Temjensangba | Searcharxiv

arXiv2026

Weak supermajorization between symplectic spectra of positive definite matrix and its pinching

Let $A = \begin{bmatrix} E & F \\ F^T & G \end{bmatrix}$ be a $2n \times 2n$ real positive definite matrix, where $E, F,$ and $G$ are $n \times n$ blocks. It is shown that $\ d(E \oplus G) \prec^w d(A)$. Here $d(A)$ denotes the $n$-vector consisting of the symplectic eigenvalues of $A$ arranged in the non-decreasing order. We also observe the following weak supermajorization relation, which is interesting on its own: $ \lambda \left( \left(\mathscr{C}(G)^{1/2} \mathscr{C}(E) \mathscr{C}(G)^{1/2}\right)^{1/2} \right) \prec^w \lambda \left( \left(G^{1/2} E G^{1/2} \right)^{1/2} \right)$. Here $\lambda \left( \left( G^{1/2}E G^{1/2} \right)^{1/2} \right)$ denotes the $n$-vector with entries given by the eigenvalues of $\left( G^{1/2}E G^{1/2} \right)^{1/2}$ in the non-decreasing order.

math.FA↗

arXiv2026

Majorization between symplectic spectra of positive semidefinite matrices

Given $2n \times 2n$ real symmetric positive semidefinite matrix $A$ with symplectic kernel, there exists a real $2n \times 2n$ \emph{symplectic matrix} $M$ such that $M^TAM= D \oplus D$, where $D$ is an $n \times n$ non-negative diagonal matrix which is unique up to permutation of its diagonal entries. The diagonal entries of $D$ are called the \emph{symplectic eigenvalues} or symplectic spectrum of $A$. In this work, we investigate some majorization and weak supermajorization relations between the symplectic spectra of two positive semidefinite matrices. More explicitly, suppose $A$ and $B$ are $2n \times 2n$ real symmetric positive semidefinite matrices with symplectic kernels. We show that if the symplectic spectrum of $A$ is majorized by the symplectic spectrum of $B$, then $A$ lies in the convex hull of the symplectic orbit of $B$. We also establish that only a weak converse of this statement holds; i.e., if $A$ lies in the convex hull of the symplectic orbit of $B$ then the symplectic spectrum of $A$ is \emph{weakly supermajorized} by the symplectic spectrum of $B$. Several consequences of our results are also presented. Our methods make use of well-known connections between the theory of majorization, doubly stochastic, doubly superstochastic, and symplectic matrices.

math.FA↗