MATLAB: Eigenvectors are not orthogonal for some skew-symmetric matrices, why

eigenvectorsskew-symmetric matrices

             0   -0.5000         0         0         0    0.5000
        0.5000         0   -0.5000         0         0         0
             0    0.5000         0   -0.5000         0         0
A =          0         0    0.5000         0   -0.5000         0
             0         0         0    0.5000         0   -0.5000
       -0.5000         0         0         0    0.5000         0
The above matrix is skew-symmetric. When I use [U E] = eig(A), to find the eigenvectors of the matrix. These eigenvectors must be orthogonal, i.e., U*U' matix must be Identity matrix. However, I am getting U*U' as
    0.9855   -0.0000    0.0410   -0.0000   -0.0265    0.0000
   -0.0000    0.9590    0.0000    0.0265   -0.0000    0.0145
    0.0410    0.0000    0.9735   -0.0000   -0.0145    0.0000
   -0.0000    0.0265   -0.0000    1.0145    0.0000   -0.0410
   -0.0265   -0.0000   -0.0145    0.0000    1.0410   -0.0000
    0.0000    0.0145    0.0000   -0.0410   -0.0000    1.0265
Here we can observe a substantial error. This happens for some other skew-symmetric matrices also. Why this large error is being observed and how do I get correct eigen-decomposition for all skew-symmetric matrices?

Best Answer

Your matrix A is "defective" , meaning that its eigenvalues are not all distinct. In fact, it has only three distinct eigenvalues. Consequently the space of eigenvectors does not fully span six-dimensional vector space. See the Wikipedia article:
http://en.wikipedia.org/wiki/Defective_matrix
What you are seeing is not an error on Matlab's part. It is a mathematical property of such matrices. You cannot achieve what you call "correct eigen-decomposition" for such matrices.
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