On linear combinations of two idempotent matrices over an arbitrary field

Notes on linear combinations of two tripotent , idempotent , and involutive matrices that commute

The aim of this paper is to provide alternate proofs of all the results of our previous paper [2] in the particular case when the given two matrices A1 and A2 in the linear combination A = c1A1 + c2A2 commute.

On idempotent matrices over semirings

Idempotent matrices play a significant role while dealing with different questions in matrix theory and its applications. It is easy to see that over a field any idempotent matrix is similar to a diagonal matrix with 0 and 1 on the main diagonal. Over a semiring the situation is quite different. For example, the matrix J of all ones is idempotent over Boolean semiring. The first characterizatio...

Involutiveness of linear combinations of a quadratic‎ ‎or tripotent matrix and an arbitrary matrix

In this article, we characterize the involutiveness of the linear combination of the forma1A1 +a2A2 when a1, a2 are nonzero complex numbers, A1 is a quadratic or tripotent matrix,and A2 is arbitrary, under certain properties imposed on A1 and A2.

The Group Involutory Matrix of the Combinations of Two Idempotent Matrices

A Note on Decomposing a Square Matrix as Sum of Two Square Nilpotent Matrices over an Arbitrary Field

Let K be an arbitrary field and X a square matrix over K. Then X is sum of two square nilpotent matrices over K if and only if, for every algebraic extension L of K and arbitrary nonzero α ∈ L, there exist idempotent matrices P 1 and P 2 over L such that X = αP 1 - αP 2.