Wavelets and Linear Algebra

Wavelets and Linear Algebra 4(1) (2017) 43-51 Wavelets and Linear Algebra Vali-e-Asr University of Rafsanan http://walavruacir Some results on the block numerical range Mostafa Zangiabadi a,, Hamid Reza Afshin b a Department of Mathematics, University of Hormozgan, Bandar Abbas, Islamic Republic of Iran b Department of Mathematics, Vali-e-Asr University, Rafsanan, Islamic Republic of Iran Article Info Article history: Received 30 October 2016 Accepted 02 March 2017 Available online 01 July 2017 Communicated by Abbas Salemi Keywords: Block numerical range, Nonnegative matrix, Perron-Frobenius theory Abstract The main results of this paper are generalizations of classical results from the numerical range to the block numerical range A different and simpler proof for the Perron-Frobenius theory on the block numerical range of an irreducible nonnegative matrix is given In addition, the Wielandt s lemma and the Ky Fan s theorem on the block numerical range are extended c (2017) Wavelets and Linear Algebra 2000 MSC: 47A12, 15A60 Corresponding author Email addresses: zangiabadi@hormozganacir (Mostafa Zangiabadi ), afshin@vruacir (Hamid Reza Afshin) http://doiorg/1022072/wala2017518091088 c (2017) Wavelets and Linear Algebra

1 Introduction Zangiabadi, Afshin/ Wavelets and Linear Algebra 4(1) (2017) 43-51 44 Let M n be the set of n n complex matrices and let A M n The numerical range of A is defined as follows (see, for instance, [2, 4]): W(A) = { x Ax : x C n, x x = 1 }, which is a useful concept in studying matrices and operators However, it does not respect the block structure of block operator matrices, thus the new concept of the quadratic numerical range was introduced in [6] The definition of the quadratic numerical range was generalized to the block numerical range in [11] as follows: Let D := (C k 1,, C k n ) be a decomposition of C m, ie, C m = C k 1 C k n With respect to this decomposition, the block matrix A has the following representation: A 11 A 1n A =, (11) A n1 A nn where A i is a k i k matrix Let us denote the spectrum of a matrix A by σ(a) For D define the block numerical range as follows: where W D (A) = {σ (Ax ) : x = (x 1,, x n ) D, x i x i = 1, i = 1,, n }, A 11 x 1, x 1 A 1n x n, x 1 A x =, x = (x 1,, x n ) D (12) A n1 x 1, x n A nn x n, x n It has been investigated in [1, 11, 10] The block numerical radius of A is defined as follows: ω D (A) = max{ λ : λ W D (A)} Apparently, for D = (C m ), ω D (A) yields the classical numerical radius of A, ie, ω (C m )(A) = ω(a) = max{ z : z W(A)} Here is a summary of the results obtained by Tretter et al in [11] Lemma 11 Let A be of the form (11) and D = (C k 1,, C k n ) Then (1) For n = 1, the block numerical range coincides with the numerical range, ie, W (C m )(A) = W(A) (2) For an n n matrix A, the block numerical range of A coincides with the spectrum of A, ie, W (C,,C) (A) = σ(a) (3) If A is a lower or upper block triangular matrix, then W D (A) = W(A 11 ) W(A nn )

Zangiabadi, Afshin/ Wavelets and Linear Algebra 4(1) (2017) 43-51 45 (4) σ(a) W D (A) W(A) (5) W D (A) is compact, but it is not necessarily convex (6) For all α, β C we have W D (αa + βi m ) = αw D (A) + β We call a matrix A M n irreducible if n = 1, or n 2 and there does not exist a permutation matrix P such that ( ) B C P T AP =, 0 D where B and D are nonempty square submatrices Given A, B M n, A is said to be diagonally similar to B if there exists a nonsingular diagonal matrix D such that A = D 1 BD; if, in addition, D can be chosen to be unitary, then we say A is unitarily diagonally similar to B By the classical Perron-Frobenius theory, if A is a (square, entrywise) nonnegative matrix, then its spectral radius ρ(a) is an eigenvalue of A and there is a corresponding nonnegative eigenvector If, in addition, A is irreducible, then ρ(a) is a simple eigenvalue and the corresponding eigenvector can be chosen to be positive Moreover, for an irreducible nonnegative matrix with index of imprimitivity h > 1 (ie, one having exactly h eigenvalues with modulus ρ(a)), Frobenius has also obtained a deeper structure theorem: The set of eigenvalues of A with modulus ρ(a) consists precisely of ρ(a) times all the h th roots of unity, the spectrum of A is invariant under a rotation about the origin of the complex plane through an angle of 2π/h and A is an h cyclic matrix, ie, there is a permutation matrix P such that P T AP is a matrix of the form 0 A 12 0 0 0 0 0 A 23 0 0, (13) 0 0 0 A h 1,h A h,1 0 0 0 where the zero blocks along the diagonal are all square This theory has been extended to the numerical range of a nonnegative matrix by Issos in his unpublished PhD thesis [5] and then completed in [7, 8] In [1], Forster et al extended the main theorem of Issos on the block numerical range of a nonnegative irreducible matrix A In Section 2, we redrive the main theorem of Issos on the block numerical range by simple method The Wielandt s lemma [9, Chapter II, Theorem 21] and also the Ky Fan s theorem [3, Theorem 8212] are the applications of the Perron-Frobenius theorem, which can be extended to the block numerical range This we do in Section 3 We first add a comment on the notation that is used We always use A = (a i ) n i, =1 to denote an n n complex matrix and use A = (A i ) n i, =1 to denote an m m (m > n) matrix with complex entries, which is partitioned in the form (11), ie, A i M ki k is the (i, ) block of A and k 1 +k 2 + +k n = m Let 0 be the symbol for real arrays with nonnegative entries, and the symbol for the induced entrywise order The symbol is reserved for the entrywise absolute value of an array For a vector x C n, we denote by x the Euclidean norm of x, ie, x = (x x) 1/2 For a matrix A M n, we denote by A the operator norm of A, ie, A = max x =1 Ax, where

Zangiabadi, Afshin/ Wavelets and Linear Algebra 4(1) (2017) 43-51 46 is the Euclidean norm Also we denote by λ max H(A) and λ min H(A) the largest eigenvalue and the smallest eigenvalue of H(A), respectively, where H(A) = (A + A )/2 denotes the Hermitian part of the square matrix A 2 Main result of Issos In this section, we show that for a nonnegative irreducible matrix A, the set of points of W D (A) with modulus ω D (A), consists precisely of ω D (A) times all the h th roots of unity The following lemma shows that if A is h cyclic, then the block numerical range of A is invariant under rotations about the origin of the complex plane through the angles 2πt/h for all t = 0, 1,, h 1 Lemma 21 Let A = (A i ) n i, =1 M m be permutationally similar to the matrix of the form (13) Then we have: (1) W D (A) = W D (e 2πti/h A), for all t = 0, 1,, h 1 (2) α W D (A) if and only if αe 2πti/h W D (A), for all t = 0, 1,, h 1 (3) If h = 2t, then W D (A) is symmetric with respect to the origin Proof (1) Let B = P T AP be an h cyclic matrix for a permutation matrix P By the proof of Theorem 6 in [5], D 1 BD = e iθ B, where D = I n1 e iθ I n2 e i(h 1)θ I nh is unitary diagonal matrix with θ = 2πt/h and n 1 + + n h = m, where n i (i = 1,, h) is the dimension of the i-th diagonal block of B This shows that e 2πti/h A = (PDP T ) 1 A(PDP T ) (21) As the block numerical range is invariant under diagonal unitary transformation [1, Equation (213)], so we have W D (A) = W D (e 2πti/h A), for all t = 0,, h 1 (2) It is clear from part (1) and Lemma 11 (3) Let h = 2t Then by part (2), α W D (A) if and only if αe 2πti/2t W D (A), ie, α W D (A) The main theorem of Issos in [5] for the block numerical range can now be stated as follows: Theorem 22 Let A = (A i ) n i, =1 M m be an irreducible nonnegative matrix with index of imprimitivity h Then {λ W D (A) : λ = ω D (A)} = {ω D (A)e 2πti/h : t = 0, 1,, h 1} Proof By the Perron-Frobenius theorem, there is a permutation matrix P such that P T AP is a matrix of the form (13) By [1, Proposition 31] ω D (A) W D (A) and then by (21) and Lemma 21 (2), we have ω D (A)e 2πti/h W D (A) for all t = 0,, h 1, ie, {ω D (A)e 2πti/h : t = 0, 1,, h 1} {λ W D (A) : λ = ω D (A)} Again by (21) and the Perron-Frobenius theorem, h is equal to the largest positive integer such that the matrix A is unitarily diagonally similar to the matrix e 2πti/h A for t = 0, 1,, h 1

Zangiabadi, Afshin/ Wavelets and Linear Algebra 4(1) (2017) 43-51 47 Therefore, the set { 0, 2π } 2π(h 1),,, h h is the cyclic group modulo 2π of the largest order, concluding that there does not exist υ = 2π p such that ω D (A)e iυ W D (A) Hence we establish the equality { λ WD (A) : λ = ω D (A) } = { ω D (A)e 2πti/h : t = 0, 1,, h 1 } < 2π h So, the proof is complete We illustrate Theorem 22 in the following example Example 23 Consider the nonnegative irreducible matrix A = 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 with index of imprimitivity h = 4 Here the spectrum of A is given by σ(a) = { 1, i, i, 1} If D = {C 2, C 2 }, then ω D (A) = 1 Figure 1 shows that the numbers 1, i, i, 1 belong to W D (A), with modulus ω D (A) = 1 Note that W(A) is the convex hull of the eigenvalues of A, Figure 1: The quadratic numerical range of the matrix A with respect to the decomposition C 4 = C 2 C 2

Zangiabadi, Afshin/ Wavelets and Linear Algebra 4(1) (2017) 43-51 48 3 The block numerical radius In this section, we generalize the Wielandt s lemma and the Ky Fan s theorem on the block numerical range For proving the main result of this section, we need the following lemma due to Wielandt [9, Chapter II, Theorem 21] Lemma 31 (Wielandt s lemma) Let A, C M n and assume that A is nonnegative If C A, then ρ(c) ρ(a) Suppose, in addition, that A is irreducible If ρ(c) = ρ(a) and ε is a unit complex number such that ερ(c) σ(c), then C = εdad 1 for some unitary diagonal matrix D Now, we can generalize the above lemma on the block numerical range Theorem 32 Assume that C = (C i ) n i, =1 M m is partitioned as in (11) Let A = (a i ) n i, =1 be such that C i a i, i, = 1,, n (31) Then ω D (C) ω(a) (32) Suppose, in addition, that A is irreducible If ω D (C) = ω(a) and ξ is a unit complex number such that ξω D (C) W D (C), then C i = a i, i, = 1,, n (33) and there exists a unitary matrix U such that U CU = ξ ( A 0 0 ) (34) Proof Let D = (C k 1,, C k n ) be a decomposition of C m In view of the Cauchy-Schwarz inequality, the entries of the matrix C x, defined by (12), satisfy the relations (C x ) i C i x x i C i, i, = 1, 2,, n, (35) for all x = (x 1,, x n ) D with x i x i = 1, i = 1, 2,, n Taking into account (31) and (35), we can apply Wielandt s lemma to C x and A, which yields the inequality ρ(c x ) ρ(a) (36) On the other hand, by definition of the block numerical range, there exists some y = (y 1,, y n ) D with y i y i = 1, i = 1, 2,, n such that ω D (C) = ρ(c y ) So by (36) and [4, Property 129], we have ω D (C) = ρ(c y ) ρ(a) ω(a), (37) which shows (32) Now assume that the matrix A is irreducible and that ω D (C) = ω(a) So the inequalities in (37) all become equalities, ie, ρ(c y ) = ρ(a) (38)

Zangiabadi, Afshin/ Wavelets and Linear Algebra 4(1) (2017) 43-51 49 In view of C y A and equality (38), by the second half of Wielandt s lemma, it follows that there is a unitary diagonal matrix D, say D = diag(d 1,, d n ), such that (C y ) i = y i C i y = ξd i a i d 1, i, = 1,, n (39) Using (39), (31) and (35), we derive (C y ) i = a i C i (C y ) i, for all i, = 1,, n and conclude that (33) holds true By setting z i = d i y i, i = 1,, n in (39), we have z i C i z = ξ a i, i, = 1,, n (310) In order to obtain (34), we form the matrix z 1 0 0 0 z 2 0 0 U 1 = 0 0 0 0 0 0 z n M m n Obviously, the columns of U 1 are orthonormal, ie, U 1 U 1 = I n Therefore, one can complete U 1 to a unitary matrix U = [U 1 U 2 ] M m Using relations (310), we derive the equality U 1 CU 1 = ξa By (33) and (310), we have C i = a i = Ci z, z i C i z z i C i, which implies that there exists α C such that C i z = α z i for all i, = 1,, n In view of the unitarity of the matrix U and the collinearity of the column vectors C i z and z i for all i, = 1,, n, we then obviously have U 2 CU 1 = 0 The remaining equality U 1 CU 2 = 0 is established in a similar way, based on the collinearity of the row vectors z i C i and z for all i, = 1,, n Remark 33 In Theorem 32, let D = C m, ie, the block numerical range coincides with the numerical range (see Lemma 11) We consider the following two cases: (1) Assume that C = (C i ) n i, =1 M m is partitioned as in (11) In this case Theorem 32 can be reformulated for the numerical range (2) Assume that C = (c i ) m i, =1 is an m m matrix In this case Theorem 32 can be reformulated in the following way, which is a version of the Wielandt s lemma on the numerical range [7, Lemma 38] Lemma 34 Let A, C M n and assume that A is nonnegative If C A, then ω(c) ω(a) Suppose, in addition, that A is irreducible If ω(c) = ω(a) and ε is a unit complex number such that εω(c) W(C), then C = εdad 1 for some unitary diagonal matrix D In the following result we present a version of the Ky Fan s theorem on the block numerical range

Zangiabadi, Afshin/ Wavelets and Linear Algebra 4(1) (2017) 43-51 50 Theorem 35 Assume that C = (C i ) n i, =1 M m is partitioned as in (11) Let A = (a i ) n i, =1 be such that C i a i, i, = 1,, n Then where W D (C) θ [0,2π] { z C : mθ cos θ Re z sin θ Im z M θ }, ( m θ = min λmin H ( ) ) ( e iθ C ω(a) + a, Mθ = max λmax H ( ) ) e iθ C + ω(a) a Proof Let D = (C k 1,, C k n ) be a decomposition of C m By (35), it is easy to see that for any θ [0, 2π] and for all x D with x i x i = 1, i = 1, 2,, n, we have H(e iθ C x ) H( C x ) H(A) Then by the Ky Fan s theorem σ ( H(e iθ C x ) ) n { z C : z Re(e iθ x C x ) } ρ(h(a)) a =1 Since ρ(h(a)) = ω(a) (see [7, Proposition 33]), it follows that Re W(e iθ C x ) [ ( ) ( ) ] min Re(e iθ x C x ) ω(a) + a, max Re(e iθ x C x ) + ω(a) a By [4, Properties 125 and 129] for all = 1,, n, we have Re(e iθ x C x ) W ( H ( e iθ C )) = [ λmin H ( e iθ C ), λmax H ( e iθ C )] which implies that ( ) ( min Re(e iθ x C x ) ω(a) + a min λmin H ( ) ) e iθ C ω(a) + a ( ) ( max Re(e iθ x C x ) + ω(a) a max λmax H ( ) ) e iθ C + ω(a) a Hence W(C x ) lies in the zone {z C : m θ Re(e iθ z) M θ } Since σ(c x ) W(C x ) and so, by definition of the block numerical range, the proof is completed by taking the intersection of these zones for all θ [0, 2π] References [1] KH Forster and N Hartanto, On the block numerical range of nonnegative matrices, Oper Theory: Adv Appl, 188 (2008), 113 133 [2] KE Gustafson and DKM Rao, Numerical range: The field of values of linear operators and matrices, Springer- Verlag, New York, 1997 [3] RA Horn and CR Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985 [4] RA Horn and CR Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991

Zangiabadi, Afshin/ Wavelets and Linear Algebra 4(1) (2017) 43-51 51 [5] JN Issos, The Field of Values of Non-Negative Irreducible Matrices, PhD Thesis, Auburn University, 1966 [6] H Langer and C Tretter, Spectral decomposition of some nonselfadoint block operator matrices, J Oper Theory, 39(2) (1998), 339 359 [7] CK Li, BS Tam and PY Wu, The numerical range of a nonnegative matrix, Linear Algebra Appl, 350 (2002), 1 23 [8] J Maroulas, PJ Psarrakos and MJ Tsatsomeros, Perron-Frobenius type results on the numerical range, Linear Algebra Appl, 348 (2002), 49 62 [9] H Mink, Nonnegative Matrices, Wiley, New York, 1988 [10] A Salemi, Total decomposition and the block numerical range, Banach J Math Anal, 5(1) (2011), 51 55 [11] C Tretter and M Wagenhofer, The block numerical range of an n n block operator matrix, SIAM J Matrix Anal Appl, 24(4) (2003), 1003 1017