On linear combinations of two idempotent matrices over an arbitrary field

TL;DR

This paper characterizes when a matrix over any field can be expressed as a linear combination of two idempotent matrices with specified coefficients, extending previous results to arbitrary fields and different coefficient cases.

Contribution

It provides necessary and sufficient conditions for such linear combinations over arbitrary fields, including characteristic 2, and covers new cases where coefficients differ.

Findings

Characterization of matrices as linear combinations of two idempotents over any field.

Extension of previous results to fields of arbitrary characteristic.

Includes cases where coefficients are different and their negatives.

Abstract

Given an arbitrary field K and non-zero scalars a and b, we give necessary and sufficient conditions for a matrix A in M_n(K) to be a linear combination of two idempotents with coefficients a and b. This extends results previously obtained by Hartwig and Putcha in two ways: the field K considered here is arbitrary (possibly of characteristic 2), and the case a is different from b and -b is taken into account.