Related papers: $m$-nil-clean nonderogatory matrices

We show that a formal power series in $2N$ non-commuting indeterminates is a positive non-commutative kernel if and only if the kernel on $N$-tuples of matrices of any size obtained from this series by matrix substitution is positive. We…

Let $S$ be an additively idempotent semiring and $\mathbf{M}_n(S)$ be the semiring of all $n\times n$ matrices over $S$. We characterize the conditions of when the semiring $\mathbf{M}_n(S)$ is congruence-simple provided that the semiring…

Let $A_1,\ldots,A_s$ be unitary commutative rings which do not have non-trivial idempotents and let $A=A_1\oplus\cdots\oplus A_s$ be their direct sum. We describe all idempotents in the $2\times 2$ matrix ring $M_2(A[[X]])$ over the ring…

We prove that if an involution in a ring is the sum of an idempotent and a nilpotent then the idempotent in this decomposition must be 1. As a consequence, we completely characterize weakly nil-clean rings introduced recently in [Breaz,…

Let $F$ be an infinite field and let $f$ be a nonzero multilinear polynomial with coefficients in $F$. We prove that for every positive integer $d$ there exists a positive integer $s$ such that $f(M_{s}(F))$, the image of $f$ in $M_{s}(F)$,…

We study a nonlinear decomposition of a positive definite matrix into two components: the inverse of another positive definite matrix and a symmetric matrix constrained to lie in a prescribed linear subspace. Equivalently, the inverse…

The following theorem is proved: Suppose $M = (a_{i,j})$ be a $k \times k$ matrix with positive entries and $a_{i,j}a_{i+1,j+1} > 4\cos ^2 \frac{\pi}{k+1} a_{i,j+1}a_{i+1,j} \quad (1 \leq i \leq k-1, 1 \leq j \leq k-1).$ Then $\det M > 0 .$…

This is the second in a series of three papers dealing with sums of squares and hypoellipticity in the infinitely degenerate regime. We give sharp conditions on the entries of a positive semidefinite NxN matrix function F on n-dimensional…

We show that for every integer $m > 0$, there is an ordinary abelian variety over ${\mathbb F}_2$ that has exactly $m$ rational points.

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…

Let $\mathbb{F}$ be an infinite field with characteristic different from two. For a graph $G=(V,E)$ with $V={1,...,n}$, let $S(G;\mathbb{F})$ be the set of all symmetric $n\times n$ matrices $A=[a_{i,j}]$ over $\mathbb{F}$ with…

Let ${\rm M}(V)={\rm M}(n,\mathbb{F}_q)$ denote the algebra of $n\times n$ matrices over $\mathbb{F}_q$, and let ${\rm M}(V)_U$ denote the (maximal reducible) subalgebra that normalizes a given $r$-dimensional subspace $U$ of…

Let F be a perfect field. Then the diagonal quadratic form $a_iX_i^2$ over $F$ is universal over $M_2(F)$ if and only if atleast two of the $a_i$ are non-zero.

Let $F$ be a field of characteristic $\ne 2,3$ and let $A$ be a unital associative $F$-algebra. Define a left-normed commutator $[a_1, a_2, \dots , a_n]$ $(a_i \in A)$ recursively by $[a_1, a_2] = a_1 a_2 - a_2 a_1$, $[a_1, \dots , a_{n-1},…

Finsler's Lemma charactrizes all pairs of symmetric $n \times n$ real matrices $A$ and $B$ which satisfy the property that $v^T A v>0$ for every nonzero $v \in \mathbb{R}^n$ such that $v^T B v=0$. We extend this characterization to all…

Nonnegative matrix factorization (NMF) is the problem of decomposing a given nonnegative $n \times m$ matrix $M$ into a product of a nonnegative $n \times d$ matrix $W$ and a nonnegative $d \times m$ matrix $H$. A longstanding open…

Suppose $F$ is an infinite field and let $f \in F\{X_1, \dots,X_m\}$ be a noncommutative polynomial. Partially answering a query of Makar-Limanov, we show that there are numbers $d$ and $m'$ such that, if $F$ is closed under taking $d$th…

Following O'Meara's result [Journal of Algebra and Its Applications Vol~\textbf{13}, No. 8 (2014)], it follows that the block matrix $A=\begin{pmatrix} B & 0 0 & 0 \end{pmatrix} \in M_{n+r}(R)$, $B\in M_n(R)$, $r\ge 1$, over a von Neumann…

We prove that the algebraic set of pairs of matrices with a diagonal commutator over a field of positive prime characteristic, its irreducible components, and their intersection are $F$-pure when the size of matrices is equal to 3.…

Let $F: C^n \rightarrow C^m$ be a polynomial map with $degF=d \geq 2$. We prove that $F$ is invertible if $m = n$ and $\sum^{d-1}_{i=1} JF(\alpha_i)$ is invertible for all $i$, which is trivially the case for invertible quadratic maps. More…