Numerical Range for the Matrix Exponential Function

Transcription

1 Electronic Journal of Linear Algebra Volume 31 Volume 31: (2016) Article Numerical Range for the Matrix Exponential Function Christos Chorianopoulos Chun-Hua Guo University of Regina, Follow this and additional works at: Part of the Physical Sciences and Mathematics Commons Recommended Citation Chorianopoulos, Christos and Guo, Chun-Hua. (2016), "Numerical Range for the Matrix Exponential Function", Electronic Journal of Linear Algebra, Volume 31, 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 NUMERICAL RANGE FOR THE MATRIX EXPONENTIAL FUNCTION CHRISTOS CHORIANOPOULOS AND CHUN-HUA GUO Abstract. For a given square matrix A, the numerical range for the exponential function e At, t C, is considered. Some geometrical and topological properties of the numerical range are presented. Key words. Matrix exponential function, Numerical range. AMS subject classifications. 15A16, 15A Introduction. The numerical range of matrices (and operators in general) has been a topic of extensive research for many decades. The numerical range of a matrix, its related notions and its extensions reveal a great deal of information about the matrix. The numerical range F(A) (also known as the field of values) of a matrix A C n n is the compact and convex set F(A) = {x Ax C : x C n, x x = 1} = {µ C : A λi n 2 µ λ, λ C}, where 2 denotes the spectral matrix norm. The latter definition for F(A) can be found in [9]. Note also that F(A) contains all eigenvalues of A. In the last few decades, the numerical range of matrix polynomials has also been studied extensively; see [8] in particular. If P(µ) = A m µ m +A m 1 µ m 1 + +A 0 is an n n matrix polynomial, then the numerical range of P(µ) is defined as W(P) = {µ C : 0 F(P(µ))}. More generally, one can define numerical ranges for analytic functions of square matrices [7]. Here, we will focus on the exponential function e At, where A C n n is fixed and t C is the variable. We will describe the set of all t values for which the numerical range of the matrix e At contains 0, i.e., the Crawford number for the Received by the editors on September 15, Accepted for publication on September 18, Handling Editor: Panayiotis Psarrakos. Papadiamanti 26, , Athens, Greece (cchorian@gmail.com). Department of Mathematics and Statistics, University of Regina, Regina, SK S4S 0A2, Canada (Chun-Hua.Guo@uregina.ca). This author was supported in part by an NSERC Discovery Grant. 633

3 634 C. Chorianopoulos and C.-H. Guo matrix e At is 0 (see [2] for a discussion of the Crawford number for powers of an operator). Note that our purpose here is not to study the exponential function e A using the numerical range of the matrix A (see [1] for some discussions along that line). Our topic here has some connection to [4], where the author studies classes of operators with 0 in the closure of their numerical range. For A C n n, the exponential of A is defined as e A = k=0 and the series always converges. We collect below some basic properties of the matrix exponential (see [5, 6]). Let A,B C n n. If AB = BA then e A e B = e A+B. It follows that e A is always invertible and (e A ) 1 = e A. If B is invertible then e BAB 1 = Be A B 1. It follows that the eigenvalues of e A are e λ, where λ are eigenvalues of A. If A is Hermitian (A = A), then e A is Hermitian positive definite. The structure of this paper is as follows: In Section 2, we define the numerical range of e At and show that the numerical range is a nonempty set if and only if A is not a scalar matrix. Some basic properties of the set are then presented in Section 3 when A is not a scalar matrix. The special case where the matrix A is Hermitian (or skew-hermitian) is treated in Section 4. Some illustrative examples are given in Section 5. A k k!, 2. Numerical range of e At. For fixed A C n n, we let E A (t) = e At, t C. Definition 1. The numerical range of E A (t) is W(E A ) = {t C : 0 F(e At )}. (2.1) Thus, we also have W(E A ) = {t C : x e At x = 0, for some nonzero x C n } = {t C : e At λi n 2 λ, λ C}. It follows immediately that for any square matrix A the origin cannot be in W(E A ) since e 0A = I n. We now determine when W(E A ) is nonempty. Theorem 1. Let A C n n. Then W(E A ) is nonempty if and only if A is not a scalar matrix.

4 Numerical Range for the Matrix Exponential Function 635 Proof. If A is a scalar matrix (A = ci n for some complex c), then W(E A ) is empty since F(e At ) = {e ct } and obviously e ct 0. Suppose that A is not a scalar matrix. We will show W(E A ) is nonempty by induction on the size n of A. Let U be unitary such that A = U RU and R is upper triangular. We have e At = U e Rt U, and thus, F(e At ) = F(e Rt ) for each t C. It follows that W(E A ) = W(E R ). Therefore, we may assume that A is already in upper triangular form. [ ] λ1 a Let A be a 2 2 complex matrix. Then by formula (10.40) in [5] 0 λ 2 e At = when λ 1 λ 2 ; if λ 1 = λ 2 = λ then e At = [ e λ1t a eλ 2 t e λ 1 t λ 2 λ 1 0 e λ2t [ e λt e λt at 0 e λt If A has two distinct eigenvalues, then since F(e At ) is convex for all t, it suffices to show that there is always a complex t for which the line segment connecting e λ1t and e λ2t contains the origin. In other words, we want a t C such that for some ρ (0,1) ρe λ1t +(1 ρ)e λ2t = 0, that is, e (λ1 λ2)t = 1 ρ ρ, or e(λ1 λ2)t (2k+1)πi = 1 ρ ρ, where k Z. Therefore, we can take t = ] ]. ln 1 ρ ρ +(2k +1)πi λ 1 λ 2, k Z. (2.2) Since ρ (0,1) can be arbitrary, we know from (2.2) that W(E A ) contains infinitely many parallel straight lines (these lines are horizontal when λ 1 λ 2 is real). If λ 1 = λ 2 = λ, then F(e At ) = {z C : z e λt at 2 eλt } and a 0 because otherwise A would be a scalar matrix. So, 0 F(e At ) if and only if t 2 a. Therefore, W(E A) = {t C : t 2 a }. Suppose that W(E A ) for every k k non-scalar upper triangular matrix A, where k 2. So, there is a t 0 C such that x 0 x eat0 0 = 0 for some nonzero x 0 C k. Now for any (k +1) (k +1) non-scalar upper triangular matrix A, there are three possibilities: A = [ Â y 0 ξ ] [ ξ y T, or A = 0 Â ],

5 636 C. Chorianopoulos and C.-H. Guo or A = λ λ... a, λ where y C k, ξ C, a 0, and  is a k k non-scalar upper triangular matrix. We will treat the first case. The second case can be treated similarly. The third case can be reduced to the first case by applying a permutation similarity that interchanges row k +1 with row 2 and column k +1 with column 2. By induction hypothesis, there is a t 0 C such that x 0 x eât0 0 = 0 for some nonzero x 0 C k. Then [ ] (At e At0 0 ) j = = eât0 z 0, j! 0 e ξt0 j=0 where z 0 C k. Now for w = [ x T 0 0 ] T, we have w e At0 w = [ x 0 0 ][ eât0 z 0 0 e ξt0 ] [ x0 0 ] = x x 0eÂt0 0 = 0. Thus, t 0 W(E A ). By an argument similar to the one at the end of the proof, we can show that W(EÂ) W(E A ) for any upper triangular matrix A and any leading principal submatrix Â. 3. Properties of W(E A ). In this section, we provide several basic properties of W(E A ). Proposition 1. Suppose A C n n is not a scalar matrix. Then W(E A ) is closed but it is never bounded. Moreover, W(E A ) always contains infinitely many parallel straight lines. Proof. To prove that it is closed suppose that there is a sequence {t n } W(E A ) with t n ˆt. Therefore, there is a sequence ofunit vectors{x n } such that x ne Atn x n = 0 for all n N. We need to prove that ˆt W(E A ). Since x n is bounded, there is a subsequence x nk converging to a unit vector ˆx. Therefore, 0 = lim n k x n k e Atn kx nk = ˆx e Aˆtˆx, so ˆt W(E A ). In the proof of Theorem 1, we have shown that W(E A ) contains infinitely many parallel straight lines for all 2 2 non-scalarmatrices A. The proof by induction there

6 Numerical Range for the Matrix Exponential Function 637 also reveals that, for n 2, W(E A ) always contains infinitely many parallel straight lines for all n n non-scalar matrices A. So, if A is not a scalar matrix a t W(E A ) can be as large in magnitude as we like. A question that naturally arises is: how small in magnitude can t be? Proposition 2. Suppose A C n n is not a scalar matrix and t W(E A ). Then ln2 t inf A ki. (3.1) n 2 k C Proof. Since t W(E A ), e At λi n 2 λ, λ C. For λ = 1 we have e At I n 2 1. Since 0 F(e At ), 0 e kt F(e At ) = F(e At ktin ), for all k C. Therefore It is known (Corollary in [6]) that e At ktin I n 2 1. (3.2) e X+Y e X 2 (e Y 2 1)e X 2, where X,Y are square matrices of the same size. For Y = At kti n and X = 0 the inequality becomes Combining inequalities (3.2) and (3.3) we have or Inequality (3.1) follows readily. e At ktin I n 2 e At ktin 2 1. (3.3) 1 e At ktin 2 1, ln2 t A ki n 2. The term inf A ki n 2 in (3.1) gives the distance from the matrix A to the set k C of all scalar matrices. The infimum is achieved for a particular scalar matrix since inf A ki n 2 = inf A ki n 2. In estimating this distance, we may reduce A k C k 2 A 2 to upper triangular form by the Schur triangularization. If A is Hermitian, then we readily find that the distance is 1 2 (λ n λ 1 ), where λ n and λ 1 are the largest and the smallest eigenvalues of A, respectively. The constant ln2 in the lower bound in (3.1) is unlikely to be sharp. But it will be seen later that the constant (valid for all n n non-scalar matrices) is at most π/2.

7 638 C. Chorianopoulos and C.-H. Guo ThenextresultisabouthowW(E A )willchangeifashiftorascalarmultiplication is applied to A. Proposition 3. Suppose A C n n is not a scalar matrix and let α C. Then W(E A+αI ) = W(E A ) and for α 0, W(E αa ) = 1 α W(E A). and Proof. W(E A+αI ) = {t C : x e t(a+αi) x = 0, for some nonzero x C n } = {t C : x e ta e tαi x = 0, for some nonzero x C n } = {t C : x e ta x = 0, for some nonzero x C n } = W(E A ), W(E αa ) = {t C : x e tαa x = 0, for some nonzero x C n } = { w α C : x e wa x = 0, for some nonzero x C n } = 1 α W(E A). We now give a condition under which W(E A ) contains some points on the real axis. Proposition 4. Let λ and µ be any two eigenvalues of A such that λ µ = a+bi with a,b R and b 0. Then W(E A ) {(2k+1)π/b : k Z}. Proof. From the proof by induction for Theorem 1, we only need to prove the result here when A is an upper triangular matrix with eigenvalues λ and µ. In this case, we know from (2.2) that W(E A ) contains all points of the form t = ln 1 ρ ρ +(2k +1)πi a+bi, ρ (0,1), k Z. We get t = (2k +1)π/b by taking ρ (0,1) with ln 1 ρ ρ = (2k +1)π a b. It is well known that the solution to the differential equation dx dt = Ax is given by x(t) = e At x(0). The above proposition says that for any matrix A with a nonzero imaginary part for the difference of any two eigenvalues, W(E A ) contains some t > 0. It follows that ˆx e Atˆx = 0 for some nonzero ˆx C n. Therefore, for x(0) = ˆx, we have x(0) x(t) = ˆx e Atˆx = 0, i.e., x(t) and x(0) are orthogonal.

8 Numerical Range for the Matrix Exponential Function 639 We now show that there are no isolated points in W(E A ). Proposition 5. Suppose A C n n is not a scalar matrix. Then W(E A ) does not have any isolated points. Proof. Let t 0 be apoint in W(E A ). We separatetwocases. FirstInt(F(e t0a )) = and then Int(F(e t0a )). Let t 0 W(E A ) and suppose that Int(F(e t0a )) =. Then F(e t0a ) is a line segment passingthrough the origin. By Theorem1.6.3of [6], the endpoints off(e t0a ) are eigenvalues of e t0a, say e t0λ1 and e t0λ2, where λ 1 and λ 2 are eigenvalues of A. So the origin cannot be an endpoint of the line segment, and there is an r (0,1) such that 0 = re t0λ1 +(1 r)e t0λ2, r 1 r = et0(λ2 λ1) < 0. (3.4) We will show that 0 F(e (t0+εeiθ )A ) for a suitable θ [0,2π] and all ε > 0. We just need to show the existence of s (0,1) such that se (t0+εeiθ )λ 1 +(1 s)e (t0+εeiθ )λ 2 = 0, i.e., s/(1 s) = e (t0+εeiθ )λ 2 /e (t0+εeiθ )λ 1 = e t0(λ2 λ1) e εeiθ (λ 2 λ 1). For this, we need to show that e t0(λ2 λ1) e εeiθ (λ 2 λ 1) is a negative real number. Since e t0(λ2 λ1) is a negative real number by (3.4), we just need to choose θ such that e εeiθ (λ 2 λ 1) is a positive real number, so we choose θ such that e iθ (λ 2 λ 1 ) = λ 2 λ 1. Now let t 0 W(E A ) and suppose that Int(F(e At0 )). Since t 0 W(E A ), we have 0 F(e At0 ), so 0 = x 0 x eat0 0 for some x 0 C n with x 0 2 = 1. Since Int(F(e At0 ) and F(e At0 ) is a convex set, we can take w 1,w 2 F(e At0 ) such that w 1 = re iθ1,w 2 = re iθ2, where r > 0 and the arguments θ 1 and θ 2 may be negative and satisfy 0 < θ 2 θ 1 < π 2. We also have w 1 = x 1 eat0 x 1, w 2 = x 2 eat0 x 2, for suitable x 1,x 2 C n with x 1 2 = x 2 2 = 1. For ǫ > 0, δ [0,2π), and j = 0,1,2, let We have w 0 (ǫ,δ) = x 0e At0 e Aǫeiδ x 0 = x 0e At0 w j (ǫ,δ) = x j ea(t0+ǫeiδ) x j. k=0 1 k! (Aǫeiδ ) k x 0 = k=0 1 k! ǫk e ikδ x 0e At0 A k x 0. If x 0 eat0 A k x 0 = 0 for all k 0, then we would have x 0 ea(t0+ǫeiδ) x 0 = 0 for all ǫ > 0 and all δ [0,2π), and in particular x 0e A0 x 0 = x 0x 0 = 0, which is impossible.

9 640 C. Chorianopoulos and C.-H. Guo We now let k 1 be the smallest integer such that x 0 eat0 A k x 0 0. Then w 0 (ǫ,δ) = 1 k! ǫk e ikδ x 0e At0 A k x 0 +o(ǫ k ). For ǫ sufficiently small, we have w 1 (ǫ,δ) r 2 and w 2(ǫ,δ) r 2, and argw 1 (ǫ,δ) θ (θ 2 θ 1 ), argw 2 (ǫ,δ) θ (θ 2 θ 1 ). Choose δ such that e ikδ x 0 eat0 A k x 0 = x 0 eat0 A k x 0 e i(θ1+θ2)/2. Then for ǫ > 0 sufficiently small, 0 is inside the triangle with vertices w j (ǫ,δ). So 0 F(e A(t0+ǫeiδ) ). Thus, t 0 +ǫe iδ W(E A ) for the chosen δ and all ǫ > 0 sufficiently small. So t 0 is not an isolated point. From the proof we know that for any t W(E A ), W(E A ) also contains a line segmentl t withtasoneofthe twoendpoints. This meansthat W(E A ) cannotcontain a circle disjoint from the rest of W(E A ). It also follows that W(E A ) = t W(E A) l t. Many of the line segments l t will merge into one line segment, but W(E A ) is still the union of infinitely many line segments by Proposition 1. In the next example, we show that a connected component of W(E A ) is not convex in general. [ ] 0 1 Example 1. Let A =. Then from the proof of Theorem 1 we already 0 0 know that W(E A ) = {t C : t 2}, which is connected but not convex. We now examine when a point t 0 is an interior point of W(E A ). Proposition 6. Suppose A C n n is not a scalar matrix. If the origin is an interior point of F(e At0 ), then t 0 is an interior point of W(E A ). Proof. We use D(c,r) to denote the open disk with centercand radiusr. Suppose 0 Int(F(e At0 )). Then there exists ǫ > 0 such that D(0,2ǫ) F(e At0 ). In particular, for j = 0,1,2, w j = ǫe 2jπ/3 = x j eat0 x j for some x j C n with x j 2 = 1. Note that 0 is inside the triangle with vertices w j = x j eat0 x j and that the functions f j (t) = x j eat x j are continuous at t 0. Then there exists δ > 0 such that for all t D(t 0,δ), 0 is inside the triangle with vertices w j = x j eat x j, so 0 F(e At ). Thus, D(t 0,δ) W(E A ). Corollary 1. Suppose A C n n is not a scalar matrix. If t 0 is a boundary point of W(E A ), then the origin is a boundary point of F(e At0 ).

10 Numerical Range for the Matrix Exponential Function 641 We end this section by one more observation about W(E A ). Proposition 7. Suppose A C n n is not a scalar matrix and t 0 W(E A ). Then (a) t 0 W(E A ), (b) t 0 W(E A ). Proof. For (a), it suffices to show that for every invertible matrix B, 0 F(B) implies that 0 F(B 1 ). But if 0 F(B) then 0 F(B ), therefore x 0 B x 0 = 0 for some unit x 0 C n. Then, so, 0 F(B 1 ) since Bx = x 0 B x 0 = x 0 B B 1 Bx 0 = (Bx 0 ) B 1 (Bx 0 ), For (b), we have for some unit x 0 C n 0 = x 0e t0a x 0 = x 0e t0a e t0a e t0a x 0 = (e t0a x 0 ) e t0a (e t0a x 0 ). Therefore, t 0 W(E A ) and the conclusion follows from (a). Part (a) means that W(E A ) is symmetric with respect to the origin. 4. The cases where A is Hermitian or skew-hermitian. We have seen earlier that W(E A ) contains infinitely many parallel straight lines for any non-scalar matrixa. When AisHermitian orskew-hermitian,we canshowthatw(e A ) consists of infinitely many parallel straight lines. In what follows, we only provide proofs for the case where A is Hermitian. If A is skew-hermitian, then ia is Hermitian and W(E A ) = iw(e ia ) by Proposition 3, so corresponding results for the skew-hermitan case will follow readily. Proposition 8. Suppose A C n n is not a scalar matrix. (a) If A is Hermitian then W(E A ) R = and t W(E A ) if and only if t+α W(E A ) for all α R. (b) If A is skew-hermitian then W(E A ) ir = and t W(E A ) if and only if t+iα W(E A ) for all α R. Proof. Let A be Hermitian. Then for r R the matrix exponential e ra is positive definite. Therefore, r / W(E A ). For any t W(E A ) there is a nonzero vector x t such that 0 = x te At x t = x te α 2 A e A(t+α) e α 2 A x t = w te A(t+α) w t, where w t = e α 2 A x t is nonzero. Thus, t+α W(E A ). Proposition 9. Suppose A C n n is not a scalar matrix.

11 642 C. Chorianopoulos and C.-H. Guo (a) If A is Hermitian then W(E A ) is symmetric with respect to the real axis. (b) If A is skew-hermitian then W(E A ) is symmetric with respect to the imaginary axis. Proof. The conclusion in (a) follows directly from Proposition 8 (a) and Proposition 7 (a). Propositions 8 and 9 reveal what W(E A ) looks like for Hermitian or skew- Hermitian A. In particular, if A is Hermitian, W(E A ) consists of disjoint complex stripes parallel to the real axis of the form {t C : γ Im(t) δ, γδ > 0}. Observe that γ and δ should have the same sign since W(E A ) has no intersection with real axis. Moreover,for each stripe in the upper plane, W(E A ) contains its symmetric stripe in the lower plane. However, it is possible to have γ = δ. In that case, the corresponding stripes are reduced to straight lines parallel to the real axis. Indeed we have the following result. Proposition 10. Suppose A C n n is Hermitian and has only two distinct eigenvalues λ and µ. Then W(E A ) = {t C : Im(t) = (2k +1)π/(λ µ), k Z}. Proof. Let U be unitary such that A = U DU and D is diagonal with diagonal entries equal to λ or µ. Then F(e At ) = F(e Dt ) is the line segment connecting e λt and e µt, by Theorem of [6]. Thus, t = a + bi W(E A ) if and only if 0 = se λt +(1 s)e µt = (se (λ µ)t +(1 s))e µt for some s (0,1), if and only if e (λ µ)t = e (λ µ)a e (λ µ)bi is a negative number, if and only if e (λ µ)bi = 1, if and only if (λ µ)b = (2k +1)π for some k Z. We also have examples showing that for an Hermitian matrix A having three distinct eigenvalues,w(e A )canhavestripeswith zerowidth andstripeswithnonzero width. In Example 2 of next section, some stripes with nonzero width are displayed. For Hermitian matrices, the lower bound in (3.1) can be improved. Proposition 11. Let A be an n n Hermitian and non-scalar matrix. Let λ 1 λ 2 λ n be the eigenvalues of A (so λ 1 < λ n since A is not scalar). Then for all t W(E A ), and the lower bound is sharp. t π λ n λ 1,

12 Numerical Range for the Matrix Exponential Function 643 Proof. By Proposition 8, min t W(EA) t is achieved at t a = ia W(E A ) for some a > 0, and ia is a boundary point of W(E A ). It follows from Proposition 6 that 0 is a boundary point of F(e iaa ). Since A is Hermitian, e iaa is normal. By Theorem of [6] and the fact that 0 is a boundary point of F(e iaa ), we know that 0 = se iaλp +(1 s)e iaλq fortwodistincteigenvaluesλ p andλ q ofa, andsomes (0,1). Thus, e ia(λp λq) is a negative real number, so a(λ p λ q ) = (2k+1)π for some k Z. Now, a = 2k+1 λ p λ q π π λ n λ 1. Therefore, t π λ n λ 1 for all t W(E A ). The lower bound is sharp since t 0 = πi λ n λ 1 W(E A ). In fact, we have e t0(λn λ1) = 1 and thus 1 2 et0λ et0λn = 0, so 0 F(e At0 ). Note that we have inf k C A ki n 2 = 1 2 (λ n λ 1 ) in Proposition11. So the constant ln2 in (3.1) has been improved to π/2 when A is Hermitian. 5. Some illustrative examples. In this section, we give three matrices and plot the numerical range of the corresponding matrix exponential function for each case. The plots are obtained using an inverse numerical range Matlab file based on the algorithm described in [3]. Example 2. Consider the Hermitian matrix A = , with eigenvalues λ 1 = 2,λ 2 = 3,λ 3 = 4. We plot W(E A ) within [ 4,4] [ 4,4] in Figure 5.1. We can see 6 stripes of W(E A ), symmetric about the real axis. The distance between the origin and W(E A ) is seen to be around 0.52, consistent with the exact distance of π/6 obtained in Proposition 11. Example 3. Consider the unitary matrix A = 2/2 0 2/2 2/2 0 2/ We plot W(E A ) within [ 7,7] [ 7,7] in Figure 5.2. The figure shows that W(E A ) containssomepoints on the realaxis. It alsosuggeststhat W(E A ) issymmetric about the origin and contains infinitely many parallel straight lines. All these are consistent with our theoretical results.

13 Electronic Journal of Linear Algebra ISSN C. Chorianopoulos and C.-H. Guo Fig Fig Example 4. Consider the randomly generated complex matrix i i i A = i i i i i i We plot W (EA ) within [ 2kAk2, 2kAk2 ] [ 2kAk2, 2kAk2 ] in Figure 5.3. The figure shows that W (EA ) contains some points on the real axis. It also suggests that W (EA )

14 Electronic Journal of Linear Algebra ISSN Numerical Range for the Matrix Exponential Function 645 is symmetric about the origin and contains infinitely many parallel straight lines. All these are consistent with our theoretical results Fig Acknowledgment. The authors thank the two referees for their helpful comments. REFERENCES [1] B. Beckermann and L. Reichel. Error estimates and evaluation of matrix functions via the Faber transform. SIAM Journal on Numerical Analysis, 47: , [2] M.-D. Choi and C.-K. Li. Numerical ranges of the powers of an operator. Journal of Mathematical Analysis and Applications, 365: , [3] C. Chorianopoulos, P. Psarrakos, and F. Uhlig. A method for the inverse numerical range problem. Electronic Journal of Linear Algebra, 20: , [4] C. Diogo. Algebraic properties of the set of operators with 0 in the closure of the numerical range. Operators and Matrices, 9:83 93, [5] N.J. Higham. Functions of Matrices. SIAM, Philadelphia, [6] R.A. Horn and C.R. Johnson. Topics in Matrix Analysis. Cambridge University Press, Cambridge, [7] P. Lancaster and P. Psarrakos. Normal and seminormal eigenvalues of matrix functions. Integral Equations Operator Theory, 41: , [8] C.-K. Li and L. Rodman. Numerical range of matrix polynomials. SIAM Journal on Matrix Analysis and Applications, 15: , [9] J.G. Stampfli and J.P. Williams. Growth conditions and the numerical range in a Banach algebra. The Tohoku Mathematical Journal, 20: , 1968.