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Nonorthogonal Eigenvectors

  • To: mathgroup at smc.vnet.net
  • Subject: [mg116374] Nonorthogonal Eigenvectors
  • From: "Kevin J. McCann" <kjm at KevinMcCann.com>
  • Date: Sat, 12 Feb 2011 05:19:39 -0500 (EST)

I have seen some threads from the past on this, but never got a 
satisfactory answer.

Suppose I have an exact matrix A:

A = {{1, 0, 0, 0, 2}, {0, 16, 0, 0, 0}, {0, 0, 9, 0, 0}, {0, 0, 0, 0,
   0}, {2, 0, 0, 0, 4}};

P = Eigenvectors[A]

produces the following

{{0,1,0,0,0},{0,0,1,0,0},{1,0,0,0,2},{-2,0,0,0,1},{0,0,0,1,0}}

which is not an unitary matrix, although the vectors are orthogonal, 
just not normal, i.e.

Transpose[P].P

is not the identity matrix.

However, if I make A numeric:

nA = A//N

then

nP = Eigenvectors[nA]

produces

{{0., 1., 0., 0., 0.}, {0., 0., 1., 0., 0.}, {0.447214, 0., 0., 0.,
   0.894427}, {-0.894427, 0., 0., 0., 0.447214}, {0., 0., 0., -1., 0.}}

and

Transpose[nP].nP

is the identity matrix.

I do not understand why making the matrix inexact produces the result 
that I would expect, but when the matrix is exact it doesn't. Also, I 
don't think the inconsistency is a useful thing.

Any ideas why someone decided to do it this way?

Kevin


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