NORMAL MATRIX POLYNOMIALS WITH NONSINGULAR LEADING COEFFICIENTS

NORMAL MATRIX POLYNOMIALS WITH NONSINGULAR LEADING COEFFICIENTS NIKOLAOS PAPATHANASIOU AND PANAYIOTIS PSARRAKOS Abstract. In this paper, we introduce the notions of weakly noral and noral atrix polynoials, with nonsingular leading coefficients. We characterize these atrix polynoials, using orthonoral systes of eigenvectors and noral eigenvalues. We also study the conditioning of the eigenvalue proble of a noral atrix polynoial, constructing an appropriate Jordan canonical for. Key words. Matrix polynoial, Noral eigenvalue, Weighted perturbation, Condition nuber. AMS subject classifications. 5A8, 5A22, 65F5, 65F35.. Introduction. In pure and applied atheatics, norality of atrices (or operators) arises in any concrete probles. This is reflected to the fact that there are nuerous ways to describe a noral atrix (or operator). A list of about ninety conditions on a square atrix equivalent to its being noral can be found in [5, 7]. The study of atrix polynoials has also a long history, especially in the context of their spectral analysis, which leads to the solutions of associated linear systes of higher order; see [6, 0, ] and the references therein. Surprisingly, it sees that the notion of norality has been overlooked by people working in the area. The only exceptions are the works of Ada and Psarrakos [] and Lancaster and Psarrakos [9]. Our present goal is to take a coprehensive look at norality of atrix polynoials. To avoid infinite eigenvalues, we restrict ourselves to atrix polynoials with nonsingular leading coefficients. The case of singular leading coefficients and infinite eigenvalues will be considered in a future work. The presentation of the article is organized as follows: In the next section, we provide the necessary theoretical background on the spectral analysis of atrix polynoials. In Section 3, we introduce the notions of weakly noral and noral atrix polynoials, and obtain necessary and sufficient conditions for a atrix polynoial to be weakly noral. In Section 4, we consider the noral eigenvalues of atrix polynoials and use the to provide sufficient conditions for a atrix polynoial to be noral. Finally, in Section Received by the editors on... Accepted for publication on... Handling Editor:... Departent of Matheatics, National Technical University of Athens, Zografou Capus, 5780 Athens, Greece (nipapath@ail.ntua.gr, N. Papathanasiou; ppsarr@ath.ntua.gr, P. Psarrakos).

2 N. Papathanasiou and P. Psarrakos 5, we investigate the Jordan structure of a noral atrix polynoial, and study the conditioning of the associated eigenvalue proble. 2. Spectral analysis of atrix polynoials. Consider an n n atrix polynoial (2.) P(λ) = A λ + A λ + + A λ + A 0, where λ is a coplex variable and A j C n n (j = 0,,...,) with deta 0. If the leading coefficient A coincides with the identity atrix I, then P(λ) is called onic. A scalar λ 0 C is said to be an eigenvalue of P(λ) if P(λ 0 )x 0 = 0 for soe nonzero x 0 C n. This vector x 0 is known as a (right) eigenvector of P(λ) corresponding to λ 0. A nonzero vector y 0 C n that satisfies y 0P(λ 0 ) = 0 is called a left eigenvector of P(λ) corresponding to λ 0. The set of all eigenvalues of P(λ) is the spectru of P(λ), naely, σ(p) = {λ C : detp(λ) = 0}, and since deta 0, it contains no ore than n distinct (finite) eleents. The algebraic ultiplicity of an eigenvalue λ 0 σ(p) is the ultiplicity of λ 0 as a zero of the (scalar) polynoial detp(λ), and it is always greater than or equal to the geoetric ultiplicity of λ 0, that is, the diension of the null space of atrix P(λ 0 ). A ultiple eigenvalue of P(λ) is called seisiple if its algebraic ultiplicity is equal to its geoetric ultiplicity. Let λ,λ 2,...,λ r σ(p) be the eigenvalues of P(λ), where each λ i appears k i ties if and only if the geoetric ultiplicity of λ i is k i (i =,2,...,r). Suppose also that for a λ i σ(p), there exist x i,,x i,2,...,x i,si C n with x i, 0, such that P (si ) (λ i ) (s i )! x i, + P (si 2) (λ i ) (s i 2)! P(λ i )x i, = 0 P (λ i ) x i, + P(λ i )x i,2 = 0!... x i,2 + + P (λ i ) x i,si + P(λ i )x i,si = 0,! where the indices denote the derivatives of P(λ) and s i cannot exceed the algebraic ultiplicity of λ i. Then the vector x i, is an eigenvector of λ i, and the vectors x i,2,x i,3,...,x i,si are known as generalized eigenvectors. The set {x i,,x i,2,...,x i,si } is called a Jordan chain of length s i of P(λ) corresponding to the eigenvalue λ i. Any eigenvalue of P(λ) of geoetric ultiplicity k has k axial Jordan chains associated to k linearly independent eigenvectors, with total nuber of eigenvectors and generalized eigenvectors equal to the algebraic ultiplicity of this eigenvalue.

Noral atrix polynoials with nonsingular leading coefficients 3 We consider now the n n atrix X = [x, x,2 x,s x 2, x r, x r,2 x r,sr ] fored by axial Jordan chains of P(λ) and the associated n n Jordan atrix J = J J 2 J r, where each J i is the Jordan block that corresponds to the X XJ Jordan chain {x i,,x i,2,...,x i,si } of λ i. Then the n n atrix Q =. XJ 0 is invertible, and we can define Y = Q.. The set (X,J,Y ) is called a 0 A Jordan triple of P(λ), and satisfies (2.2) P(λ) = X(λI J) Y ; λ / σ(p). The set {x,, x,2,..., x,s, x 2,,..., x r,, x r,2,..., x r,sr } is known as a coplete syste of eigenvectors and generalized eigenvectors of P(λ). 3. Weakly noral and noral atrix polynoials. In [], the ter noral atrix polynoial has been used for the atrix polynoials that can be diagonalized by a unitary siilarity. For atrix polynoials of degree 2, this definition does not ensure the seisiplicity of the eigenvalues, and hence, it is necessary to odify it. Consider, for exaple, the diagonal atrix polynoials (3.) and (3.2) P(λ) = R(λ) = (λ 2)(λ ) 0 0 0 λ(λ ) 0 0 0 (λ + )(λ + 2) (λ ) 2 0 0 0 λ(λ 2) 0 0 0 (λ + )(λ + 2) which have exactly the sae eigenvalues (counting ultiplicities), 2,, 0, (double) and 2. The eigenvalue λ = is seisiple as an eigenvalue of P(λ) with algebraic and geoetric ultiplicities equal to 2. On the other hand, λ = is not seisiple as an eigenvalue of R(λ) since its algebraic ultiplicity is 2 and its geoetric ultiplicity is. Definition 3.. The atrix polynoial P(λ) in (2.) is called weakly noral if there is a unitary atrix U C n n such that U P(λ)U is diagonal for all λ C. If,,

4 N. Papathanasiou and P. Psarrakos in addition, every diagonal entry of U P(λ)U is a polynoial with exactly distinct zeros, or equivalently, all the eigenvalues of P(λ) are seisiple, then P(λ) is called noral. Clearly, the atrix polynoial P(λ) in (3.) is noral, and the atrix polynoial R(λ) in (3.2) is weakly noral (but not noral). Note also that the notions of weakly noral and noral atrix polynoials coincide for atrices and for linear pencils of the for P(λ) = A λ + A 0. The next two leas are necessary to characterize weakly noral atrix polynoials. Lea 3.2. Let A,B C n n be noral atrices such that AB = B A. Then the atrices A + B and AB are also noral. Lea 3.3. Suppose that for every µ C, the atrix P(µ) is noral. Then for every i,j = 0,,...,, it holds that A i A j = A j A i. In particular, all coefficient atrices A 0,A,...,A are noral. Proof. Let P(µ) be a noral atrix for every µ C. Then P(0) = A 0 is noral, i.e. A 0 A 0 = A 0A 0. Fro the proof of [4, Lea 6], we have that A 0 A i = A i A 0 for every i =,2,...,. Thus, P(µ)A 0 = A 0P(µ) for every µ C. By Lea 3.2, it follows that for the atrix polynoials P 0 (λ) = A λ + + A 2 λ + A and P(λ) A 0 = λp 0 (λ), the atrices P 0 (µ) and P(µ) A 0 = µp 0 (µ) are noral for every µ C. Siilarly, by [4, Lea 6] and the fact that P 0 (µ) is noral for every µ C, we have that P 0 (0) = A is also noral and A A i = A i A for every i = 2,3,...,. Hence, as before, P (µ) = A µ 2 + + A 3 µ + A 2 and P 0 (µ) A = µp (µ) are noral atrices for every µ C. Repeating the sae process, we coplete the proof. Theore 3.4. The atrix polynoial P(λ) in (2.) is weakly noral if and only if for every µ C, the atrix P(µ) is noral. Proof. If the atrix polynoial P(λ) is weakly noral, then it is apparent that for every µ C, the atrix P(µ) is noral. For the converse, suppose that for every µ C, the atrix P(µ) is noral. The next assuption is necessary. Assuption. Suppose that there is a coefficient atrix A i with s 2 distinct eigenvalues. Then without loss of generality, we ay assue that A i = λ i I k λ i2 I k2 λ is I ks

Noral atrix polynoials with nonsingular leading coefficients 5 and A j = A j A j2 A js ; j i, where the eigenvalues λ i,λ i2,...,λ is of A i are distinct and nonzero, with ultiplicities k,k 2,...,k s, respectively, and A j C k k, A j2 C k2 k2,..., A js C ks ks. Proof of Assuption. Since A i is noral, there is a unitary atrix U C n n such that U A i U = λ i I k λ i2 I k2 λ is I ks. We observe that for any µ,a C, the atrix P(µ) is noral if and only if the atrix U P(µ)U + aiµ i is noral. Thus, without loss of generality, we ay assue that all λ il s are nonzero. By Lea 3.3, it follows that for every j i, or equivalently, A i A j = A ja i, A j = A i A ja i. By straightforward calculations, we coplete the proof of the assuption. We proceed now with the proof of the converse, which is by induction with respect to the order n of P(λ). Clearly, for n =, there is nothing to prove. If n = 2, and there is a coefficient atrix with distinct eigenvalues, then by Assuption, all A 0,A,...,A are diagonal. If there is no coefficient atrix of P(λ) with distinct eigenvalues, then each A i (i = 0,,...,) is noral with a double eigenvalue, and hence, it is scalar, i.e., A i = a i I. As a consequence, P(λ) is diagonal. Assue now that for any n = 3,4,...,k, every n n atrix polynoial P(λ) such that the atrix P(µ) is noral for any µ C, is weakly noral. Let P(λ) = A λ + A λ + + A λ + A 0 be a k k atrix polynoial, and suppose that there is a A i with s 2 distinct eigenvalues. By Assuption, and for every j i, Then A i = λ i I k λ i2 I k2 d is I ks, A j = A j A j2 A js. P(λ) = P (λ) P 2 (λ) P s (λ),

6 N. Papathanasiou and P. Psarrakos where the atrix polynoials P (λ),p 2 (λ),...,p s (λ) are weakly noral. Hence, there are unitary atrices U t C kt kt, t =,2,...,s, such that U t P t (λ)u t, t =,2,...,s, are diagonal. Thus, for the k k unitary atrix U = U U 2 U s, the atrix polynoial U P(λ)U is diagonal. Suppose that there is no A i with at least two distinct eigenvalues. Then each coefficient atrix A i is noral with exactly one eigenvalue (of algebraic ultiplicity k), and hence, it is scalar, i.e., A i = a i I. As a consequence, P(λ) is diagonal. By this theore, it follows that the atrix polynoial P(λ) is weakly noral if and only if P(λ)[P(λ)] [P(λ)] P(λ) = 0 for every λ C. We observe that each entry of the atrix function P(λ)[P(λ)] [P(λ)] P(λ) is of the for χ(α,β) + iψ(α,β), where α and β are the real and iaginary parts of variable λ, respectively, and χ(α,β) and ψ(α,β) are real polynoials in α,β R of total degree at ost 2. As a consequence, Lea 3. of [2] yields the following. Corollary 3.5. The atrix polynoial P(λ) in (2.) is weakly noral if and only if for any distinct real nubers s,s 2,...,s 4 2 +2+, the atrices P(s j + ) (j =,2,...,4 2 + 2 + ) are noral. is 2+ j By Theore 3.4, Corollary 3.5 and [6] (see also the references therein), the next corollary follows readily. Corollary 3.6. Let P(λ) = A λ + + A λ + A 0 be an n n atrix polynoial as in (2.). Then the following are equivalent: (i) The atrix polynoial P(λ) is weakly noral. (ii) For every µ C, the atrix P(µ) is noral. (iii) For any distinct real nubers s,s 2,...,s 4 2 +2+, the atrices P(s j +i s 2+ j ) (j =,2,...,4 2 + 2 + ) are noral. (iv) The coefficient atrices A 0,A,...,A are noral and utually couting, i.e., A i A j = A j A i for i j. (v) All the linear cobinations of the coefficient atrices A 0,A,...,A are noral atrices. (vi) The coefficient atrices A 0,A,...,A are noral and satisfy property L, that is, there exists an ordering of the eigenvalues λ (j),λ(j) 2,...,λ(j) n of A j (j = 0,,...,) such that for all scalars t 0,t,...,t C, the eigenvalues of t 0 A 0 +t A + +t A are t 0 λ (0) i +t λ () i + +t λ () i (i =,2,...,n). (vii) There exists a unitary atrix U C n n such that U A j U is diagonal for every j = 0,,...,. 4. Noral eigenvalues. In the atrix case, it is well known that norality (or diagonalizibility) is equivalent to the orthogonality (resp., linear independence) of eigenvectors. In the atrix polynoial case, it is clear (by definition) that any n n

Noral atrix polynoials with nonsingular leading coefficients 7 noral atrix polynoial of degree has an orthogonal syste of n eigenvectors, such that each one of these eigenvectors corresponds to exactly distinct eigenvalues. Proposition 4.. Consider a atrix polynoial P(λ) as in (2.), with all its eigenvalues seisiple. Suppose also that P(λ) has a coplete syste of eigenvectors where each vector of a basis {g,g 2,...,g n } of C n appears exactly ties. Then there exists a diagonal atrix polynoial D(λ) such that where G = [g g 2 g n ] C n n. P(λ) = A GD(λ)G, Proof. Each vector g i (i =,2,...,n) appears exactly ties as an eigenvector of distinct eigenvalues of P(λ), say λ i,λ i2,...,λ i. By [8, Theore ], we have that P(λ)g i = (λ λ ij )g i ; i =,2,...,n. Thus, P(λ)[g g 2 g n ] = A (λ λ j )g (λ λ 2j )g 2 (λ λ nj )g n = A [g g 2 g n ]diag (λ λ j )g, Consequently, P(λ) = A G diag (λ λ j ), and the proof is coplete. (λ λ 2j )g 2,..., (λ λ 2j ),..., (λ λ nj )g n. (λ λ nj ) G, Corollary 4.2. Under the hypothesis of Proposition 4., it holds that: (i) If the atrix G A G is diagonal, then the atrix polynoial G P(λ)G is diagonal. (ii) If G is unitary, then there exists a diagonal atrix polynoial D(λ) such that P(λ) = A GD(λ)G. (iii) If G is unitary and the atrix G A G is diagonal, then the atrix polynoial G P(λ)G is diagonal, i.e., P(λ) is noral.

8 N. Papathanasiou and P. Psarrakos Note that if P(λ) = Iλ A, then in Corollary 4.2, G A G = G I G = I for nonsingular G, and G A G = G I G = I for unitary G. This eans that all the parts of the corollary are direct generalizations of standard results on atrices. The following definition was originally introduced in [9]. Definition 4.3. Let P(λ) be an n n atrix polynoial as in (2.). An eigenvalue λ 0 σ(p) of algebraic ultiplicity k is said to be noral if there exists a unitary atrix U C n n such that U P(λ)U = [(λ λ 0 )D(λ)] Q(λ), where the atrix polynoial D(λ) is k k diagonal and λ 0 / σ(d) σ(q). By this definition and Definition 3., it is obvious that every noral atrix polynoial has all its eigenvalues noral. In the sequel, we obtain the converse. Proposition 4.4. Suppose that all the eigenvalues of P(λ) in (2.) are seisiple, and that λ,λ 2,...,λ s are noral eigenvalues of P(λ) with ultiplicities, 2,..., s, respectively, such that + 2 +... + s = n. If P(λ) satisfies U j P(λ)U j = [(λ λ j )D j (λ)] Q j (λ) ; j =,2,...,s, where for each j, the atrix U j C n n is unitary, the atrix polynoial D j (λ) is j j diagonal and λ i σ(q j )\σ(d j ) (i = j+,...,s), then the atrix polynoial P(λ) is noral. Proof. For the eigenvalue λ, by hypothesis, we have that U P(λ)U = [(λ λ )D (λ)] Q (λ), where U is unitary and D (λ) is diagonal. The first coluns of U are an orthonoral syste of eigenvectors of λ. Fro hypothesis we also have that deta 0 and all the eigenvalues of P(λ) are seisiple. As a consequence, each one of the eigenvectors of λ is an eigenvector for exactly eigenvalues of P(λ) (counting λ ). Moreover, since λ i σ(q )\σ(d ), i = 2,3,...,s, these eigenvectors of λ, are orthogonal to the eigenspaces of the eigenvalues λ 2,λ 3,...,λ s. Siilarly, for the eigenvalue λ 2, we have U 2 P(λ)U 2 = [(λ λ 2 )D 2 (λ)] Q 2 (λ), where U 2 is unitary and D 2 (λ) is 2 2 diagonal. As before, the first 2 coluns of U 2 are an orthonoral syste of eigenvectors of λ 2. In addition, each one of these 2 eigenvectors of λ 2 is an eigenvector for exactly eigenvalues of P(λ) (counting λ 2 ). Since λ i σ(q 2 )\σ(d 2 ), i = 3,4,...,s, these 2 eigenvectors of λ 2 are orthogonal to the eigenspaces of the eigenvalues λ 3,λ 4,...,λ s.

Noral atrix polynoials with nonsingular leading coefficients 9 Repeating this process for the eigenvalues λ 3,λ 4,...,λ s, we construct an orthonoral basis of C n, u,u 2,...,u,u +,u +2,...,u +2,...,u n s+,u n s+2,...,u n, }{{}}{{}}{{} of λ of λ 2 of λ s where each vector is an eigenvector for distinct eigenvalues of P(λ) and an eigenvector of the leading coefficient A. By Corollary 4.2, P(λ) is a noral atrix polynoial. The next lea follows readily and it is necessary for our discussion. Lea 4.5. If an n n atrix polynoial P(λ) has a noral eigenvalue of ultiplicity n or n, then it is weakly noral. Theore 4.6. Consider a atrix polynoial P(λ) as in (2.). If all its eigenvalues are noral, then P(λ) is noral. Proof. Let λ,λ 2,...,λ s be the distinct eigenvalues of P(λ) with corresponding ultiplicities k,k 2,...,k s (k +k 2 + +k s = n), and suppose that they are noral. It is easy to see that s. If s = then all the eigenvalues have ultiplicity n and by Lea 4.5, the theore follows. Suppose that s > and for every j = 0,,...,s, there is a unitary atrix U j = [u j u j2... u jn ] such that U j P(λ)U j = [(λ λ j )D j (λ)] Q j (λ), where D j (λ) is k j k j diagonal and λ j / σ(d j ) σ(q j ). Then the first k j coluns of U j (j =,2,...,s) are right and left eigenvectors of P(λ), and also of A 0,A,...,A. The set of all vectors u,, u,2,..., u,k, u 2,, u 2,2,..., u 2,k2,... u s,, u s,2,..., u s,ks for a coplete syste of eigenvectors of P(λ). So, by [3], there is a basis of C n {u,u 2,...,u n } {u,u 2,...,u k, u 2,u 22,...,u 2k2,...,u s,u s2,...,u sks }. We also observe that the vectors u,u 2,...,u n are linearly independent right and left coon eigenvectors of the coefficient atrices A 0,A,...,A. Keeping in ind that any left and right eigenvectors (of the sae atrix) corresponding to distinct eigenvalues are orthogonal, [7, Condition 3] iplies that all A j s are noral. Moreover, any two vectors u i, u j (i j) that correspond to distinct eigenvalues of a coefficient atrix are also orthogonal. Hence, it is straightforward to see that there exists a unitary atrix U such that all U A j U s are diagonal. As a consequence, the atrix polynoial P(λ) is weakly noral, and since all its eigenvalues are seisiple, P(λ) is noral.

0 N. Papathanasiou and P. Psarrakos 5. Weighted perturbations and condition nubers. Let P(λ) be a atrix polynoial as in (2.). We are interested in perturbations of P(λ) of the for (5.) Q(λ) = P(λ) + (λ) = (A j + j )λ j, j=0 where the atrices 0,,..., C n n are arbitrary. For a given paraeter ε > 0 and a given set of nonnegative weights w = {w 0,w,...,w } with w 0 > 0, we define the set of adissible perturbed atrix polynoials (5.2) B(P,ε,w) = {Q(λ) as in (5.) : j εw j, j = 0,,...,}, where denotes the spectral atrix nor (i.e., that nor subordinate to the euclidean vector nor). The weights w 0,w,...,w allow freedo in how perturbations are easured, and the set B(P,ε,w) is convex and copact [3] with respect to the ax nor P(λ) = ax 0 j A j. In [5], otivated by (2.2) and the work of Chu [4], for a Jordan triple (X,J,Y ) of P(λ), the authors introduced the condition nuber of the eigenproble of P(λ), that is, k(p) = X Y. Furtherore, they applied the Bauer-Fike technique [2, 4] and used k(p), to bound eigenvalues of perturbations of P(λ). Denoting w(λ) = w λ + + w λ + w 0, one of the results of [5] is the following. Proposition 5.. Let (X,J,Y ) be a Jordan triple of P(λ), and let Q(λ) B(P,ε,w) for soe ε > 0. If the Jordan atrix J is diagonal, then for any µ σ(q)\σ(p), in µ λ k(p)εw( µ ). λ σ(p) As in the atrix case, we say that a atrix polynoial eigenvalue proble is well-conditioned (or ill-conditioned) if its condition nuber is sufficiently sall (resp., sufficiently large). In the reainder of this section, we confine our discussion to noral atrix polynoials. Recall that for an n n noral atrix polynoial P(λ), there is a unitary atrix U C n n such that U P(λ)U is diagonal and all its diagonal entries Note that the definition of k(p) clearly depends on the choice of the triple (X, J, Y ), but to keep things siple, the Jordan triple will not appear explicitly in the notation.

Noral atrix polynoials with nonsingular leading coefficients are polynoials of degree exactly, with distinct zeros. Moreover, for any Jordan triple (X,J,Y ) of P(λ), the Jordan atrix J is diagonal. In the sequel, we derive soe bounds for the condition nuber k(p). The following lea is a siple exercise. Lea 5.2. Let λ,λ 2,...,λ be distinct scalars. Then it holds that = (λ λ (λ λ j ) j ) (λ i λ j ). i j Next we copute a Jordan triple of a onic noral atrix polynoial P(λ) and the associated condition nuber k(p). Proposition 5.3. Let P(λ) be an n n onic noral atrix polynoial of degree, and let U P(λ)U = (Iλ J ) (Iλ J 2 ) (Iλ J ) for soe unitary U C n n and diagonal atrices J,J 2,...,J. Then a Jordan triple (X,J,Y ) of P(λ) is given by [ ] 2 X = U I (J 2 J ) (J J i ) (J J i ), i= J = J J 2 J and Y = i= (J J i ) (J 2 J i ) i=2 i=3. (J J ) I U. Proof. 2 By Lea 5.2 and straightforward calculations, we see that (Iλ J j ) = (Iλ J j ) i J j ), i j(j or equivalently, P(λ) = X(Iλ J) Y. 2 Originally, we have derived another (constructive) proof of the proposition, which explains the choice of atrices X and Y. That proof is inductive with respect to the degree of P(λ), and uses Corollary 3.3 of [6]. It also requires soe coputations, and hence, we decided to present this short proof.

2 N. Papathanasiou and P. Psarrakos Theore 2.6 of [6] copletes the proof. Theore 5.4. Let P(λ) be an n n onic noral atrix polynoial of degree, and let (X,J,Y ) be the Jordan triple given by Proposition 5.3. If we denote J i = diag{λ (i),λ(i) 2,...,λ(i) n } ; i =,2,...,, then the condition nuber of the eigenproble of P(λ) is k(p) = ( + ax s=,2,...,n ( { + ax s=,2,...,n λ (2) s λ () { λ () s s + + 2 i= λ () s λ ( ) s + + 2 i=2 λ (i) s 2 }) /2 λ (i) s λ () s 2 }) /2. Proof. Since P(λ) is noral, it follows λ s (i) that [ 2 X = U I (J 2 J ) i= λ (j) s, i j, s =,2,...,n. Recall ] (J J i ) (J J i ), and observe that ( XX = U I + (J 2 J ) (J 2 J ) + + (J J i ) (J J i ) )U. If we denote by λ ax ( ) the largest eigenvalue of a square atrix, then { X 2 = λ ax (XX ) = + ax s=,2,...,n λ (2) s λ () s + + 2 Siilarly, we verify that Y 2 = λ ax (Y Y ) = + and the proof is coplete. ax s=,2,...,n { λ () s i= i= i= λ ( ) s + + 2 i=2 λ () s λ (i) s 2 λ (i) s λ () s 2 It is worth noting that since a (onic) noral atrix polynoial is essentially diagonal, the condition nuber of its eigenproble depends on the eigenvalues and not on the eigenvectors. Furtherore, by the above theore, it is apparent that if 2 and the utual distances of the eigenvalues of the onic atrix polynoial P(λ) are sufficiently large, then the condition nuber k(p) is relatively close to, i.e., the eigenproble of P(λ) is well-conditioned. On the other hand, if 2 and } }.,

Noral atrix polynoials with nonsingular leading coefficients 3 the utual distances of the eigenvalues are sufficiently sall, then k(p) is relatively large, i.e., the eigenproble of P(λ) is ill-conditioned. Theore 5.4 iplies practical lower and upper bounds for the condition nuber k(p). For convenience, we denote Θ = ax λ, ˆλ σ(p) λ ˆλ λ ˆλ and θ = in λ, ˆλ σ(p) λ ˆλ λ ˆλ, and assue that Θ,θ. Corollary 5.5. Let P(λ) be an n n onic noral atrix polynoial of degree, and let (X,J,Y ) be the Jordan triple given by Proposition 5.3. Then the condition nuber k(p) satisfies Θ 2 Θ 2 Θ 2( ) k(p) θ 2 θ 2 θ 2( ). Consider the atrix polynoial P(λ) in (2.), and recall that its leading coefficient A is nonsingular. By [6, 0], (X,J,Y ) is a Jordan triple of the onic atrix polynoial ˆP(λ) = A P(λ) = Iλ + A A λ + + A A λ + A A 0 if and only if (X,J,Y A ) is a Jordan triple of P(λ). This observation and the proof of Theore 5.4 yield the next results. Corollary 5.6. Let P(λ) = A λ + A 0 be an n n noral linear pencil, and let U P(λ)U be diagonal for soe unitary U C n n. Then a Jordan triple of P(λ) is (X,J,Y ) = (U,J,UA ), and k(p) = A. Theore 5.7. Let P(λ) in (2.) be noral, and let (X,J,Y ) be the Jordan triple of the onic atrix polynoial ˆP(λ) = A P(λ) given by Proposition 5.3. Then for the condition nuber k(p) = X Y A, we have A Θ 2 k(p) A Θ 2 Θ 2( ) θ 2 θ 2 θ 2( ). Proof. As entioned above, (X,J,Y ) is a Jordan triple of the onic atrix polynoial ˆP(λ) if and only if (X,J,Y A ) is a Jordan triple of P(λ). Notice also that ˆP(λ) is noral, and by the proof of Theore 5.4, Θ 2 X, Y Θ 2 Θ 2( ) θ 2 θ 2 θ. 2( ) Furtherore, there is an n n unitary atrix U such that D(λ) = U P(λ)U and D = U A U are diagonal. As a consequence, A = D and A = D.

4 N. Papathanasiou and P. Psarrakos Since the atrix Y U C n n is a block-colun of diagonal atrices of order n, it is straightforward to see that Y A = Y UD U = Y UD = Y D. The atrix Y Y is also diagonal, and thus, Y D ( ) = D Y Y D /2 = Y Y ( D ) D /2. Hence, it follows Y D Y D Y D. By Corollary 5.5, the proof is copleted. Proposition 5. iplies directly the following. Corollary 5.8. Let P(λ) in (2.) be noral, and let Q(λ) B(P,ε,w) for soe ε > 0. Then for any µ σ(q)\σ(p), it holds that in µ λ εw( µ ) A λ σ(p) θ 2 θ 2 θ 2( ). Reark 5.9. In the construction of the above bounds, we have assued that Θ and θ are different than. Suppose that this assuption fails for a noral atrix polynoial P(λ). Then, keeping in ind the Jordan triple (X,J,Y ) of ˆP(λ) = A P(λ) given by Proposition 5.3 and the proofs of Theores 5.4 and 5.7, one can easily see that k(p) A when Θ =, and k(p) A when θ =. Finally, as an exaple, recall the onic noral atrix polynoial P(λ) in (3.). For the weights w 0 = w = w 2 =, Theore 5.4 yields k(p) = 2, i.e., the eigenproble of P(λ) is well-conditioned. Note also that θ = and the value 2 coincides with the upper bound given by Reark 5.9. REFERENCES [] M. Ada and P. Psarrakos. On a copression of noral atrix polynoials. Linear and Multilinear Algebra, 52:25 263, 2004. [2] F.L. Bauer and C.T. Fike. Nors and exclusion theores. Nuer. Math., 2:37 44, 960. [3] L. Boulton, P. Lancaster, and P. Psarrakos. On pseudospecta of atrix polynoials and their boundaries. Math. Cop., 77:33 334, 2008. [4] K.-W.E. Chu. Perturbation of eigenvalues for atrix polynoials via the Bauer-Fike theores. SIAM J. Matrix Anal. Appl., 25:55 573, 2003.

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