Matrix equations with diagonalizable coefficients

Abstract

Let A and A' be diagonalizable matrices. The paper deals with the equation

∑_j=0^m₁ ∑_k=0^m₂ c_jk A^jX(A')^k = M

where X should be found, c_jk are numbers and M is a given matrix. A solution of this equation is represented as a function of two non-commuting matrices. We establish a norm estimate for a function of A and A' defined on their spectra. That estimate gives us a norm estimate for X. In addition, we suggest a new bound for the similarity condition number of a diagonalizable matrix.