Eigen Values and Eigen Vectors Matlab

Question

I have a matrix A

A = [ 124.6,95.3,42.7 ; 95.3,55.33,2.74 ; 42.7,2.74,33.33 ]

The eigenvalues and vectors:

[V,D] =         


        

Answer 1

To compute all dot products between columns of V:

M = squeeze(sum(bsxfun(@times, conj(V), permute(V, [1 3 2]))));

The columns of V (eigenvectors) will be orthogonal if the above M is diagonal.

Answer 2

To show orthogonality

>> V'*V-eye(size(V))

ans =

   1.0e-15 *

    0.2220    0.1110    0.2498
    0.1110   -0.4441    0.1388
    0.2498    0.1388    0.4441

To show that the definition of the eigendecomposition is satisfied,

>> A*V - V*D

ans =

   1.0e-13 *

    0.4086    0.0400    0.8527
    0.3908    0.0355    0.5684
    0.1954    0.0355         0

The results won't be exactly zero, because digital computers don't do exact math, but you can see that they're pretty close.