Factorization of matrices into quadratic ones. II Original Research Article

This paper is mainly concerned with characterizations of complex matrices which are expressible as a product of finitely many specified quadratic matrices. The complex matrices are characterized under the condition that the specified quadratic matrices T with spectrum σ(T) = {α, β} satisfy (1) α = β = 1, (2) α = β = 1, (3) α = 1 and β = p, or (4) α = 1 and β = p, where p is a positive number different from 1. Moreover, the minimal number of required quadratic matrices is determined in terms of their determinants and sizes. On the other hand, products of two invertible quadratic matrices are also characterized, and various necessary or sufficient conditions are obtained for products of three invertible quadratic matrices.