#### 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

#### Recommended Citation

Nikiforov, V.
(2008). Numerical radius and zero pattern of matrices.* Journal of Mathematical Analysis and Applications**, 337* (1), 739-743.
https://doi.org/10.1016/j.jmaa.2007.04.042