If any, please provide explanation (examples are best welcomed) and/or references.
Gro Hovhannisyan
Hello Shu-Chang,
Every diagonalizable matrix A can be decomposed to A=Q L Q^{-1} where L is
diagonal eigenvalue matrix, Q is eigenvector matrix.
Otherwise:
Every 2n × 2n real skew-symmetric matrix can be written in the form A = Q Σ Q^T where Q is orthogonal. See for details:
Youla, D. C. (1961). "A normal form for a matrix under the unitary congruence group". Canad. J. Math. 13: 694–704.
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Shuchang Zhang
Hi Gro,
Thank you for your answer. 2n × 2n real skew-symmetric matrix could be decomposed to something like symplectic matrix, what would it be if order of matrix is 2n+1?
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Gro Hovhannisyan
In the odd-dimensional case the same decomposition is possible, but Σ has at least one row and column of zeros.
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