Title

Numerical radius and zero pattern of matrices

Abstract

Let A be an n × n complex matrix and r be the maximum size of its principal submatrices with no off-diagonal zero entries. Suppose A has zero main diagonal and x is a unit n-vector. Then, letting {norm of matrix} A {norm of matrix} be the Frobenius norm of A, we show that| 〈 A x, x 〉 |2 ≤ (1 - 1 / 2 r - 1 / 2 n) {norm of matrix} A {norm of matrix}2 . This inequality is tight within an additive term O (r n-2). If the matrix A is Hermitian, then| 〈 A x, x 〉 |2 ≤ (1 - 1 / r) {norm of matrix} A {norm of matrix}2 . This inequality is sharp; moreover, it implies the Turán theorem for graphs. © 2007 Elsevier Inc. All rights reserved.

Publication Title

Journal of Mathematical Analysis and Applications

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