The moment and Gram matrices, distinct eigenvalues and zeroes, and rational criteria for diagonalizability
Let \(A\) be an \(n\times n\) complex matrix with \(d\) distinct eigenvalues. The moment matrix \(K_m\) of order \(m\) associated with \(A\) has entries \(\operatorname {tr} A^{i+j-2}\), where \(1\leq i,j\leq m\). It is shown that \(d= \operatorname {rank} (K_n)= \max \{m: K_m\) nonsingular\}, and that there is a unique \((d+1)\)-vector \(a\) such that the last entry of \(a\) is 1 and \(k_{d+1}a= 0\). The entries of \(a\) are the coefficients of the unique monic polynomial of degree \(d\) whose zeros are the \(d\) distinct eigenvalues of \(A\). This polynomial can be easily computed and annihilates \(A\) iff \(A\) is diagonalizable. The minimal polynomial of \(A\) has distinct zeros iff the moment matrix of its companion matrix is nonsingular. The Gram matrix \(L_m\) of order \(m\) associated with \(A\) has entries \(\operatorname {tr} A^{i-1} (A^*)^{j-1}\), where \(1\leq i,j\leq m\). It is shown that the minimal polynomial of \(A\) has degree \(\mu= \operatorname {rank} (L_n)\), and that its coefficients are the entries of the unique \((\mu+1)\)-vector \(b\) such that the last entry of \(b\) is 1 and \(L_{\mu+1} b=0\). Properties of \(K_m\) and \(L_m\) coalesce when \(A\) is normal.
scientific article; zbMATH DE number 1399448