Abstract: Firstly, by using the relationship between generalized quadratic matrix A and idempotent matrix P, which was expressed as (A-dP)(A-eP)=0, we discussed the basic properties of generalized polynomial f_P(A) of A, and proved the rank invariance of operations for generalized polynomials. The results show that ranks of generalized polynomials are not only independent of the choice of combination coefficients, but also independent of the choice of polynomials in most cases. Secondly, as applications, the relevant results of idempotent matrix, involutory matrix, quadratic matrix and generalized quadratic matrix were summarized and generalized.

Key words: generalized quadratic matrix, generalized polynomial, invariance, commutator