On the diagonal of the matrices in a similarity class

TL;DR

This paper characterizes when a matrix can be similar to one with prescribed diagonal entries, showing that the sum of these entries must equal the trace of the original matrix, except for scalar multiples of the identity.

Contribution

It provides a necessary and sufficient condition for the existence of a similar matrix with a given diagonal, extending understanding of matrix similarity classes.

Findings

The sum of the desired diagonal entries must equal the trace of the matrix.

The condition is both necessary and sufficient unless the matrix is a scalar multiple of the identity.

The result holds over arbitrary fields.

Abstract

Let A be an n by n matrix with entries in an arbitrary field, and c_1,...,c_n be scalars. We prove that if A is not a scalar multiple of the identity matrix, then the condition c_1+...+c_n=tr(A) is necessary and sufficient for A to be similar to a matrix with diagonal entries c_1,...,c_n.