Variational principles for symplectic eigenvalues

Abstract

AbstractIf A is a real $2n \times 2n$ positive definite matrix, then there exists a symplectic matrix M such that $M^TAM=\text {diag}(D, D),$ where D is a positive diagonal matrix with diagonal entries $d_1(A)\leqslant \cdots \leqslant d_n(A).$ We prove a maxmin principle for $d_k(A)$ akin to the classical Courant–Fisher–Weyl principle for Hermitian eigenvalues and use it to derive an analogue of the Weyl inequality $d_{i+j-1}(A+B)\geqslant d_i(A)+d_j(B).$