Rectangular random matrices. Related convolution

Infinitely divisible distributions for rectangular free convolution: classification and matricial interpretation

In a previous paper (called "Rectangular random matrices. Related covolution"), we defined, for $λ\in [0,1]$, the rectangular free convolution with ratio $λ$. Here, we investigate the related notion of infinite divisiblity, which happens to be closely related to the classical infinite divisibility: there exists a bijection between the set of classical symmetric infinitely divisible distributions and the set of distributions infinitely divisible with respect to this convolution, which preserves limit theorems. We give an interpretation of this correspondance in term of random matrices: we const…

Rectangular random matrices, related free entropy and free Fisher's information

We prove that independent rectangular random matrices, when embedded in a space of larger square matrices, are asymptotically free with amalgamation over a commutative finite dimensional subalgebra $D$ (under an hypothesis of unitary invariance). Then we consider elements of a finite von Neumann algebra containing $D$, which have kernel and range projection in $D$. We associate them a free entropy with the microstates approach, and a free Fisher's information with the conjugate variables approach. Both give rise to optimization problems whose solutions involve freeness with amalgamation over $…

Böttcher-Wenzel不等式对三个矩形矩阵的推广

A Generalization of the Böttcher-Wenzel inequality for three rectangular matrices

本文针对三个矩形矩阵定义广义换位子ABC-CBA，估计其Frobenius范数并得到推广Böttcher-Wenzel不等式的结果，且在一维情形给出更紧上界。

Nonsurjective maps between rectangular matrix spaces preserving disjointness, triple products, or norms

Let $M_{m,n}$ be the space of $m\times n$ real or complex rectangular matrices. Two matrices $A, B \in M_{m,n}$ are disjoint if $A^*B = 0_n$ and $AB^* = 0_m$. In this paper, a characterization is given for linear maps $Φ: M_{m,n} \rightarrow M_{r,s}$ sending disjoint matrix pairs to disjoint matrix pairs, i.e., $A, B \in M_{m,n}$ are disjoint ensures that $Φ(A), Φ(B) \in M_{r,s}$ are disjoint. More precisely, it is shown that $Φ$ preserves disjointness if and only if $Φ$ is of the form $$Φ(A) = U\begin{pmatrix} A \otimes Q_1 & 0 & 0 \cr 0 & A^t \otimes Q_2 & 0 \cr 0 & 0 & 0 \cr\end{pmatrix}V$$…

Free Deterministic Equivalents, Rectangular Random Matrix Models, and Operator-Valued Free Probability Theory

Motivated by the asymptotic collective behavior of random and deterministic matrices, we propose an approximation (called "free deterministic equivalent") to quite general random matrix models, by replacing the matrices with operators satisfying certain freeness relations. We comment on the relation between our free deterministic equivalent and deterministic equivalents considered in the engineering literature. We do not only consider the case of square matrices, but also show how rectangular matrices can be treated. Furthermore, we emphasize how operator-valued free probability techniques can…

有限自由概率的奇异值理论

A theory of singular values for finite free probability

本文提出适用于矩形矩阵的有限自由概率理论，将其转化为多项式奇异值操作，证明其能复制自由概率变换并渐近收敛，同时在新框架下显式化大数定律与中心极限定理。