,,,m µ γ (λ) = m γ,j λ j j= We observe that γ (µ) = γ, and for i =,,,r, (i) γ (µ) = m ( ) w j µ j j (j ) (j i + ) γ µ j i µ j=i = γ = γ w (i) ( µ ) m ( µ j (j ) (j i + )w j µ j=i ( ) µ i = (i!)φ i γ µ ) i ( ) µ j i µ j i µ Since u(γ), v(γ) C rn are a left and a right singular vector of s rn r+ (F r [P(µ); γ]), respectively, it holds that F r [P(µ); γ] v(γ) = s rn r+ (F r [P(µ); γ])u(γ) (7) 8

9 Denote also by e, e, the vectors of the standard basis, ie, the columns of the identity matrix If rank(v (γ)) = r ( {, 3,,n}), or equivalently, if V (γ) V (γ) = I r (the r r identity matrix), then the perturbed matrix polynomial satisfies Q γ (λ) = P(λ) + γ (λ) = m (A j + γ,j )λ j (8) j= Q γ (µ)v (γ) = P(µ)v (γ) + γ v (γ) = s rn r+ (F r [P(µ); γ])u (γ) s rn r+ (F r [P(µ); γ])u(γ)e = s rn r+ (F r [P(µ); γ])u (γ) s rn r+ (F r [P(µ); γ])u (γ) = By straightforward computations, setting φ = and recalling (6) and (7), we verify that for every i =,,,r, = j= j= j! γj Q (j) γ (µ)v i j+ (γ) j! γj P (j) (µ)v i j+ (γ) + = s rn r+ (F r [P(µ); γ])u i+ (γ) j= = s rn r+ (F r [P(µ); γ]) u i+ (γ) = s rn r+ (F r [P(µ); γ]) u i+ (γ) = s rn r+ (F r [P(µ); γ]) u i+ (γ) j! γj (j) γ (µ)v i j+ (γ) γ j φ j γ v i j+ (γ) j= γ j φ j U(γ)Θ γ V (γ) v i j+ (γ) j= γ j φ j U(γ)Θ γ e i j+ j= j= i j+ ξ= = s rn r+ (F r [P(µ); γ]) u i+ (γ) θ i+,i+ u i+ (γ) γ j φ j θ ξ,i j+ u ξ (γ) i+ = s rn r+ (F r [P(µ); γ]) γ i j+ φ i j+ θ ξ,j u ξ (γ) = Dividing by γ i yields j= ξ= j=ξ i+ γ i j+ φ i j+ θ ξ,j u ξ (γ) ξ= ( ) j! Q(j) γ (µ) γ (i j) v i j+ (γ) = 9 j=ξ

10 As a consequence, if rank(v (γ)) = r ( {, 3,,n}), then µ is a defective eigenvalue of Q γ (λ) with { v (γ), γ v (γ), γ v 3 (γ),, γ (r ) v r (γ) } as an associated Jordan chain of length r (recall the definition of Jordan chains in ()) Furthermore, we see that γ (µ) = γ = s rn r+ (F r [P(µ); γ]) U(γ)Θ γ V (γ) and γ,j = w j s rn r+ (F r [P(µ); γ]) U(γ)Θ γ V (γ) ; j =,,,r Hence, we have the next result, which is a direct generalization of the second part of Theorem in [3] Theorem 3 Let P(λ) be a matrix polynomial as in (), µ C, and r {, 3,,n} Then for every γ > such that rank(v (γ)) = r, it holds that E r (µ) s rn r+(f r [P(µ); γ]) U(γ)Θ γ V (γ), ( and Q γ (λ) in (8) lies on the boundary of B P, s rn r+(f r[p(µ);γ]) U(γ)Θ γ V (γ) ), w and has µ as a defective eigenvalue with a (not necessarily maximal) Jordan chain of length r We remark that for r 3, it is not easy to find values of γ for which the condition rank(v (γ)) = r is ensured, as it was done in [3] for r = On the other hand, in all our experiments, this rank condition appears to hold generically 4 Numerical examples To illustrate the proposed (lower and upper) bounds and their tightness, we begin with the special case of constant matrices Example 4 Consider the 6 6 smoke matrix that can be generated by the Matlab command gallery( smoke,6), the corresponding linear pencil L(λ) = I 6 λ S, the weights w = {w, w } = {, }, and the scalar µ = 384+i 6767 By [], we know that E 3 (384 + i6767) = E 3,3 (384 + i 6767) = 37 The graphs of our lower bound for E 3,3 (384 + i 6767) (see Corollary 5) and our upper bound for E 3 (384 + i 6767) (see Theorem 3) are depicted in Figure for γ (, 5] For γ = 6748, Corollary 5 implies the lower bound 345 Moreover, for γ = 54, Theorem 3 yields the upper bound 4694 and an associated perturbed matrix S + with = 4694, which has µ = i 6767 as a defective eigenvalue of algebraic multiplicity 3 and geometric multiplicity

11 6 4 upper bound lower bound Figure : Lower and upper bounds for E3 (384 + i 6767), < γ upper bound 5 lower bound Figure : Lower bound for E3,3 ( 5) and upper bound for E3 ( 5), < γ 5 For our second example, we consider a real quadratic matrix polynomial Example 4 Let P (λ) = λ + 3 λ and w = {w, w, w } = {, 68, 3} (the norms of the coefficient matrices) For the scalar µ = 5, by Example of [3] (see also Proposition 7 and Theorem 8 in []), we know that E ( 5) = E, ( 5) = E ( 5) = (9) The graphs of our lower bound for E3,3 ( 5) and our upper bound for E3 ( 5) are illustrated in Figure for γ (, 5] Setting γ = 553 and γ = 658, we get the lower bound 48 and the upper bound 377, respectively It is worth noting that these bounds are clearly compatible with (9) Furthermore, Theorem 3 yields

12 upper bound 5 5 lower bound Figure 3: Lower bound for E 3, () and upper bound for E 3 (), < γ 5 the perturbed matrix polynomial Q(λ) = λ that lies on the boundary of B (P, 377, w), and has µ = 5 as a defective eigenvalue of algebraic [ multiplicity ] 3 and geometric multiplicity, with an associated 957 eigenvector x = The matrix polynomial in our last example is triangular, and hence, we can directly compute a (non optimal) perturbation with the desired properties Example 43 Consider the 4 4 quadratic matrix polynomial λ 5λ + 4 λ + λ + 6 P(λ) = λ + λ 5 λ λ i8 iλ λ λ + 5 and the weights w = {w, w, w } = {,, } The graphs of the proposed lower bound for E 3, () and upper bound for E 3 () are plotted in Figure 3 for γ (, 5] As γ + and for γ = 87, we get the lower bound 493 for E 3, () and the upper bound 6357 for E 3 (), respectively The exact values of the distances E 3, () and E 3 () are not known, but these bounds are consistent with the fact that the matrix polynomial R(λ) = λ 4λ + 4 λ + λ + 6 λ + λ 6 λ λ i 9 iλ λ + 6 λ

13 lies on the boundary of B (P,, w), and has µ = as a defective eigenvalue of algebraic multiplicity 3 and index of annihilation Recalling the comment in the last paragraph of Section (and applying the methodology in [3, Section 3]), we also observe that the lower bound 493 coincides with the distance E, () Corollary 5 and Theorem 3 provide an infinite number of lower bounds of E r,k (µ) and upper bounds of E r (µ), respectively, in such a way that the best lower/upper bound is given by the maximum/minimum of these bounds over all γ > Since these optimization problems are over only one real variable, in our examples above, we apply the standard grid search with respect to γ (, 5] The difficulty with this brute-force approach is that it usually requires too many bound evaluations Alternatively, we can use golden section search [3, pp ], or Brent s method [, Chapter 5] The latter algorithm is based on the combination of inverse quadratic interpolation and golden section search, and one of its implementations is the Matlab function fminbnd As it is expected, these two derivative-free optimization methods compute at best a local extremum, and there is no guarantee that the global extremum will be found (so, the grid search with a few grid points can be used to initiate an interval of the global optimum) For example, we consider the upper bound of the distance E 3 ( 5) in Example (see Figure ) Applying fminbnd on the interval (, ], we get the global minimum 377 at γ = 658 after 8 iterations Over the interval (, 5], fminbnd finds the local minimum 4449 at γ = 995 after iterations Acknowledgment The author is grateful to the anonymous referees for their valuable comments and suggestions References [] L Boulton, P Lancaster and P Psarrakos, On pseudospecta of matrix polynomials and their boundaries, Math Comp, 77 (8), [] RP Brent, Algorithms for Minimization without Derivatives, Prentice-Hall, Englewood Cliffs, New Jersey, 973 [3] G Dahlquist and A Björck, Numerical Methods in Scientific Computing, Society for Industrial and Applied Mathematics, Philadelphia, 8 [4] I Gohberg, P Lancaster and L Rodman, Matrix Polynomials, Academic Press, New York, 98 [5] I Gohberg and L Rodman, Analytic matrix functions with prescribed local data, J Anal Math, 4 (98), 9 8 [6] KhD Ikramov and AM Nazari, On the distance to the closest matrix with triple zero eigenvalue, Math Notes, 73 (3), 5 5 [7] G Kalogeropoulos, P Psarrakos and N Karcanias, On the computation of the Jordan canonical form of regular matrix polynomials, Linear Algebra Appl, 385 (4), 7 3 [8] P Lancaster and M Tismenetsky, The Theory of Matrices, nd edition, Academic Press, Orlando, 985 3

14 [9] AN Malyshev, A formula for the -norm distance from a matrix to the set of matrices with a multiple eigenvalue, Numer Math, 83 (999), [] AS Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Amer Math Society, Providence, Translations of Math Monographs, Vol 7, 988 [] E Mengi, Locating a nearest matrix with an eigenvalue of prescribed algebraic multiplicity, Numer Math, 8 (), 9 35 [] L Qiu, B Bernhardsson, A Rantzer, EJ Davison, PM Young and JC Doyle, A formula for computation of the real stability radius, Automatika, 3 (995), [3] N Papathanasiou and P Psarrakos, The distance from a matrix polynomial to matrix polynomials with a prescribed multiple eigenvalue, Linear Algebra Appl, 49 (8), [4] JH Wilkinson, The Algebraic Eigenvalue Problem, Claredon Press, Oxford, 965 [5] JH Wilkinson, Note on matrices with a very ill-conditioned eigenprolem, Numer Math, 9 (97),