Path: utzoo!utgpu!jarvis.csri.toronto.edu!cs.utexas.edu!mailrus!accuvax.nwu.edu!tank!eecae!cps3xx!cps3xx.egr.msu.edu!ben From: ben@nsf1.mth.msu.edu (Ben Lotto) Newsgroups: comp.theory Subject: Re: Matrix Properties Message-ID: Date: 29 Nov 89 17:14:20 GMT References: <1989Nov29.155406.23647@ee.rochester.edu> Sender: usenet@cps3xx.UUCP Reply-To: ben@nsf1.mth.msu.edu Followup-To: comp.theory Organization: Michigan State University Lines: 23 In-reply-to: bobm@ee.rochester.edu's message of 29 Nov 89 15:54:06 GMT >>>>> On 29 Nov 89 15:54:06 GMT, bobm@ee.rochester.edu (Bob Molyneaux) said: Bob> Is there a classification for the square matrix A for which A^k Bob> = A for not all but some value(s) of k? Let k be the smallest integer greater than 1 such that A^k = A. The minimal polynomial of A must then divide x^k - x. The roots of the minimal polynomial are either 0 or a (k-1)st root of unity. Since any root has multiplicity 1, A is diagonalizable. The characterization: A is such a matrix if and only if A is diagonalizable and every eigenvalue of A either equals 0 or an integral root of unity. The smallest k that works is 1 greater than the least common multiple of the order of the root of unity eigenvalues. A reference: Herstein's *Topics in Algebra*, chapter on linear transformations. In particular, the section on Jordan canonical form. Any good linear algebra book that covers Jordan canonical form should cover this material as well. -- -B. A. Lotto (ben@nsf1.mth.msu.edu) Department of Mathematics/Michigan State University/East Lansing, MI 48824 Brought to you by Super Global Mega Corp .com