### A counting theorem in the semigroup of circulant Boolean matrices

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We investigate the problem of counting the real or complex Hadamard matrices which are circulant, by using analytic methods. Our main observation is the fact that for $|{q}_{0}|=...=|{q}_{N-1}|=1$ the quantity $\Phi ={\sum}_{i+k=j+l}\frac{{q}_{i}{q}_{k}}{{q}_{j}{q}_{l}}$ satisfies $\Phi \ge {N}^{2}$, with equality if and only if $q=\left({q}_{i}\right)$ is the eigenvalue vector of a rescaled circulant complex Hadamard matrix. This suggests three analytic problems, namely: (1) the brute-force minimization of $\Phi $, (2) the study of the critical points of $\Phi $, and (3) the computation of the moments of $\Phi $. We explore here these questions,...

Drew, Johnson and Loewy conjectured that for n ≥ 4, the CP-rank of every n × n completely positive real matrix is at most [n2/4]. In this paper, we prove this conjecture for n × n completely positive matrices over Boolean algebras (finite or infinite). In addition,we formulate various CP-rank inequalities of completely positive matrices over special semirings using semiring homomorphisms.

Let $A$ be a square $(0,1)$-matrix. Then $A$ is a Hall matrix provided it has a nonzero permanent. The Hall exponent of $A$ is the smallest positive integer $k$, if such exists, such that ${A}^{k}$ is a Hall matrix. The Hall exponent has received considerable attention, and we both review and expand on some of its properties. Viewing $A$ as the adjacency matrix of a digraph, we prove several properties of the Hall exponents of line digraphs with some emphasis on line digraphs of tournament (matrices).

The set of all $m\times n$ Boolean matrices is denoted by ${\mathbb{M}}_{m,n}$. We call a matrix $A\in {\mathbb{M}}_{m,n}$ regular if there is a matrix $G\in {\mathbb{M}}_{n,m}$ such that $AGA=A$. In this paper, we study the problem of characterizing linear operators on ${\mathbb{M}}_{m,n}$ that strongly preserve regular matrices. Consequently, we obtain that if $min\{m,n\}\le 2$, then all operators on ${\mathbb{M}}_{m,n}$ strongly preserve regular matrices, and if $min\{m,n\}\ge 3$, then an operator $T$ on ${\mathbb{M}}_{m,n}$ strongly preserves regular matrices if and only if there are invertible matrices $U$ and $V$ such that $T\left(X\right)=UXV$ for all $X\in {\mathbb{M}}_{m,n}$, or $m=n$ and $T\left(X\right)=U{X}^{T}V$ for all $X\in {\mathbb{M}}_{n}$.

The Boolean rank of a nonzero $m\times n$ Boolean matrix $A$ is the minimum number $k$ such that there exist an $m\times k$ Boolean matrix $B$ and a $k\times n$ Boolean matrix $C$ such that $A=BC$. In the previous research L. B. Beasley and N. J. Pullman obtained that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks $1$ and $2$. In this paper we extend this characterizations of linear operators that preserve the Boolean ranks of Boolean matrices. That is, we obtain that a linear operator preserves Boolean rank...

Let $A$ be a Boolean $\{0,1\}$ matrix. The isolation number of $A$ is the maximum number of ones in $A$ such that no two are in any row or any column (that is they are independent), and no two are in a $2\times 2$ submatrix of all ones. The isolation number of $A$ is a lower bound on the Boolean rank of $A$. A linear operator on the set of $m\times n$ Boolean matrices is a mapping which is additive and maps the zero matrix, $O$, to itself. A mapping strongly preserves a set, $S$, if it maps the set $S$ into the set $S$ and the complement of...

Let ${\mathbb{B}}_{k}$ be the general Boolean algebra and $T$ a linear operator on ${M}_{m,n}\left({\mathbb{B}}_{k}\right)$. If for any $A$ in ${M}_{m,n}\left({\mathbb{B}}_{k}\right)$ (${M}_{n}\left({\mathbb{B}}_{k}\right)$, respectively), $A$ is regular (invertible, respectively) if and only if $T\left(A\right)$ is regular (invertible, respectively), then $T$ is said to strongly preserve regular (invertible, respectively) matrices. In this paper, we will give complete characterizations of the linear operators that strongly preserve regular (invertible, respectively) matrices over ${\mathbb{B}}_{k}$. Meanwhile, noting that a general Boolean algebra ${\mathbb{B}}_{k}$ is isomorphic...

Let ${\mathbb{Z}}_{+}$ be the semiring of all nonnegative integers and $A$ an $m\times n$ matrix over ${\mathbb{Z}}_{+}$. The rank of $A$ is the smallest $k$ such that $A$ can be factored as an $m\times k$ matrix times a $k\times n$ matrix. The isolation number of $A$ is the maximum number of nonzero entries in $A$ such that no two are in any row or any column, and no two are in a $2\times 2$ submatrix of all nonzero entries. We have that the isolation number of $A$ is a lower bound of the rank of $A$. For $A$ with isolation number $k$, we investigate the possible values of the rank of $A$...

This is a presentation of recent work on quantum permutation groups, complex Hadamard matrices, and the connections between them. A long list of problems is included. We include as well some conjectural statements about matrix models.

Je zcela běžné, že speciální třídy matic jsou pojmenovány podle matematika, který je buď poprvé představil nebo podstatně přispěl k jejich studiu. Článek je věnován třem třídám matic nesoucích ve svých názvech jména čtyř matematiků: Sylvesterovým–Hadamardovým maticím, Kravčukovým maticím a Sylvesterovým–Kacovým maticím. Přestože na první pohled nemají uvedené třídy příliš společného, jsou v textu ukázány jejich vzájemné souvislosti.

We characterize matrices whose powers coincide with their Hadamard powers.