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

✍ Scribed by Roger A. Horn; Alexey K. Lopatin

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 94 KB
• Volume: 299
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(99)00169-x

✦ Synopsis

The moment matrix of order m associated with the n-by-n complex matrix A is K m ≡ trA i+j -2 m i,j =1

. We show that d ≡ rank K n is the number of distinct eigenvalues of A, d = max {m = 1, . . . , n: K m is nonsingular}, and there is a unique (d + 1)-vector a ≡ a i-1 d+1 i=1 such that K d+1 a = 0 and a d = 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 asso-

. We observe that µ ≡ rank L n is the degree of the minimal polynomial of A, whose coefficients are the entries of the unique (µ + 1)vector b = b i-1 µ+1 i=1 such that L µ+1 b = 0 and b µ = 1. Properties of the moment and Gram matrices coalesce when A is normal.