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
