The uniqueness of expression for generalized quadratic matrices

It is shown that the expression as A(2) = alpha A + beta P for generalized quadratic matrices is not unique by numerical examples. Then it is proven that the uniqueness of expression for generalized quadratic matrices is concerned not only with the properties of A but also with the rank of P . Furthermore, the sufficient and necessary conditions for the uniqueness of generalized quadratic matrices'expression are obtained. Finally, some related discussions about generalized quadratic matrices are also given.