Gershgorin type sets for eigenvalues of matrix polynomials

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1 Electronic Journal of Linear Algebra Volume 34 Volume 34 (18) Article Gershgorin type sets for eigenvalues of matrix polynomials Christina Michailidou National Technical University of Athens, Panayiotis Psarrakos National Technical University of Athens, Follow this and additional works at: Part of the Algebra Commons Recommended Citation Michailidou, Christina and Psarrakos, Panayiotis. (18), "Gershgorin type sets for eigenvalues of matrix polynomials", Electronic Journal of Linear Algebra, Volume 34, pp DOI: This Article is brought to you for free and open access by Wyoming Scholars Repository. It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Repository. For more information, please contact

2 GERSHGORIN TYPE SETS FOR EIGENVALUES OF MATRIX POLYNOMIALS CHRISTINA MICHAILIDOU AND PANAYIOTIS PSARRAKOS Abstract. New localization results for polynomial eigenvalue problems are obtained, by extending the notions of the Gershgorin set, the generalized Gershgorin set, the Brauer set and the Dashnic-Zusmanovich set to the case of matrix polynomials. Key words. Eigenvalue, Matrix polynomial, The Gershgorin set, The generalized Gershgorin set, The Brauer set, The Dashnic-Zusmanovich set. AMS subject classifications. 15A18, 15A. 1. Introduction. For an arbitrary square complex matrix A C n n with standard spectrum σ), the celebrated Gershgorin (Geršgorin) circle theorem [8, 1] implies n easily computable disks, in the complex plane, centered at the diagonal entries of the matrix, whose union contains σ). The simplicity and the applications of the Gershgorin circle theorem have inspired further research in this area, resulting in hundreds of papers on Gershgorin disks and related sets such as the generalized Gershgorin set (known also as the A-Ostrowski set), the Brauer set, the Dashnic-Zusmanovich set and others (see [3, 5, 6, 18, 1] and the references therein), which are widely used in the analysis of ordinary eigenvalue problems. In this paper, we consider n n matrix polynomials of the form (1.1) P (λ) = A m λ m + A m 1 λ m A 1 λ + A, where λ is a complex variable, A, A 1,..., A m C n n with A m, and the determinant det P (λ) is not identically zero. The study of matrix polynomials, especially with regard to their spectral analysis and the solution of higher order linear differential (or difference) systems with constant coefficients, has a long history and important applications; see [9, 14, 15, 16, 17] and the references therein. A scalar µ C is called an eigenvalue of P (λ) if the system P (µ)x = has a nonzero solution x C n. This solution x is known as an eigenvector of P (λ) corresponding to the eigenvalue µ. The set of all finite eigenvalues of P (λ), σ(p ) = µ C : det P (µ) = } = µ C : σ(p (µ))}, is the finite spectrum of P (λ). The algebraic multiplicity of an eigenvalue µ σ(p ) is the multiplicity of µ as a root of the (scalar) polynomial det P (λ), and it is always greater than or equal to the geometric multiplicity of µ, that is, the dimension of the null space of matrix P (µ). Furthermore, it is said that µ = is an eigenvalue of P (λ) exactly when is an eigenvalue of the reverse matrix polynomial ˆP (λ) = λ m P (1/λ) = A λ m + A 1 λ m A m 1 λ + A m. Received by the editors on March 17, 18. Accepted for publication on November, 18. Handling Editor: Michael Tsatsomeros. Corresponding Author: Panayiotis Psarrakos. Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografou Campus, 1578 Athens, Greece (xristina.michailidou@gmail.com, ppsarr@math.ntua.gr). The research of the first author was supported by the Research Committee of the National Technical University of Athens (ELKE PhD Scholarships).

3 653 Gershgorin Type Sets for Eigenvalues of Matrix Polynomials In this case, the algebraic multiplicity and the geometric multiplicity of the eigenvalue µ = of P (λ) are defined as the algebraic multiplicity and the geometric multiplicity of the eigenvalue of ˆP (λ), respectively. In the next sections, we extend the notions of the Gershgorin set (Section ), the generalized Gershgorin set (Section 3), the Brauer set (Section 4) and the Dashnic-Zusmanovich set (Section 5) to the case of matrix polynomials; see also [1, 1, 13]. In particular, in each section, we define an eigenvalues inclusion set, present basic topological and geometrical properties of this set, and give illustrative examples to verify our results. Our approach is direct, does not involve any kind of linearizations, and it is motivated by the following simple observation: Suppose that F is a set-valued function of n n matrices which associates each matrix A C n n to a region F ) C that contains the spectrum σ). If, for an n n matrix polynomial P (λ) as in (1.1), we define the set F (P ) = µ C : F (P (µ))}, then σ(p ) = µ C : σ(p (µ))} µ C : F (P (µ))} = F (P ).. The Gershgorin set..1. Definitions. Consider a square complex matrix A C n n, define the set N = 1,,..., n}, and let ) i,j denote the (i, j)-th entry of A, i, j N. For any i N, define also the nonnegative quantity r i ) = ) i,j and the i-th row Gershgorin disk G i ) = µ C : µ ) i,i r i )}. The j N \i} Gershgorin circle theorem [8, 1] states that the spectrum σ) lies in the union G) = G i ), which is known as the Gershgorin set of A. Consider now an n n matrix polynomial P (λ) as in (1.1), and define the nonnegative functions r i (P (λ)) = (P (λ)) i,j, i N, the i-th row Gershgorin sets of P (λ) j N \i} G i (P ) = µ C : G i (P (µ))} = µ C : (P (µ)) i,i r i (P (µ))}, i N, and the Gershgorin set of P (λ) It is easy to see that and G(P ) = µ C : G(P (µ))} = i N G i (P ). G i ( ˆP )\} = µ C\} : G i ( P ( µ 1 ))} = µ C\} : µ 1 G i (P ) }, i N, G( ˆP )\} = µ C\} : G ( P ( µ 1))} = µ C\} : µ 1 G(P ) }. As in the case of the eigenvalues of P (λ), we say that µ = lies in G i (P ) (resp., in G(P )) exactly when lies in G i ( ˆP ) (resp., in G( ˆP )). The Gershgorin circle theorem [8, 1] is directly generalized to the case of matrix polynomials. Theorem.1. All the (finite and infinite) eigenvalues of the matrix polynomial P (λ) lie in the Gershgorin set G(P ). Proof. For any finite eigenvalue µ σ(p ), it is apparent that σ(p (µ)) G(P (µ)), and hence, µ lies in G(P ). Moreover, if µ = is an eigenvalue of P (λ), then σ( ˆP ) G( ˆP ), and consequently, µ = lies in G(P ). i N

4 Christina Michailidou and Panayiotis Psarrakos 654 By the original proof of the Gershgorin circle theorem [8] (see also [1, Theorem 1.1]), it follows that if λ is an eigenvalue of a matrix A C n n and x = [ x 1 x x n ] T is an eigenvector of A corresponding to λ, with < x k = max x 1, x,..., x n } for some k N, then λ lies in the k-th row Gershgorin disk G k ). As a consequence, if µ is an eigenvalue of a matrix polynomial P (λ) and x = [ x 1 x x n ] T is an eigenvector of P (λ) corresponding to µ, with < x k = max x 1, x,..., x n } for some k N, then is an eigenvalue of the matrix P (µ ) with x as an associated eigenvector, and thus, G k (P (µ )), or equivalently, µ G k (P )... Basic properties. For an n n matrix polynomial P (λ) as in (1.1), it is easy to verify the following properties. Proposition.. Let i N. (i) G i (P ) is a closed subset of C. (ii) For any scalar b C\}, consider the matrix polynomials Q 1 (λ) = P (bλ), Q (λ) = bp (λ) and Q 3 (λ) = P (λ + b). Then G i (Q 1 ) = b 1 G i (P ), G i (Q ) = G i (P ) and G i (Q 3 ) = G i (P ) b. (iii) If the (i, i)-th entry of P (λ), (P (λ)) i,i, is (identically) zero, then G i (P ) = C. (iv) If all the coefficient matrices A, A 1,..., A m have their i-th rows real, then G i (P ) is symmetric with respect to the real axis. Proof. (i) Consider a scalar µ / G i (P ). Then (P (µ)) i,i > r i (P (µ)), and continuity implies that for any scalar ˆµ sufficiently close to µ, (P (ˆµ)) i,i > r i (P (ˆµ)), i.e., ˆµ / G i (P ). Thus, the set C\G i (P ) is open. (ii) Recall that µ G i (P ) if and only if G i (P (µ)), and observe that µ } G i (Q 1 ) = µ C : G i (P (bµ))} = b C : G i(p (µ)), G i (Q ) = µ C : G i (bp (µ))} = µ C : G i (P (µ))} and G i (Q 3 ) = µ C : G i (P (µ + b))} = µ b C : G i (P (µ))}. (iii) If the (i, i)-th entry of P (λ) is identically zero, then for any µ C, the origin is the center of the Gershgorin disk G i (P (µ)). Hence, the inequality (P (µ)) i,i r i (P (µ)) is satisfied trivially for all µ C. (iv) Suppose that all the coefficient matrices A, A 1,..., A m have their i-th rows real. If µ G i (P ), then m k ) i,i µ k m k ) i,j µ k, or equivalently, or equivalently, This means that µ G i (P ). m k ) i,i µ k m k ) i,i µ k m k ) i,j µ k, m k ) i,j µ k.

5 655 Gershgorin Type Sets for Eigenvalues of Matrix Polynomials Proposition.3. For any i N, the set µ C : (P (µ)) i,i < r i (P (µ))} lies in the interior of G i (P ); as a consequence, G i (P ) µ C : (P (µ)) i,i = r i (P (µ))} = µ C : G i (P (µ))}. Proof. Consider a scalar µ C such that (P (µ)) i,i < r i (P (µ)). By continuity, there is a real ε > such that for every ˆµ C with µ ˆµ ε, (P (ˆµ)) i,i r i (P (ˆµ)). Thus, µ is an interior point of G i (P ). By applying the proof technique of Theorem 3.1 in [1], we obtain a necessary condition for the isolated points. Theorem.4. If a scalar µ C is an isolated point of G i (P ), then µ is a common root of all polynomials (P (λ)) i,j, j N ; as a consequence, µ is an eigenvalue of P (λ). Proof. For the sake of contradiction, assume that (P (µ)) i,i. Since µ is an isolated point of G i (P ), it belongs to the boundary G i (P ), and there exists an ε > such that the disk D(µ, ε) = λ C : λ µ ε} contains no other point of G i (P ). The i-th row Gershgorin set can be written as (.) Define the function G i (P ) = µ C : (P (µ)) i,i r i (P (µ))} } r i (P (µ)) = µ C : (P (µ)) i,i 1 = µ C : log r } i(p (µ)) (P (µ)) i,i. ϕ(λ) = log r i(p (λ)), λ D(µ, ε) (P (λ)) i,i [ = log (P (λ))i,1,..., (P (λ)) i,i 1, (P (λ)) i,i+1,..., (P (λ)) ] i,n, λ D(µ, ε), (P (λ)) i,i (P (λ)) i,i (P (λ)) i,i (P (λ)) i,i 1 and observe that it is subharmonic and satisfies the maximum principle [, 4]. By definition, ϕ(λ) is equal to zero on the boundary of G i (P ), nonnegative in the interior of G i (P ) and negative elsewhere; see Proposition.3. Since µ G i (P ), it follows (P (µ)) i,i = r i (P (µ)) = (P (µ)) i,j. j N \i} So, the function ϕ(λ) is equal to zero at the center µ of D(µ, ε) and negative in the rest of the disk. Since ϕ(λ) satisfies the maximum principle, it should take it s maximum value on the boundary of the disk; this is a contradiction. As a consequence, = (P (µ)) i,i = (P (µ)) i,j, j N \i} and µ is a common root of all polynomials (P (λ)) i,j, j N. Let Ω be a closed subset of C, and let µ Ω. The local dimension of the point µ in Ω is defined as the limit lim dimω D(µ, h)} (h R, h > ), where dim } denotes the topological dimension [11]. Any h + isolated point in Ω has local dimension. Any non-isolated point in Ω has local dimension if it belongs to the closure of the interior of Ω, and 1 otherwise.

6 Christina Michailidou and Panayiotis Psarrakos 656 Theorem.5. Any point of the i-th row Gershgorin set G i (P ) has local dimension either or (isolated point); in other words, the Gershgorin set of P (λ) cannot have parts with (nontrivial) curves. Proof. For the sake of contradiction, assume that the local dimension of a point µ G i (P ) is equal to 1. Then µ G i (P ) and there is an ε > such that G i (P ) λ C : λ µ ε} G i (P ). Thus, the function ϕ(λ) in (.), defined in the disk λ µ ε, takes its maximum value in (infinitely many) interior points of the disk. Since ϕ(λ) is subharmonic and satisfies the maximum principle, this is a contradiction. Hence, the local dimension of µ is either or. Next we obtain necessary and sufficient conditions for the row Gershgorin sets to be bounded. For any i N, we define the sets (.3) β i = j N : m ) i,j } and β i = N \β i = j N : m ) i,j = }. It is worth mentioning that a scalar µ C is a common root of all polynomials (P (λ)) i,1, (P (λ)) i,,..., (P (λ)) i,n if and only if the i-th row of matrix P (µ) is zero, or equivalently, if and only if G i (P (µ)) = }. By Theorem.4, if the origin is an isolated point of the row Gershgorin set G i (P ), then the i-th row of the coefficient matrix A is zero. As a consequence, if β i, then the origin is not an isolated point of G i ( ˆP ), i.e., G i (P ) is not the union of a bounded set and. Theorem.6. Suppose that for an i N, β i. (i) If i β i, then the i-th row Gershgorin set G i (P ) is unbounded if and only if G i m ). (ii) If i β i, then the i-th row Gershgorin set G i (P ) is unbounded and G i m ). Proof. (i) Suppose that i β i, i.e., m ) i,i. Let G i (P ) be unbounded. Since β i, the origin is not an isolated point of G i ( ˆP ) and there is a sequence µ l } l N in G i (P )\} such that µ l +. Then, for every l N, m k ) i,i µ k l m k ) i,j µ k l, or As l +, it follows and hence, G i m ). m µ k l k ) i,i m µ k l k ) i,j m ) i,i m ) i,j, j β i\i}. For the converse, suppose that G i m ) (or equivalently, m ) i,i r i m )). Then G i ( ˆP ) and, by definition, G i (P ) (where by hypothesis, is a non-isolated point of G i ( ˆP ), and hence, is a non-isolated point of G i (P )). (ii) Suppose that i β i, i.e., m ) i,i =.

7 657 Gershgorin Type Sets for Eigenvalues of Matrix Polynomials It is clear that G i m ), and m 1 G i (P )\} = µ C\} : k ) i,i µ k m k ) i,j µ k m 1 = µ C\} : µ k k ) i,i m µ k k ) i,j µ m 1 µ m 1, where at least one of the coefficients m ) i,j, j N \i}, is nonzero. As a consequence, for large enough µ, µ lies in G i (P ). Thus, there exists a real number M > such that every scalar µ C with µ M lies in G i (P ), i.e., µ C : µ M} G i (P ). Remark.7. Suppose that G i m ), i.e., G i ( ˆP ). If the origin is a non-isolated point of G i ( ˆP ), then there exists a sequence µ l } l N G i ( ˆP )\} that converges to. This means that the sequence µ 1 l } l N G i (P ) is unbounded. Hence, the i-th row Gershgorin set G i (P ) is also unbounded and does not have the infinity as an isolated point. On the other hand, if the origin is an isolated point of G i ( ˆP ) (which yields β i = ), then G i (P )\} = µ C\} : µ 1 G i ( ˆP } )\} }, where the set µ C\} : µ 1 G i ( ˆP } )\} is bounded. Remark.8. Suppose that i β i and the origin is an interior point of G i m ) (or equivalently, m ) i,i and m ) i,i r i m ) < ). Then, for any sequence µ l } l N in C\} such that µ l +, and for sufficiently large l, m µ k l k ) i,i m µ k l k ) i,j, or or m k ) i,i µ k l m k ) i,j µ k l, µ l G i (P ). As a consequence, G i (P ) is unbounded, and there exists a real number M > such that µ C : µ M} G i (P ). The next result, as well as Theorem 3.7 at the end of the next section, is a special case of Theorem 3.1 in the work of D. Bindel and A. Hood [1] (see also Theorem in [13]). For clarity, we give an elementary proof. Theorem.9. Suppose that A m has no zero rows, and the Gershgorin set G(P ) is bounded. Then the number of the connected components of G(P ) is less than or equal to nm. Moreover, if G is a connected component of G(P ) constructed by ξ connected components of the row Gershgorin sets, then the total number of roots of the polynomials (P (λ)) 1,1, (P (λ)),,..., (P (λ)) n,n in G is at least ξ and it is equal to the number of eigenvalues of P (λ) in G, counting multiplicities. Proof. Since G(P ) is bounded, Theorem.6 (i) implies that for every i N, / G i m ), i β i and the polynomial (P (λ)) i,i is of degree m (i.e., it has exactly m roots, counting multiplicities). For all coefficient matrices of P (λ), A, A 1,..., A m, consider the splitting A j = D j + F j, j =, 1,..., m,

8 Christina Michailidou and Panayiotis Psarrakos 658 where D j is the diagonal part of A j (i.e., D j is a diagonal matrix and its diagonal coincides with the diagonal of A j ) for every j =, 1,..., m. Define also the family of matrix polynomials P t (λ) = (D m + tf m )λ m + (D m 1 + tf m 1 )λ m (D 1 + tf 1 )λ + (D + tf ) = D m λ m + D m 1 λ m D 1 λ + D + t ( F m λ m + F m 1 λ m F 1 λ + F ) for t [, 1]. We observe that (.4) G(P t1 ) G(P t ), t 1 t 1. Indeed, if < t 1 t 1 and a scalar µ C lies in G i (P t1 ) for some i N, then (P t (µ)) i,i = (P t1 (µ)) i,i r i (P t1 (µ)) = t1 t r i (P t (µ)) r i (P t (µ)), and thus, µ G i (P t ). Moreover, it is apparent that if = t 1 t 1 and µ G i (P ), then µ is a root of (P (λ)) i,i and (P t (µ)) i,i = (P (µ)) i,i = r i (P (µ)) = r i (P t (µ)). By Proposition. (i), Theorem.6 and (.4), it follows that the family G(P t ), t [, 1] is a family of nondecreasing compact sets (with respect to t [, 1]). Furthermore, G(P 1 ) = G(P ), and G(P ) coincides with the roots of the polynomials (P (λ)) 1,1, (P (λ)),,..., (P (λ)) n,n (which are eigenvalues of the diagonal matrix polynomial D m λ m +D m 1 λ m D 1 λ + D ). Keeping in mind that all polynomials det P t (λ), t [, 1], are of degree nm, the continuity of the eigenvalues of P t (λ) in t [, 1] implies that each root of the polynomials (P (λ)) 1,1, (P (λ)),,..., (P (λ)) n,n is connected to an eigenvalue of P (λ) with a continuous curve in G(P ). Hence, any (compact) connected component of G(P ) constructed by ξ connected components of the row Gershgorin sets, G, contains at least ξ roots of the polynomials (P (λ)) 1,1, (P (λ)),,..., (P (λ)) n,n, and each root of (P (λ)) 1,1, (P (λ)),,..., (P (λ)) n,n in G is connected to an eigenvalue of P t (λ), t [, 1], with a continuous curve in G..3. Examples. The Gershgorin set is simple and gives better results than known bounds based on norms. In the next example, we compare it to some bounds given in [1]; in particular, we draw and compare the Gershgorin set to the bounds given by Lemmas. and.3 of [1]. For a matrix polynomial P (λ) as in (1.1), consider the n n matrices U i = A 1 m A i (i =, 1,..., m 1) and L i = A 1 A i (i = 1,,..., m). Then, by Lemma. of [1], every eigenvalue λ of P (λ) satisfies 1 m m 1 (.5) 1 + L j p λ 1 + U j p, 1 p. j=1 Define also the n nm matrices U = [ U U 1 U m 1 ] and L = [ L m L m 1 L 1 ]. Then, by Lemma.3 of [1], we have the following bounds: } 1 } (.6) max L m 1, 1 + max L i 1 λ max U 1, 1 + max U i 1, i=1,,...,m 1 i=1,,...,m 1 (.7) max L, 1} 1 λ max U, 1}) and (.8) I + LL 1/ λ I + UU 1/. j=

9 659 Gershgorin Type Sets for Eigenvalues of Matrix Polynomials (a) The set G(P ) and the bounds of [1, Lemma.] for norm-. (b) The set G(P ) and the bounds of [1, Lemma.3] for norm-. (c) The set G(P ) and the bounds of [1, Lemma.3] for norm-. (d) The set G(P ). (e) The set G 1(P ). (f) The set G (P ). Figure 1: Comparing the Gershgorin set to the bounds of Lemmas. and.3 of [1]. Example.1. Consider the (real) matrix polynomial [ ] 5λ 6 + λ λ P (λ) = λ λ λ The upper bound given by (.5) for norm- is approximately equal to and can be seen in Figure 1 (a) (dashed line) compared to the Gershgorin set G(P ). The lower bound of (.5) is equal to.9 and since it is relatively small, its illustration is omitted in the figure. The lower and upper bounds given by (.6) (.8) are.181 and 3.65 for norm-1,.1111 and 3.5 for norm-, and.1394 and.678 for norm-. The upper bounds for norm- and norm- compared to the Gershgorin set G(P ) are illustrated in parts (b) and (c) of the figure, respectively. The sets G(P ), G 1 (P ) (with exactly six connected components) and G (P ) (with exactly six connected components) are given (magnified) in parts (d), (e) and (f) of Figure 1, respectively. Here and elsewhere, the eigenvalues are marked with dots. Note that Proposition. (iv) (symmetry with respect to the real axis), Theorem.6 (boundedness conditions) and Theorem.9 (distribution of eigenvalues in the connected components) are clearly verified.

10 Christina Michailidou and Panayiotis Psarrakos (a) The set G(Q) (b) The sets G1 (Q) and G (Q) (c) The set G3 (Q). Figure : The Gershgorin set of Q(λ) = Iλm A. For a matrix A Cn n, consider the matrix polynomial Q(λ) = Iλm A. Then, for any i N, Gi (Q) = µ C : λm )i,i ri )} = µ C : µ1/m Gi ) is bounded and its shape depends on the location of the origin with respect to Gi ); in particular, we have three cases illustrated in the next example. Example.11. Consider the (complex) matrix polynomial 1 3.5i 3i λ + 3.5i 3i. Q(λ) = Iλ1 A = Iλ1 i = λ1 i.5 i 7i 5 + 3i.5 + i 7i λ i The Gershgorin set G(Q) = G1 (Q) G (Q) G3 (Q), the union of the sets G1 (Q) and G (Q), and the set G3 (Q) are illustrated in parts (a), (b) and (c) of Figure, respectively. Observe that / G1 ) (since 3.5i > 3i ), and thus, G1 (Q) consists of ten connected components located symmetrically around the origin. Concerning the second row, notice that i = and G ), and hence, the set G (Q) is connected and daisy shaped. Finally, in the third row, we have 5 + 3i <.5 i + 7i (i.e., the origin lies in the interior of G3 )), and G3 (Q) is connected, with smooth boundary. 3. The generalized Gershgorin set. In this section, we extend the notion of the generalized Gershgorin set (or the A-Ostrowski set) of matrices to the case of matrix polynomials Definitions. For a square complex matrix A Cn n, denote ci ) = ri T ), i N, and for any a [, 1], define the disks Ai,a ) = µ C : µ )i,i ri )a ci )1 a, i N. Then the spectrum S of matrix A lies in the union Aa ) = Ai,a ), and consequently, it lies in the intersection A) = i N T Aa ) [18, 1], which is known as the generalized Gershgorin set (or the A-Ostrowski set) of A. a [,1] Similarly, for a matrix polynomial P (λ) as in (1.1), we define Ai,a (P ) = µ C : Ai,a (P (µ))} = µ C : P (µ)i,i ri (P (µ))a ci (P (µ))1 a

11 661 Gershgorin Type Sets for Eigenvalues of Matrix Polynomials for any i N and a [, 1], and the generalized Gershgorin set of P (λ) A a (P ) = µ C : A a (P (µ))} = i N A(P ) = µ C : A(P (µ))} = A i,a (P ), a [, 1], a [,1] A a (P ). As before, we say that µ = lies in A i,a (P ) (resp., in A a (P ), or in A(P )) exactly when lies in A i,a ( ˆP ) (resp., in A a ( ˆP ), or in A( ˆP )). Theorem 3.1. The generalized Gershgorin set of a matrix polynomial P (λ) is a subset of the Gershgorin set of P (λ) that contains all the (finite and infinite) eigenvalues of P (λ). Proof. By definition, a scalar µ C lies in A(P ) (resp., in σ(p ), or in G(P )) if and only if lies in A(P (µ)) (resp., in σ(p (µ)), or in G(P (µ))). Since σ(p (µ)) A(P (µ)) G(P (µ)) and σ( ˆP (µ)) A( ˆP (µ)) G( ˆP (µ)) [18, 1], the proof follows readily. 3.. Basic properties. Let P (λ) be an n n matrix polynomial as in (1.1). It is easy to verify the following properties. Proposition 3.. Let a [, 1] and i N. (i) A i,a (P ) is a closed subset of C. (ii) For any scalar b C\}, consider the matrix polynomials Q 1 (λ) = P (bλ), Q (λ) = bp (λ) and Q 3 (λ) = P (λ + b). Then A i,a (Q 1 ) = b 1 A i,a (P ), A i,a (Q ) = A i,a (P ) and A i,a (Q 3 ) = A i,a (P ) b. (iii) If the (i, i)-th entry of P (λ), (P (λ)) i,i, is (identically) zero, then A i,a (P ) = C. (iv) If a (, 1), and all the coefficient matrices A, A 1,..., A m have their i-th rows and i-th columns real, then A i,a (P ) is symmetric with respect to the real axis (for a = 1 or, see Proposition. (iv) applied to P (λ) or its transpose, respectively). Proof. The proof is similar to the proof of Proposition.. Proposition.3 for the row Gershgorin sets can be readily generalized for A i,a (P ). Proposition 3.3. For any a [, 1] and i N, the set µ C : (P (µ)) i,i < r i (P (µ)) a c i (P (µ)) 1 a} lies in the interior of A i,a (P ); as a consequence, A i,a (P ) µ C : (P (µ)) i,i = r i (P (µ)) a c i (P (µ)) 1 a} = µ C : A i,a (P (µ))}. Theorems.4 and.5 can also be extended. Theorem 3.4. (For a = 1 or, see Theorem.4 applied to P (λ) or its transpose, respectively.) Let a (, 1) and i N. If µ is an isolated point of A i,a (P ), then µ is a common root of all polynomials (P (λ)) i,j, j N, or a common root of all polynomials (P (λ)) p,i, p N ; as a consequence, µ is an eigenvalue of P (λ). Proof. For the sake of contradiction, assume that (P (µ)) i,i. Since µ is an isolated point, it belongs to the boundary of A i,a (P ), and there exists an ε > such that the disk D(µ, ε) = λ C : λ µ ε}

12 Christina Michailidou and Panayiotis Psarrakos 66 contains no other point of A i,a (P ). The set A i,a (P ) can be written as A i,a (P ) = µ C : (P (µ)) i,i r i (P (µ)) a c i (P (µ)) 1 a} = µ C : r i(p (µ)) a c i (P (µ)) 1 a } 1 (P (µ)) i,i = µ C : log r i(p (µ)) a c i (P (µ)) 1 a } (P (µ)) i,i = µ C : a log r i(p (µ)) (P (µ)) i,i + (1 a) log c } i(p (µ)) (P (µ)) i,i. Consider the function (3.9) φ(λ) = a log r i(p (λ)) (P (λ)) i,i + (1 a) log c i(p (λ)) (P (λ)) i,i, λ D(µ, ε), and observe that it is subharmonic (as a sum of subharmonic functions) and satisfies the maximum principle [, 4]. By definition, φ(λ) is equal to zero on the boundary of A i,a (P ), nonnegative in the interior of A i,a (P ) and negative elsewhere; see Proposition 3.3. Furthermore, since µ A i,a (P ), it follows that the function φ(λ) is equal to zero at the center µ of D(µ, ε) and negative in the rest of the disk; this is a contradiction because φ(λ) satisfies the maximum principle. As a consequence, = (P (µ)) i,i = and the result follows readily. j N \i} a (P (µ)) i,j p N \i} (P (µ)) p,i Theorem 3.5. Let a [, 1] and i N. Any point of the i-th generalized Gershgorin set A i,a (P ) has local dimension either or (isolated point); in other words, the generalized Gershgorin set of P (λ) cannot have parts with (nontrivial) curves. Proof. For a = 1 or, see Theorem.5 applied to P (λ) or its transpose, respectively. For a (, 1), the proof is similar to the proof of Theorem.5, using the (subharmonic) function φ(λ) in (3.9) instead of ϕ(λ) in (.). Recalling the definition of β i and β i in (.3), we define similarly the sets (3.1) γ j = p N : m ) p,j } and γ j = N \γ j = p N : m ) p,j = }, and we obtain (as in Theorem.6) necessary and sufficient conditions for the sets A i,a (P ) to be bounded. Note that, by Theorem 3.4, if µ C is an isolated point of A i,a (P ) for some a (, 1) and i N, then µ is a common root of all polynomials (P (λ)) i,j, j N, or a common root of all polynomials (P (λ)) p,i, p N. As a consequence, if the sets β i and γ i are nonempty, then the origin is not an isolated point of A i,a ( ˆP ), i.e., A i,a (P ) is not the union of a bounded set and. Theorem 3.6. (For a = 1 or, see Theorem.6 applied to P (λ) or its transpose, respectively.) Let a (, 1), and suppose that for an i N, the sets β i and γ i are nonempty. (i) If i β i (thus, i γ i ), then A i,a (P ) is unbounded if and only if A i,a m ). (ii) If i β i (thus, i γ i ), then A i,a (P ) is unbounded and A i,a m ). 1 a,

13 663 Gershgorin Type Sets for Eigenvalues of Matrix Polynomials Proof. (i) Suppose that i β i, i.e., m ) i,i. Let A i,a (P ) be unbounded. Since the sets β i and γ i are nonempty, the origin is not an isolated point of A i,a ( ˆP ) and there is a sequence µ l } l N in A i,a (P )\} such that µ l +. Then, for every l N, m k ) i,i µ k l a m k ) i,j µ k l p N \i} m k ) p,i µ k l 1 a, or m µ k l k ) i,i As l +, it follows m ) i,i m µ k l k ) i,j a m ) i,j a j β i\i} p γ i\i} p N \i} m µ k l k ) p,i 1 a m ) p,i, 1 a, and hence, A i,a m ). For the converse, suppose that A i,a m ) (or equivalently, m ) i,i r i m ) a c i m ) 1 a ). Then A i,a ( ˆP ) and, by definition, A i,a (P ). (ii) Suppose that i β i, i.e., m ) i,i =. It is clear that A i,a m ). Moreover, A i,a (P )\} m 1 = µ C\} : k ) i,i µ k m 1 µ k = µ C\} : k ) i,i µ m a m k ) i,j µ k m µ k k ) i,j µ m p N \i} a p N \i} 1 a m k ) p,i µ k 1 a m µ k k ) p,i, where at least one of the coefficients m ) i,j, j N \i}, and one of the coefficients m ) p,i, p N \j}, are nonzero. As a consequence, for large enough µ, µ lies in A i,a (P ). Thus, there exists a real number M > such that every scalar µ C with µ M lies in A i,a (P ), i.e., µ C : µ M} A i,a (P ). As mentioned in the paragraph before Theorem.9, we conclude this section with a result which is a special case of Theorem 3.1 of [1]. Theorem 3.7. (For a = 1 or, see Theorem.9 applied to P (λ) or its transpose, respectively.) Let a (, 1), and suppose that for every i N, the sets β i and γ i are nonempty. If A a (P ) is bounded, then the number of connected components of A a (P ) is less than or equal to nm. Moreover, if G is a connected component of A a (P ) constructed by ξ connected components of the sets A 1,a (P ), A,a (P ),..., A n,a (P ), then the total number of roots of the polynomials (P (λ)) 1,1, (P (λ)),,..., (P (λ)) n,n in G is at least ξ and it is equal to the number of eigenvalues of P (λ) in G, counting multiplicities. µ m

14 Christina Michailidou and Panayiotis Psarrakos 664 (a) The Gershgorin set. (b) The generalized Gershgorin set. Figure 3: Comparing the Gershgorin set and the generalized Gershgorin set. Proof. Since A a (P ) is bounded, Theorem 3.6 (i) implies that for every i N, i β i γ i and the polynomial (P (λ)) i,i is of degree m (i.e., it has exactly m roots, counting multiplicities). Based on the continuity of the eigenvalues of a matrix polynomial with respect to its matrix coefficients and the boundedness of A a (P ), one can complete the proof, following the arguments of the proof of Theorem Examples. Example 3.8. Consider the matrix polynomial of Example.1. In Figure 3, one can see the improvement of the Gershgorin set due to the use of column sums in addition to row sums 1. Moreover, Proposition 3. (iv) (symmetry with respect to the real axis), Theorem 3.6 (boundedness conditions) and Theorem 3.7 (distribution of eigenvalues in the connected components) are verified. Example 3.9. Consider the matrix polynomial 4.λ i 4λ P (λ) = λ 3 λ + 4. λ + i λ + λ 1 In Figure 4, the Gershgorin and generalized Gershgorin sets of P (λ) are illustrated. It is worth mentioning that µ 1 = and µ = ( are isolated points of A(P ), and the third columns of the matrices P (µ 1 ) = ) ( ) P and P (µ ) = P are zero; see Theorems.4 and In Figures 3 (b) and 4 (b), the generalized Gershgorin set A(P ) is estimated by the intersection of the sets A a(p ), a =,.5,.1,...,.95, 1.

15 665 Gershgorin Type Sets for Eigenvalues of Matrix Polynomials (a) The Gershgorin set (b) The generalized Gershgorin set. Figure 4: Comparing the Gershgorin set and the generalized Gershgorin set. 4. The Brauer set. In this section, we introduce the Brauer set of matrix polynomials, similarly to the Gershgorin set and the generalized Gershgorin set Definitions. The Brauer set of a matrix A C n n is defined as B) = i 1 i N j=1 B i,j ), where B i,j ) = µ C : µ ) i,i µ ) j,j r i )r j )}. The Brauer set consists of n(n 1) Cassini ovals, lies in the Gershgorin set, and contains all the eigenvalues of the matrix [3, 1]. For a matrix polynomial P (λ) as in (1.1) and i, j N with j < i, we define the (i, j)-th Brauer set of P (λ) B i,j (P ) = µ C : B i,j (P (µ))} = µ C : (P (µ)) i,i (P (µ)) j,j r i (P (µ))r j (P (µ))}, and the Brauer set of P (λ) B(P ) = µ C : B(P (µ))} = i 1 i N j=1 B i,j (P ). We also say that µ = lies in B i,j (P ) (resp., in B(P )) exactly when lies in B i,j ( ˆP ) (resp., in B( ˆP )). The Brauer theorem for matrices [3, 1] is generalized to the case of matrix polynomials as well as the Gershgorin circle theorem. Theorem 4.1. The Brauer set of a matrix polynomial P (λ) is a subset of the Gershgorin set of P (λ) that contains all the (finite and infinite) eigenvalues of P (λ). Proof. By definition, a scalar µ C lies in B(P ) (resp., in σ(p ), or in G(P )) if and only if lies in B(P (µ)) (resp., in σ(p (µ)), or in G(P (µ))). Since σ(p (µ)) B(P (µ)) G(P (µ)) and σ( ˆP (µ)) B( ˆP (µ)) G( ˆP (µ)) [3, 1], the proof follows readily.

16 Christina Michailidou and Panayiotis Psarrakos Basic properties. Let P (λ) be an n n matrix polynomial as in (1.1). It is easy to verify the following properties. Proposition 4.. Consider two integers i, j N with j < i. (i) B i,j (P ) is a closed subset of C. (ii) For any scalar b C\}, consider the matrix polynomials Q 1 (λ) = P (bλ), Q (λ) = bp (λ) and Q 3 (λ) = P (λ + b). Then B i,j (Q 1 ) = b 1 B i,j (P ), B i,j (Q ) = B i,j (P ) and B i,j (Q 3 ) = B i,j (P ) b. (iii) If the (i, i)-th or the (j, j)-th entry of P (λ) is (identically) zero, then B i,j (P ) = C. (iv) If all the coefficient matrices A, A 1,..., A m have their i-th and j-th rows real, then B i,j (P ) is symmetric with respect to the real axis. Proof. The proof is similar to the proof of Proposition.. Propositions.3 and 3.3 for the Gershgorin set and generalized Gershgorin set, respectively, can be readily extended to the case of the Brauer set. Proposition 4.3. For any i, j N with j < i, the set µ C : (P (µ)) i,i (P (µ)) j,j < r i (P (µ)) r j (P (µ))} lies in the interior of B i,j (P ); as a consequence, B i,j (P ) µ C : (P (µ)) i,i (P (µ)) j,j = r i (P (µ))r j (P (µ))} = µ C : B i,j (P (µ))}. As in Theorem.4 for the Gershgorin set and Theorem 3.4 for the generalized Gershgorin set, we obtain a necessary condition for the isolated points of B i,j (P ). Theorem 4.4. If µ is an isolated point of the (i, j)-th Brauer set B i,j (P ), 1 j < i n, then (a) µ is a root of (P (λ)) i,i or (P (λ)) j,j, and (b) µ is a common root of all polynomials (P (λ)) i,p, p N \i}, or a common root of all polynomials (P (λ)) j,p, p N \j}. Proof. For the sake of contradiction, assume that (P (µ)) i,i and (P (µ)) j,j. Since µ is an isolated point, it belongs to the boundary of the Brauer set, and also there exists an ε > such that the disk D(µ, ε) = λ C : λ µ ε} contains no other point of B i,j (P ). The Brauer set can be written as Consider the function B i,j (P ) = µ C : (P (µ)) i,i (P (µ)) j,j r i (P (µ))r j (P (µ))} } r i (P (µ))r j (P (µ)) = µ C : (P (µ)) i,i (P (µ)) j,j 1 } r i (P (µ))r j (P (µ)) = µ C : log (P (µ)) i,i (P (µ)) j,j = µ C : log r i(p (µ)) (P (µ)) i,i + log r } j(p (µ)) (P (µ)) j,j. (4.11) ϑ(λ) = log r i(p (λ)) (P (λ)) i,i + log r j(p (λ)) (P (λ)) j,j, λ D(µ, ε), and observe that it is subharmonic (as a sum of subharmonic functions) and satisfies the maximum principle [, 4]. By definition, ϑ(λ) is equal to zero on the boundary of B i,j (P ), nonnegative in the interior of B i,j (P )

17 667 Gershgorin Type Sets for Eigenvalues of Matrix Polynomials and negative elsewhere. Since µ B i,j (P ), it follows (P (µ)) i,i (P (µ)) j,j = r i (P (µ))r j (P (µ)) = p N \i} (P (µ)) i,p p N \j} (P (µ)) j,p. Hence, the function ϑ(µ) is equal to zero at the center µ of D(µ, ε) and negative in the rest of the disk; this is a contradiction because ϑ(λ) satisfies the maximum principle. As a consequence, = (P (µ)) i,i (P (µ)) j,j = (P (µ)) i,p (P (µ)) j,p, and the proof is complete. p N \i} p N \j} Theorem 4.5. Any point of the (i, j)-th Brauer set B i,j (P ), 1 j < i n, has local dimension either or (isolated point); in other words, the Brauer set cannot have parts with (nontrivial) curves. Proof. The proof is similar to the proof of Theorem.5, using the (subharmonic) function ϑ(λ) in (4.11) instead of ϕ(λ) in (.). Similarly to Theorem.6 for the Gershgorin set and Theorem 3.6 for the generalized Gershgorin set, and recalling the definition of β i and β i in (.3), we obtain necessary and sufficient conditions for B i,j (P ) to be bounded. By Theorem 4.4, if µ C is an isolated point of B i,j (P ) for some i, j N with j < i, then µ is a root of (P (λ)) i,i or (P (λ)) j,j, and a common root of polynomials (P (λ)) i,p, p N \i}, or of polynomials (P (λ)) j,p, p N \j}. As a consequence, if i β i and j β j, or β i \i} = and β j \j} =, then the origin is not an isolated point of B i,j ( ˆP ), i.e., B i,j (P ) is not the union of a bounded set and. Theorem 4.6. Suppose that for some integers i, j N with j < i, it holds that i β i and j β j, or β i \i} and β j \j}. (i) If i β i and j β j, then B i,j (P ) is unbounded if and only if B i,j m ). (ii) If i β i and j β j, then B i,j (P ) is unbounded and B i,j m ). Proof. (i) Suppose that i β i, i.e., m ) i,i, and j β j, i.e., m ) j,j. Let B i,j (P ) be unbounded. By hypothesis, the origin is not an isolated point of B i,j ( ˆP ), and there is a sequence µ l } l N in B i,j (P )\} such that µ l +. Then, for every l N, m k ) i,i µ k m l k ) j,j µ k l m k ) i,p µ k l m k ) j,p µ k l, or m µ k l m µ k l k ) i,i k ) j,j As l +, it follows m ) i,i m ) j,j p N \i} p N \i} p β i\i} m µ k l k ) i,p m ) i,p p β j\j} p N \j} p N \j} m µ k l k ) j,p. m ) j,p,

18 Christina Michailidou and Panayiotis Psarrakos 668 and hence, B i,j m ). For the converse, suppose that B i,j m ) (or equivalently, m ) i,i m ) j,j r i m )r j m )). Then B i,j ( ˆP ) and, by definition, B i,j (P ). (ii) Suppose that i β i and j β j, i.e., m ) i,i = m ) j,j =. Then, it is clear that B i,j m ). Moreover, m 1 m 1 B i,j (P )\} = µ C\} : k ) i,i µ k k ) j,j µ k m k ) i,p µ k m k ) j,p µ k p N \i} p N \j} m 1 µ k m 1 µ k = µ C\} : k ) i,i µ m k ) j,j µ m m µ k k ) i,p m µ k k ) j,p, p N \i} µ m p N \j} where at least one of the coefficients m ) i,p (p N \i}) and one of the coefficients m ) j,p (p N \j}) are nonzero. As a consequence, for large enough µ, µ lies in B i,j (P ). Thus, there exists a real number M > such that every scalar µ C with µ M lies in B i,j (P ), i.e., µ C : µ M} B i,j (P ). Theorems.9 and 3.7 can be easily extended to the case of the Brauer set. Theorem 4.7. Suppose that for every i, j N with j < i, i β i and j β j. If the Brauer set B(P ) is bounded, then the number of connected components of B(P ) is less than or equal to nm. Moreover, if G is a connected component of B(P ) constructed by ξ connected components of (i, j)-th Brauer sets, then the total number of roots of the polynomials (P (λ)) 1,1, (P (λ)),,..., (P (λ)) n,n in G is at least ξ and it is equal to the number of eigenvalues of P (λ) in G, counting multiplicities. Proof. By hypothesis, for every i, j N with j < i, the polynomial (P (λ)) i,i (P (λ)) j,j is of degree m (i.e., it has exactly m roots, counting multiplicities). Based on the continuity of the eigenvalues of a matrix polynomial with respect to its matrix coefficients and the boundedness of B(P ), one can complete the proof, following the arguments of the proof of Theorem.9. µ m 4.3. Examples. In the case of a matrix polynomial P (λ), the relation det P (λ) = apparently yields (P (λ)) 1,1 (P (λ)), = (P (λ)) 1, (P (λ)),1, and consequently, the eigenvalues appear on the boundary of the Brauer set. Example 4.8. Consider the matrix polynomial P (λ) of Example.1. In Figure 5, the Gershgorin set G(P ) is illustrated in part (a) and the Brauer set B(P ) = B,1 (P ) is illustrated in part (b). It is clearly verified that the Brauer set contains the eigenvalues of P (λ) and lies in the Gershgorin set. As expected, the eigenvalues lie on the boundary of the Bruer set. Moreover, Proposition 4. (iv) (symmetry with respect to the real axis), Theorem 4.4 (isolated points.935±.58i and.98±1.573i), Theorem 4.6 (boundedness conditions) and Theorem 4.7 (distribution of eigenvalues in the connected components) are also confirmed.

19 669 Gershgorin Type Sets for Eigenvalues of Matrix Polynomials (a) The Gershgorin set. (b) The Brauer set. Figure 5: Comparing the Gershgorin set and the Brauer set. (a) The Gershgorin set. (b) The Brauer set. Figure 6: Comparing the Gershgorin set and the Brauer set. Example 4.9. Consider the 3 3 matrix polynomial P (λ) of Example 3.9. In Figure 6, the Gershgorin set G(P ) is illustrated in part (a) and the Brauer set B(P ) is illustrated in part (b). It is once again verified that the Brauer set contains the eigenvalues of P (λ) and lies in the Gershgorin set.

20 Christina Michailidou and Panayiotis Psarrakos The Dashnic-Zusmanovich set. In this section, we introduce the Dashnic-Zusmanovich set of matrix polynomials Definitions. The Dashnic-Zusmanovich set of a matrix A C n n is defined as D) = D i,j ), where D i,j ) = µ C : µ ) i,i ( µ ) j,j r j ) + ) j,i ) r i ) ) j,i } i N for distinct integers i, j N. The Dashnic-Zusmanovich set D) is determined by n(n 1) oval sets, lies in the Brauer set B), and contains all the eigenvalues of matrix A [6, 1]. For a matrix polynomial P (λ) as in (1.1), we define D i,j (P ) = µ C : D i,j (P (µ))} = µ C : (P (µ)) i,i ( (P (µ)) j,j r j (P (µ)) + (P (µ)) j,i ) r i (P (µ)) (P (µ)) j,i } for distinct integers i, j N, the union D i (P ) = and the Dashnic-Zusmanovich set of P (λ) D(P ) = µ C : D(P (µ))} = D i,j (P ), i N, i N D i,j (P ) = i N D i (P ). We also say that µ = lies in D i,j (P ) (resp., in D i (P ), or in D(P )) exactly when lies in D i,j ( ˆP ) (resp., in D i ( ˆP ), or in D( ˆP )). Theorem 5.1. The Dashnic-Zusmanovich set of a matrix polynomial P (λ) is a subset of the Brauer set of P (λ) that contains all the (finite and infinite) eigenvalues of P (λ). Proof. By definition, a scalar µ C lies in D(P ) (resp., in σ(p ), or in B(P )) if and only if lies in D(P (µ)) (resp., in σ(p (µ)), or in B(P (µ))). Since σ(p (µ)) D(P (µ)) B(P (µ)) and σ( ˆP (µ)) D( ˆP (µ)) B( ˆP (µ)) [6, 1], the proof follows readily. 5.. Basic properties. Let P (λ) be an n n matrix polynomial as in (1.1). It is easy to verify the following properties. Proposition 5.. Consider two distinct integers i, j N. (i) D i,j (P ) and D i (P ) are closed subsets of C. (ii) For any scalar b C\}, consider the matrix polynomials Q 1 (λ) = P (bλ), Q (λ) = bp (λ) and Q 3 (λ) = P (λ + b). Then D i,j (Q 1 ) = b 1 D i,j (P ), D i,j (Q ) = D i,j (P ) and D i,j (Q 3 ) = D i,j (P ) b. (iii) If the (i, i)-th or the the (j, j)-th entry of P (λ) is (identically) zero, then D i,j (P ) = D i (P ) = C. (iv) If all the coefficient matrices A, A 1,..., A m have their i-th and j-th rows real, then D i,j (P ) is symmetric with respect to the real axis. Proof. The proof is similar to the proof of Proposition.. Similarly to Theorems.6, 3.6 and 4.6, we obtain necessary and sufficient conditions for the Dashnic- Zusmanovich set to be bounded. Theorem 5.3. Suppose that for some distinct i, j N, the sets β i and β j are nonempty, and the origin is not an isolated point of D i,j ( ˆP ), i.e., D i,j (P ) is not the union of a bounded set and.

21 671 Gershgorin Type Sets for Eigenvalues of Matrix Polynomials (i) If i β i and j β j, then D i,j (P ) is unbounded if and only if D i,j m ). (ii) If i β i β j and j β j, then D i,j (P ) is unbounded and D i,j m ). Proof. (i) Suppose that i β i, i.e., m ) i,i and j β j, i.e., m ) j,j. Let D i,j (P ) be unbounded. Since the origin is not an isolated point of D i,j ( ˆP ), there is a sequence µ l } l N in D i,j (P )\} such that µ l +. Then, for every positive integer l, m k ) i,i µ k l m k ) j,j µ k l m k ) j,p µ k l + m k ) j,i µ k l p N \j} m k ) i,p µ k l m k ) j,i µ k l, p N \i} or m µ k l k ) i,i µ m m µ k l l k ) j,j µ m l m µ k l k ) j,p µ m p N \j} l m µ k l k ) i,p m µ k l k ) j,i, p N \i} + m µ k l k ) j,i As l +, it follows m ) i,i m ) j,j and thus, D i,j m ). p β i\i} m ) i,p + m ) j,i p β i\i} m ) i,p m ) j,i, For the converse, suppose that D i,j m ) (or equivalently, m ) i,i ( m ) j,j r j m ) + m ) j,i ) m ) j,i r j m ) ). Then D i,j ( ˆP ) and, by definition, D i,j (P ). (ii) Suppose that i β i, j β j and i β j, i.e., m ) i,i =, m ) j,j = and m ) j,i. Then, it is clear that D i,j m ), and m 1 D i,j (P )\} = µ C\} : m 1 k ) i,i µ k k ) j,j µ k m k ) j,p µ k p N \j} ) m + k ) j,i µ k m k ) i,p µ k m k ) j,i µ k p N \i} m 1 = µ C\} : µ k m 1 k ) i,i µ m 1 µ k k ) j,j µ m 1 m µ k k ) j,p p N \j} ) m µ k + k ) j,i m µ k k ) i,p m µ k k ) j,i, µ m 1 p N \i} µ m 1 µ m 1 µ m 1

22 Christina Michailidou and Panayiotis Psarrakos (a) The Gershgorin set (b) The Brauer set (c) The D.-Z. set. Figure 7: Comparing the Gershgorin set, the Brauer set and the D.-Z. set. where m ) j,i and at least one of the coefficients m ) i,p, p N \i}, is nonzero. As a consequence, for large enough µ, µ lies in D i,j (P ). Thus, there exists a real number M > such that every scalar µ C with µ M lies in D i,j (P ), i.e., µ C : µ M} D i,j (P ) Examples. Example 5.4. Consider the matrix polynomial [ ] 8iλ iλ + iλ + iλ + (1 + i) P (λ) = 3iλ + 5iλ + (1 + i) 6iλ. + 3iλ + 4i In Figure 7, the Gershgorin set, the Brauer set and the Dashnic-Zusmanovich (D.-Z.) set of P (λ) are drawn. Notice that the Brauer set and the Dashnic-Zusmanovich set are the same (because the matrix polynomial has only rows) and lie in the Gershgorin set. In the following two examples, we consider two 3 3 quadratic matrix polynomials, with bounded (Example 5.5) and unbounded (Example 5.6) eigenvalues inclusion sets. Example 5.5. Consider the matrix polynomial 8iλ iλ + iλ + iλ + (1 + i) ( 1 + i)λ + λ + P (λ) = 3iλ + 5iλ + (1 + i) 8iλ + 3iλ + 4i ( i)λ 4λ 5i. (.8 i)λ + iλ + (1 i).6iλ iλ (6 i)λ + i The Gershgorin set, the Brauer set and the Dashnic-Zusmanovich set are illustrated in Figure 8. It is clear that these three sets are bounded, confirming Theorems.6, 4.6 and 5.3, and D(P ) B(P ) G(P ). Example 5.6. Consider the matrix polynomial 8iλ iλ + iλ + iλ + (1 + i) ( 1 + i)λ + λ + P (λ) = 3iλ + 5iλ + (1 + i) 6iλ + 3iλ + 4i ( i)λ 4λ 5i. (.8 i)λ + iλ + (1 i) 6iλ iλ (6 i)λ + i

23 673 Gershgorin Type Sets for Eigenvalues of Matrix Polynomials (a) The Gershgorin set (b) The Brauer set (c) The D.-Z. set. Figure 8: Comparing the Gershgorin set, the Brauer set and the D.-Z. set (a) The Gershgorin set (b) The Brauer set (c) The D.-Z. set. Figure 9: Comparing the Gershgorin set, the Brauer set and the D.-Z. set. The Gershgorin set, the Brauer set and the Dashnic-Zusmanovich set are drawn in Figure 9, and they are unbounded, verifying Theorems.6, 4.6 and 5.3. Notice once again that D(P ) B(P ) G(P ). Acknowledgments. The authors wish to thank the anonymous referees for valuable comments and suggestions. REFERENCES [1] D. Bindel and A. Hood. Localization theorems for nonlinear eigenvalue problems. SIAM Journal on Matrix Analysis and Applications, 34(6): , 13. [] S. Boyd and C.A. Desoer. Subharmonic functions and performance bounds on linear time-invariant feedback systems. IMA Journal of Mathematical Control and Information, :153 17, 1985.

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