Products of positive definite symplectic matrices
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.
scientific article; zbMATH DE number 7371456
scientific article