A combinatorial method for determining the spectrum of the linear combinations of finitely many diagonalizable matrices that mutually commute

Abstract

Let $X_i$, $i=1,2,...,m$, be diagonalizable matrices  that mutually commute. This paper provides a combinatorial method to handle the problem of when a linear combination matrix $X=\sum_{i=1}^{m}c_iX_i$ is a matrix such that $\sigma(X)\subseteq\{\lambda_1, \lambda_2,..., \lambda_{n}\}$, where $c_i$, $i=1,2,...,m$, are nonzero complex scalars and $\sigma(X)$ denotes the spectrum of the matrix $X$. If the spectra of the matrices $X$ and $X_i$, $i=1,2,...,m$, are chosen as subsets of some particular sets, then this problem is equivalent to the problem of characterizing all situations in which a linear combination of some commuting special types of matrices, e.g. the matrices such that $A^k=A$, $k=2,3,...$, is also a special type of matrix. The method developed in this note makes it possible to solve such characterization problems for the linear combinations of finitely many special types of matrices. Moreover, the method is illustrated by considering the problem, which is one of the open problems left in [Linear Algebra Appl. 437 (2012) 2091-2109], of characterizing all situations in which a linear combination $X=c_1X_1+c_2X_2+c_3X_3$ is a tripotent matrix when $X_1$ is an involutory matrix and both $X_2$ and $X_3$ are tripotent matrices that mutually commute. The results obtained cover those established in the reference above.