Sums of diagonalizable matrices (Q1583714)

The problem of decomposition of an arbitrary \(n\times n\)-matrix into a sum of diagonalizable matrices is considered. It is shown that every square matrix \(A\) over an arbitrary field \(F\) is a sum of two diagonalizable matrices, except for special choice of the field \(F\) \((F=GF(2))\). In the last case the matrix \(A\) is, generally, a sum of three diagonalizable matrices or of two ones, under fixed conditions. The extent to which the ranks in a diagonalizable sum decomposition of a matrix \(A\) can be prescribed is also investigated. It is proved that the necessary conditions for a matrix \(A\) over an arbitrary field \(F\) to be a sum of two diagonalizable matrices with prescribed ranks are also sufficient when \(F\) is infinite.