Sums and products of two quadratic matrices Original Research Article

Necessary and sufficient conditions are given for a complex matrix T to be the sum of an idempotent and a square-zero matrix. These conditions are expressed in terms of (1) the block structure of T under similarity, as well as (2) the elementary-divisor structure of T. This leads to a complete characterization of which complex matrices are a sum of two quadratic matrices. Also, using a similar scheme, we characterize when a matrix is the product of an involution and a unipotent matrix of index 2.