The moment and Gram matrices, distinct eigenvalues and zeroes, and rational criteria for diagonalizability
type: Item
instance of: scholarly article
title: The moment and Gram matrices, distinct eigenvalues and zeroes, and rational criteria for diagonalizability
zbMATH Open document ID: 0943.15016
DOI: 10.1016/S0024-3795(99)00169-X
author: Alexey K. Lopatin
author: Roger A. Horn
published in: Linear Algebra and its Applications
publication date: 2000-09-05T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 1399448
zbMATH Keywords: eigenvalues
zbMATH Keywords: diagonalizability
zbMATH Keywords: moment matrix
zbMATH Keywords: minimal polynomial
zbMATH Keywords: Gram matrix
reviewed by: Rabe von Randow
MaRDI profile type: Publication
Wikidata QID: Q127203037
description: scientific article; zbMATH DE number 1399448