The moment and Gram matrices, distinct eigenvalues and zeroes, and rational criteria for diagonalizability Original Research Article

The moment matrix of order m associated with the n-by-n complex matrix A is Km≡[trAi+j−2]i,j=1m. We show that d≡rank Kn is the number of distinct eigenvalues of A, d=max{m=1,…,n:Km isnonsingular}, and there is a unique (d+1)-vector a≡[ai−1]i=1d+1 such that Kd+1a=0 and ad=1. The entries of a are the coefficients of the unique monic polynomial of degree d whose zeroes are exactly the distinct eigenvalues of A. This polynomial, which can be computed rationally by Gaussian elimination, annihilates A if and only if A is diagonalizable. The minimal polynomial of A has distinct zeroes if and only if the moment matrix of its companion matrix is nonsingular. The Gram matrix of order m associated with A is Lm≡[trAi−1(A*)j−1]i,j=1m. We observe that μ≡rank Ln is the degree of the minimal polynomial of A, whose coefficients are the entries of the unique (μ+1)-vector b=[bi−1]i=1μ+1 such that Lμ+1b=0 and bμ=1. Properties of the moment and Gram matrices coalesce when A is normal.