Motivated both by the work of Anstee, Griggs, and Sali on forbidden submatrices and also by the extremal sat-function for graphs, we introduce sat-type problems for matrices. Let F be a family of k-row matrices. A matrix M is called F-admissible if M contains no submatrix F ∈ F (as a row and column permutation of F ). A matrix M without repeated columns is F-saturated if M is F-admissible but t...

Suppose $textbf{M}_{n}$ is the vector space of all $n$-by-$n$ real matrices, and let $mathbb{R}^{n}$ be the set of all $n$-by-$1$ real vectors. A matrix $Rin textbf{M}_{n}$ is said to be $textit{row substochastic}$ if it has nonnegative entries and each row sum is at most $1$. For $x$, $y in mathbb{R}^{n}$, it is said that $x$ is $textit{sut-majorized}$ by $y$ (denoted by $ xprec_{sut} y$) if t...

Matrices of dimensions m × 1 and 1 × n are called column and row vectors, respectively. We will typically denote column and row vectors by lower case Latin letters, e.g. a, b, x, y and other matrices by upper case Latin letters, e.g. A, B, X, Y . The scalars (or 1 × 1 matrices) will be frequently denoted by Greek letters α, β, λ, μ, etc. Unless stated otherwise, all scalars will be real numbers...

In this paper we use the Gray code representation of the genetic code C=00, U=10, G=11 and A=01 (C pairs with G, A pairs with U) to generate a sequence of genetic code-based matrices. In connection with these code-based matrices, we use the Hamming distance to generate a sequence of numerical matrices. We then further investigate the properties of the numerical matrices and show that they are d...

For $A,Bin M_{nm},$ we say that $A$ is left matrix majorized (resp. left matrix submajorized) by $B$ and write $Aprec_{ell}B$ (resp. $Aprec_{ell s}B$), if $A=RB$ for some $ntimes n$ row stochastic (resp. row substochastic) matrix $R.$ Moreover, we define the relation $sim_{ell s} $ on $M_{nm}$ as follows: $Asim_{ell s} B$ if $Aprec_{ell s} Bprec_{ell s} A.$ This paper characterizes all linear p...

This paper studies the set of n × n matrices for which all row and column sums equal zero. By representing these matrices in a lower dimensional space, it is shown that this set is closed under addition and multiplication, and furthermore is isomorphic to the set of arbitrary (n−1)×(n−1) matrices. The Moore-Penrose pseudoinverse corresponds with the true inverse, (when it exists), in this lower...

Recently, Jeĺınek derived that the number of self-dual interval orders of reduced size n is twice the number of row-Fishburn matrices of size n by using generating functions. In this paper, we present a bijective proof of this relation by establishing a bijection between two variations of upper-triangular matrices of nonnegative integers. Using the bijection, we provide a combinatorial proof of...

The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance $G$ keeping largest nonzero entries in each row and column leaving zeros remaining ones. $\mathcal{E}$-eigenvalues are eigenvalues $\mathcal{E}(G)$. modulus an eigenvalue $\mathcal{E}$-spectral radius $G$. $\mathcal{E}$-energy sum absolute values all In this article, we study some extrem...