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From: pumphrey@ttidcb.UUCP (Larry Pumphrey)
Newsgroups: net.math
Subject: Re: I need an A in linear algebra!  Help!
Message-ID: <714@ttidcb.UUCP>
Date: Wed, 12-Mar-86 11:44:25 EST
Article-I.D.: ttidcb.714
Posted: Wed Mar 12 11:44:25 1986
Date-Received: Sat, 15-Mar-86 20:13:08 EST
References: <967@nmtvax.UUCP>
Organization: TTI, Santa Monica, CA.
Lines: 36


   I think you need to supply more information as there appear to be
   counter-examples as the problem is stated.  The following points
   should be clarified.

	1. Over what field do the elements a   of the matrices M
					    ij                  i
	   range (and hence k in the companion matrix A)?  i.e., the
	   rationals, reals, complexes?

	2. Any restrictions on the value of k?

	3. Can any of the M  be the identity matrix?
			   i
   Counter-example:
   ----------------
   If there are no restrictions on the value of k in the A matrix
   then let k=1 and A becomes the identity matrix.  Consider the 8
   3X3 matrices, M  as follows:
		  i
       [1 0 0]      [1 0  0]      [1  0 0]      [1  0  0]
   M = [0 1 0]  M = [0 1  0]  M = [0 -1 0]  M = [0 -1  0]
    1  [0 0 1]   2  [0 0 -1]   3  [0  0 1]   4  [0  0 -1]

       [-1 0 0]      [-1 0  0]      [-1  0 0]      [-1  0  0]
   M = [ 0 1 0]  M = [ 0 1  0]  M = [ 0 -1 0]  M = [ 0 -1  0]
    5  [ 0 0 1]   6  [ 0 0 -1]   7  [ 0  0 1]   8  [ 0  0 -1]


   Note that all 8 matrices are distinct, each M  squared is the
						i
   identity and their product is A = I.  Also note that M = I.
							 1
   It's an interesting problem, but something appears to be missing.

					       - Larry