Multiplication of matrices over lattices

Algebras of cubic matrices

We consider algebras of $m\times m\times m$-cubic matrices (with $m=1,2,\dots$). Since there are several kinds of multiplications of cubic matrices, one has to specify a multiplication first and then define an algebra of cubic matrices (ACM) with respect to this multiplication. We mainly use the associative multiplications introduced by Maksimov. Such a multiplication depends on an associative binary operation on the set of size $m$. We introduce a notion of equivalent operations and show that such operations generate isomorphic ACMs. It is shown that an ACM is not baric. An ACM is commutative…

On the Associativity of Infinite Matrix Multiplication

A natural definition of the product of infinite matrices mimics the usual formulation of multiplication of finite matrices with the caveat (in the absence of any sense of convergence) that the intersection of the support of each row of the first factor with the support of each column of the second factor must be finite. Multiplication is hence not completely defined, but restricted to a specific relation on infinite matrices. In order for the product of three infinite matrices $A$, $B$, and $C$ to behave in an associative manner, the middle factor, $B$, must link $A$ and $C$ in three ways: (i)…

Construction of flows of finite-dimensional algebras

Recently, we introduced the notion of flow (depending on time) of finite-dimensional algebras. A flow of algebras (FA) is a particular case of a continuous-time dynamical system whose states are finite-dimensional algebras with (cubic) matrices of structural constants satisfying an analogue of the Kolmogorov-Chapman equation (KCE). Since there are several kinds of multiplications between cubic matrices one has fix a multiplication first and then consider the KCE with respect to the fixed multiplication. The existence of a solution for the KCE provides the existence of an FA. In this paper our …

矩阵乘法由正交性和迹决定

Matrix multiplication is determined by orthogonality and trace

本文证明特征非2域上n阶矩阵的结合双线性乘法若保持相同正交关系与迹，则只能是普通矩阵乘法或其相反乘法。

Index of Hadamard multiplication by positive matrices II

Given a definite nonnegative matrix $A \in M_n (C)$, we study the minimal index of A: $I(A) = \max \{λ\ge 0 : A\circ B \ge λB$ for all $0\le B\}$, where $A\circ B$ denotes the Hadamard product $(A\circ B)_{ij} = A_{ij} B_{ij}$. For any unitary invariant norm N in $M_n(C)$, we consider the N-index of A: $I(N,A) = \min\{N(A\circ B) : B\ge 0$ and $N(B) = 1 \}$. If A has nonnegative entries, then $I(A) = I(\| \cdot \|_{sp}, A)$ if and only if there exists a vector u with nonnegative entries such that $Au = (1, >..., 1)^T$. We also show that $I(\| \cdot \|_{2}, A)= I(\| \cdot \|_{sp}, {\bar A}\circ…

Orthogonality for $(0,-1)$ tropical normal matrices

We study pairs of mutually orthogonal normal matrices with respect to tropical multiplication. Minimal orthogonal pairs are characterized. The diameter and girth of three graphs arising from the orthogonality equivalence relation are computed.