MathGroup Archive 2001

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Eigenvalues of a Special Matrix

  • To: mathgroup at smc.vnet.net
  • Subject: [mg28006] Eigenvalues of a Special Matrix
  • From: Joel Storch <jstorch at earthlink.net>
  • Date: Wed, 28 Mar 2001 02:40:34 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Consider the symmetric, Toeplitz matrix A=[a_ij]=cos[(i-j) w]
{i=1,2,...,n ;j=1,2, ....,n}

This matrix has rank 2 and hence at most 2 nonzero eigenvalues.
Generating the eigenvalues for explicit values of n, it is conjectured
that the general expression for the two non-zero eigenvalues are:

lam1=n/2 +1/2 csc w  sin(n w)                    lam2=n/2 ?1/2 csc w
sin(n w)

Can someone produce a formal proof ?


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