Related papers: On $k$-idempotent 0-1 matrices

Given $n$ integer, let $X$ be either the set of hermitian or real $n\times n$ matrices of rank at least $n-1$. If $n$ is even, we give a sharp estimate on the maximal dimension of a real vector subspace of $X\cup\{0\}$. The rusults are…

A 0-1 matrix $M$ contains a 0-1 matrix pattern $P$ if we can obtain $P$ from $M$ by deleting rows and/or columns and turning arbitrary 1-entries into 0s. The saturation function $\mathrm{sat}(P,n)$ for a 0-1 matrix pattern $P$ indicates the…

We investigate the number of symmetric matrices of non-negative integers with zero diagonal such that each row sum is the same. Equivalently, these are zero diagonal symmetric contingency tables with uniform margins, or loop-free regular…

We determine sufficient conditions for certain classes of $(n+k) \times n$ matrices $E$ to have all order-$n$ minors to be nonzero. For a special class of $(n+1) \times n$ matrices $E,$ we give the formula for the order-$n$ minors. As an…

A real symmetric n times n matrix is called copositive if the corresponding quadratic form is non-negative on the closed first orthant. If the matrix fails to be copositive there exists some non-negative certificate for which the quadratic…

The practically important classes of equal-input and of monotone Markov matrices are revisited, with special focus on embeddability, infinite divisibility, and mutual relations. Several uniqueness results for the classic Markov embedding…

We introduce a sorting machine consisting of $k+1$ stacks in series: the first $k$ stacks can only contain elements in decreasing order from top to bottom, while the last one has the opposite restriction. This device generalizes \cite{SM},…

We study the problem of when a periodic square matrix of order $n$ over an arbitrary field $\mathbb{F}$ is decomposable into the sum of a square-zero matrix and a torsion matrix, and show that this decomposition can always be obtained for…

Two matrices are said non-overlapping if one of them can not be put on the other one in a way such that the corresponding entries coincide. We provide a set of non-overlapping binary matrices and a formula to enumerate it which involves the…

Consider the set of scalars $\alpha$ for which the $\alpha$th Hadamard power of any $n\times n$ positive semi-definite (p.s.d.) matrix with non-negative entries is p.s.d. It is known that this set is of the form $\{0, 1, \dots, n-3\}\cup…

This note presents explicit matrix expressions of a class of recently-discovered non-diagonal K-matrices for the trigonometric $A^{(1)}_{n-1}$ vertex model. From these explicit expressions, it is easily seen that in addition to a {\it…

A Lie algebra $L$ is said to be of breadth $k$ if the maximal dimension of the images of left multiplication by elements of the algebra is $k$. In this paper we give characterization of finite dimensional nilpotent Lie algebras of breadth…

For a fixed $n\ge2$, consider an $n\times n$ matrix $M$ whose entries are random integers bounded by $k$ in absolute value. In this paper, we examine the probability that $M$ is singular (hence has eigenvalue 0), and the probability that…

An n\times n matrix M is called a fooling-set matrix of size n, if its diagonal entries are nonzero, whereas for every k\ne \ell we have M_{k,\ell} M_{\ell,k} = 0. Dietzfelbinger, Hromkovi\v{c}, and Schnitger (1996) showed that n \le (\rk…

We study {\em sign-restricted matrices} (SRMs), a class of rectangular $(0, \pm 1)$-matrices generalizing the alternating sign matrices (ASMs). In an SRM each partial column sum, starting from row 1, equals 0 or 1, and each partial row sum,…

Let K be a (commutative) field with characteristic not 2, and V be a linear subspace of n by n matrices that have at most two eigenvalues in K (respectively, at most one non-zero eigenvalue in K). We prove that the dimension of V is less…

A $0$-$1$ matrix $M$ is saturating for a $0$-$1$ matrix $P$ if $M$ does not contain a submatrix that can be turned into $P$ by changing some $1$ entries to $0$ entries, and changing an arbitrary $0$ to $1$ in $M$ introduces such a submatrix…

A 0-1 matrix $M$ contains another 0-1 matrix $P$ if some submatrix of $M$ can be turned into $P$ by changing any number of $1$-entries to $0$-entries. $M$ is $\mathcal{P}$-saturated where $\mathcal{P}$ is a family of 0-1 matrices if $M$…

In this paper, by using matix representation for quaternions andoctonions, we provide a procedure to obtain some example of k potent matrices of order 4 or 8, over the real field or over the field ZP, with p a prime number.

For a fixed integer $k$, we consider the set of noncrossing partitions, where both the block sizes and the difference between adjacent elements in a block is $1\bmod k$. We show that these $k$-indivisible noncrossing partitions can be…