Inequalities for the spread of a matrix and partitioned matrices

Abstract

Let A be an n × n complex matrix. The spread of the matrix A, denotes by sp(A), is the largest distance between any two eigenvalues of A. It was introduced by L. Mirsky. In the present work we give new lower and upper bounds for the spread of A. Some of these bounds are explicit functions of the entries of A, others involve the rows and the columns of A. Bounds for the spread of the power of A are obtained too. Block matrices arise in many aspects of matrix theory, we present some bounds for the spread of the general 2 × 2 block matrix.

Subject Classification:

• 15A18
• 15A60
• 15B57

Keywords:

• Spread of matrix
• Trace of matrix
• Frobenius norm
• Hermitian matrix
• Normal matrix