On the diagonalizability of a matrix by a symplectic equivalence, similarity or congruence transformation. (English) Zbl 1332.15030
Summary: A symplectic matrix \(S\in\mathbb C^{2n\times 2n}\) satisfies \(S=J^{-1}S^T\) \(J\) for \(J=\left[\begin{smallmatrix} 0 & I_n \\ -I_n & 0\end{smallmatrix}\right]\in\mathbb R^{2n\times 2n}\). We will consider symplectic equivalence, similarity and congruence transformations and answer the question under which conditions a \(2n\times 2n\) matrix is diagonalizable under one of these transformations. In particular, we will give symplectic analogues of the singular value decomposition and the Takagi factorization.
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