Block diagonalization of a matrix

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To block diagonalize the given 4x4 matrix, it is essential to identify the eigenvalues and corresponding eigenvectors accurately. The user is struggling to find the correct method, questioning whether a commuting matrix is necessary for the process. They have attempted to use resources like the Jordan normal form but are encountering issues with the matrix's square yielding no new equations for eigenvectors. The discussion highlights the importance of understanding the relationship between eigenvalues, eigenvectors, and block diagonalization techniques. Further exploration of reliable mathematical resources is recommended for clarity on the topic.
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Hi. i have a 4x4 matrix
\begin{pmatrix}
0 & 1 & 1 & 1\\
1 & 0 & i & -i\\
1 & -i & 0 & i\\
1 & i & -i & 0\\
\end{pmatrix}
it has 2 eigenvalues
and i want to block diagonalize it into a 2x2 block diagonal matrix.
i can't seem to find the proper way to do that. do i need to have a commuting matrix in order to preform block diagonalization?
iv'e tried to follow this
http://en.wikipedia.org/wiki/Jordan_normal_form
but the square of the matrix gives me the same matrix with a constant factor, so i don't get any new equation for the eigenvectors.
 

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