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The \(J\)-Householder matrices. (English) Zbl 1236.15029

Summary: Let \(J=\left[\begin{smallmatrix} 0&I\\ -I&0\end{smallmatrix}\right]\in M_{2n}(\mathbb C)\). Let \(0\neq u\in\mathbb C^{2n}\) be given. A \(J\)-Householder matrix corresponding to \(u\) is \(H_{u}\equiv I-uu^TJ\). We show that every symplectic matrix is a product of \(J\)-Householder matrices. We present properties of \(J\)-Householder matrices, and we also present the possible Jordan canonical forms of products of two \(J\)-Householder matrices.

MSC:

15A21 Canonical forms, reductions, classification
15B10 Orthogonal matrices
15B57 Hermitian, skew-Hermitian, and related matrices
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