Odd-even partitioning

Because of the symmetry of the 5x5 matrix, when we multiply the first component by the matrix we get a symmetric result, and when we multiply the second component by the matrix we get an antisymmetric result. We can thus look at each of these cases. 

The next simplification of the calculations is by exploiting the block structure of these matrices. We use the property that if a matrix is block lower triangular, its eigenvalues are those of the diagonal blocks. 

The eigenvalues of the upper, symmetric, case are given by those of 6 2] 

Observe that the matrices containing the eigenvectors have the same lower triangular block structure as the originals. 

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