Some inequalities for sector matrices

Abstract

Abstract. Two new inequalities are proved for sector matrices. The first one complements a recent result in [Oper. Matrices, 8 (2014) 1143-1148; the second one is an analogue of the AM-GM inequality, where the geometric mean for two sector matrices was introduced in [Linear Multilinear Algebra 63 (2015) 296-301]. As an application of the second inequality, we present similar inequalities for singular values or norms.Mathematics subject classification (2010): 15A45, 15A42, 47A30.

On the phases of a complex matrix

“…If 0 / ∈ W (C), then W (C) is completely contained in an open half plane by its convexity. In this case, C is said to be a sectorial matrix [7], also called sector matrix in [8]. The sectorial matrices have been widely studied in the literature [9,10,11].…”

“…Clearly, sectorial matrices constitute a special type of unitoid matrices. A nonsectorial unitoid matrix admits a factorization of the form (8). However, in this case, the eigenvalues of D lie on an arc of the unit circle of length no less than π.…”

On the logarithmic mean of accretive matrices

“…In [9], if W(A) ⊂ S θ , then A is called a sector matrix. Clearly, if W(A) ⊂ S θ , then A is positive definite.…”

“…Clearly, if W(A) ⊂ S θ , then A is positive definite. Some recent studies of sector matrices can be found in [2,9,18,20].…”

Singular value inequalities for sector matrices

“…then A is nonsingular. A matrix A ∈ M n is said to be sector matrix if its numerical range is contained in S α , for some α ∈ [0, π 2 ) (see [2]). Recently, many interesting articles have been devoted to study the singular value inequalities and unitarily invariant norm inequalities for sector matrices, see [3][4][5][6][7] and references therein.…”