Related papers: On symplectic eigenvalues of positive definite mat…

We investigate the structure of $k$-positivity and Schmidt numbers for classes of linear maps and bipartite quantum states exhibiting symplectic group symmetries. Specifically, we consider (1) linear maps on $M_d(\mathbb{C})$ which are…

A $n\times n$ matrix $A$, which has a certain sign-symmetric structure ($J$--sign-symmetric), is studied in this paper. It is shown that such a matrix is similar to a nonnegative matrix. The existence of the second in modulus positive…

The probability that all eigenvalues of a product of $m$ independent $N \times N$ sub-blocks of a Haar distributed random real orthogonal matrix of size $(L_i+N) \times (L_i+N)$, $(i=1,\dots,m)$ are real is calculated as a multi-dimensional…

To better understand the algebra $\mathcal{M}_n$ of all $n\times n$ complex matrices, we explore the class of accretive matrices. This class has received renowned attention in recent years due to its role in complementing those results…

We study the symplectic semi-characteristic of a closed 4n-dimensional symplectic manifold. First, using the even-degree part of the primitive cohomology, we define the symplectic semi-characteristic. Second, using a vector field with…

A matrix convex set is a set of the form $\mathcal{S} = \cup_{n\geq 1}\mathcal{S}_n$ (where each $\mathcal{S}_n$ is a set of $d$-tuples of $n \times n$ matrices) that is invariant under UCP maps from $M_n$ to $M_k$ and under formation of…

We consider sample covariance matrices $S_N=\frac{1}{p}\Sigma_N^{1/2}X_NX_N^* \Sigma_N^{1/2}$ where $X_N$ is a $N \times p$ real or complex matrix with i.i.d. entries with finite $12^{\rm th}$ moment and $\Sigma_N$ is a $N \times N$…

Let $\mathbb{A}= \begin{pmatrix} A & 0 \\ 0 & A \\ \end{pmatrix} $ be a $2\times2$ diagonal operator matrix whose each diagonal entry is a bounded positive (semidefinite) linear operator $A$ acting on a complex Hilbert space $\mathcal{H}$.…

In this work some results on the structure-preserving diagonalization of selfadjoint and skewadjoint matrices in indefinite inner product spaces are presented. In particular, necessary and sufficient conditions on the symplectic…

We define a new average - termed the resolvent average - for positive semidefinite matrices. For positive definite matrices, the resolvent average enjoys self-duality and it interpolates between the harmonic and the arithmetic averages,…

Let $G$ be a compact connected Lie group of dimension $m$. Once a bi-invariant metric on $G$ is fixed, left-invariant metrics on $G$ are in correspondence with $m\times m$ positive definite symmetric matrices. We estimate the diameter and…

For any primitive matrix $M\in\mathbb{R}^{n\times n}$ with positive diagonal entries, we prove the existence and uniqueness of a positive vector $\mathbf{x}=(x_1,\dots,x_n)^t$ such that $M\mathbf{x}=(\frac{1}{x_1},\dots,\frac{1}{x_n})^t$.…

We introduce a new notion of the determinant, called symmetrized determinant, for a square matrix with the entries in an associative algebra $\AA$. The monomial expansion of the symmetrized determinant is obtained from the standard…

Sylvester's criterion characterizes positive definite (PD) and positive semidefinite (PSD) matrices without the need of eigendecomposition. It states that a symmetric matrix is PD if and only if all of its leading principal minors are…

We study the real algebraic variety of real symmetric matrices with eigenvalue multiplicities determined by a partition. We present formulas for the dimension and Euclidean distance degree. We give a parametrization by rational functions.…

The largest eigenvalue of a matrix is always larger or equal than its largest diagonal entry. We show that for a large class of random Laplacian matrices, this bound is essentially tight: the largest eigenvalue is, up to lower order terms,…

A "mysterious" relation between the number variance and the variance of the $L$-th ordered eigenvalue, first suggested by French et al. [Ann. Phys. 113, 277 (1978)], is revisited and proven to be asymptotically exact for the $\beta=2$ Dyson…

Symplectic self-adjointness of Hamiltonian operator matrices is studied, which arises in symplectic elasticity and optimal control. For the cases of diagonal domain and off-diagonal domain, necessary and sufficient conditions are shown. The…

The principal minors of a symmetric $n{\times}n$-matrix form a vector of length $2^n$. We characterize these vectors in terms of algebraic equations derived from the $ 2{\times}2{\times}2$-hyperdeterminant.

The class of stochastic matrices that have a stochastic $c$-th root for infinitely many natural numbers $c$ is introduced and studied. Such matrices are called arbitrarily finely divisible, and generalise the class of infinitely divisible…