Products of positive definite symplectic matrices
type: Item
instance of: scholarly article
title: Products of positive definite symplectic matrices
zbMATH Open document ID: 1472.15015
author: Daryl Q. Granario
author: Tin-Yau Tam
published in: Linear Algebra and its Applications
publication date: 2021-07-14T00:00:00Z
review text: Let us recall that the complex symplectic group \(\mathrm{Sp}(2n,\mathbb C)\) is defined as follows: \[\mathrm{Sp}(2n,\mathbb C)=\left\{A\in\mathrm{GL}(2n,\mathbb C): A^TJ_nA=J_n\right\}\text{ with }J_n=\begin{bmatrix}0 & I_n\\ -I_n & 0\\ \end{bmatrix}, \] and that a matrix is called positive definite if its spectrum is contained in \((0,\infty)\). \textit{C. S. Ballantine} [Pac. J. Math. 23, 427--433 (1968; Zbl 0211.35302); J. Algebra 10, 174--182 (1968; Zbl 0225.15012); Linear Algebra Appl. 3, 79--114 (1970; Zbl 0192.37002)] proved that every matrix from \(\mathrm{GL}(n,\mathbb C)\) with a positive determinant is the product of five positive definite matrices. In the present work, the authors study the same problem in \(\mathrm{Sp}(2n,\mathbb C)\). They show that every symplectic matrix is the product of five symplectic positive definite matrices. The proof uses canonical forms of symplectic matrices. It is also shown that there exist symplectic matrices that cannot be written as a product of a lower number of symplectic positive definite matrices. Additionally, the authors characterize products of two symplectic positive definite matrices. The paper contains a very comprehensive introduction into the topic and an up-to-date list of references.
reviewed by: Roksana Słowik
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A23
zbMATH DE Number: 7371456
zbMATH Keywords: canonical form
zbMATH Keywords: matrix decomposition
zbMATH Keywords: symplectic matrix
MaRDI profile type: Publication
full work available at URL: j.laa.2021.05.016
OpenAlex ID: W3168924283
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DOI: 10.1016/J.LAA.2021.05.016
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description: scientific article; zbMATH DE number 7371456