Suppose that $A$ is an $n\times n$ nonnegative matrix whose eigenvalues are $\lambda =\rho \left(A\right),{\lambda }_2,...,{\lambda }_n$. Fiedler and others have shown that $det(\lambda I-A)\le {\lambda }^n-{\rho }^n$, for all $\lambda >\rho$, with equality for any such $\lambda$ if and only if $A$ is the simple cycle matrix. Let $a_i$ be the signed sum of the determinants of the principal submatrices of $A$ of order $i\times i$, $i=1,...,n-1$. We use similar techniques to Fiedler to show that Fiedler’s inequality can be strengthened to: $det(\lambda I-A)+{\sum }_{i=1}^{n-1}{\rho }^{n-2i}\left|a_i\right|{(\lambda -\rho )}^i\le {\lambda }^n-{\rho }^n$, for all $\lambda \ge \rho$. We use this inequality to derive the inequality that: ${\prod }_2^n(\rho -{\lambda }_i)\le {\rho }^{n-2}{\sum }_{i=2}^n(\rho -{\lambda }_i)$. In the spirit of a celebrated conjecture due to Boyle-Handelman,...

In this note we explain why the group of n×n upper triangular matrices is defined usually over commutative ring while the full general linear group is defined over any associative ring.

The paper presents an iterative algorithm for computing the maximum cycle mean (or eigenvalue) of $n\times n$ triangular Toeplitz matrix in max-plus algebra. The problem is solved by an iterative algorithm which is applied to special cycles. These cycles of triangular Toeplitz matrices are characterized by sub-partitions of $n-1$.

This paper is dealing with solvability of interval systems of linear equations in max-min algebra. Max-min algebra is the algebraic structure in which classical addition and multiplication are replaced by $\oplus$ and $\otimes$, where $a\oplus b=max\{a,b\},a\otimes b=min\{a,b\}$. The notation $𝔸\otimes x=𝕓$ represents an interval system of linear equations, where $𝔸=[\underline{A},\overline{A}]$ and $𝕓=[\underline{b},\overline{b}]$ are given interval matrix and interval vector, respectively. We can define several types of solvability of interval systems. In this paper, we define the T4 and T5 solvability and give necessary and...