### ${\mathbb{Z}}_{k+l}\times {\mathbb{Z}}_{2}$-graded polynomial identities for ${M}_{k,l}\left(E\right)\otimes E$

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An $n\times n$ sign pattern $\mathcal{A}$ is said to be potentially nilpotent if there exists a nilpotent real matrix $B$ with the same sign pattern as $\mathcal{A}$. Let ${\mathcal{D}}_{n,r}$ be an $n\times n$ sign pattern with $2\le r\le n$ such that the superdiagonal and the $(n,n)$ entries are positive, the $(i,1)$$(i=1...$

A sign pattern $A$ is a $\pm $ sign pattern if $A$ has no zero entries. $A$ allows orthogonality if there exists a real orthogonal matrix $B$ whose sign pattern equals $A$. Some sufficient conditions are given for a sign pattern matrix to allow orthogonality, and a complete characterization is given for $\pm $ sign patterns with $n-1\le {N}_{-}\left(A\right)\le n+1$ to allow orthogonality.

We analyze a boundary-value problem for linear partial differential algebraic equations, or PDAEs, by using the method of the separation of variables. The analysis is based on the Kronecker-Weierstrass form of the matrix pencil[A,-λ_n B]. A new theorem is proved and two illustrative examples are given.

Given a square matrix A, a Brauer’s theorem [Brauer A., Limits for the characteristic roots of a matrix. IV. Applications to stochastic matrices, Duke Math. J., 1952, 19(1), 75–91] shows how to modify one single eigenvalue of A via a rank-one perturbation without changing any of the remaining eigenvalues. Older and newer results can be considered in the framework of the above theorem. In this paper, we present its application to stabilization of control systems, including the case when the system...

Let $\mathbb{N}$ be the set of nonnegative integers and $\mathbb{Z}$ the ring of integers. Let $\mathcal{B}$ be the ring of $N\times N$ matrices over $\mathbb{Z}$ generated by the following two matrices: one obtained from the identity matrix by shifting the ones one position to the right and the other one position down. This ring plays an important role in the study of directly finite rings. Calculation of invertible and idempotent elements of $\mathcal{B}$ yields that the subrings generated by them coincide. This subring is the sum of the ideal $\mathcal{F}$ consisting of...