Related papers: The formula ABA=Tr(AB)A for matrices
We characterize the unit stable range one 2x2 and 3x3 matrices over commutative rings. In particular, we characterize the 2x2 matrices which satisfy the Goodearl-Menal condition. For 2x2 integral matrices we show that the stable range one…
Within the framework of the theory of the column and row determinants, we obtain explicit representation formulas (analogs of Cramer's rule) for the minimum norm least squares solutions of quaternion matrix equations ${\bf A} {\bf X} = {\bf…
A real square matrix is algebraically positive if there exists a real polynomial $f$ such that $f(A)$ is a positive matrix. In this paper, we give a sufficient condition for a sign pattern matrix to allow algebraic positivity, and give some…
We show an explicit formula, with a quite easy deduction, for the exponential matrix $e^{tA}$ of a real square matrix $A$ of order $n\times n$. The elementary method developed requires neither Jordan canonical form, nor eigenvectors, nor…
Let $\chi(A)$ denote the characteristic polynomial of a matrix $A$ over a field; a standard result of linear algebra states that $\chi(A^{-1})$ is the reciprocal polynomial of $\chi(A)$. More formally, the condition $\chi^n(X)…
The book is devoted to investigation of arithmetic of the matrix rings over certain classes of commutative finitely generated principal ideals domains. We mainly concentrate on constructing of the matrix factorization theory. We reveal a…
In this paper we find the explicit formulas of two dimensional commuting ($2\times 2$)-matrix differential operators which were introduced by Nakayashiki. The common eigen functions and eigen values of these operators are parametrized by…
This is a revised version of the previous version with a new appendix consisting of characteristic two case. We define quasi-quadratic modules in a commutative ring generalizing the notion of quadratic modules. The main theorem is a…
We extend Latimer and MacDuffee's theorem to a general commutative domain and apply this result to study similarity of matrices over integral rings of number fields. We also conjecture similarity over discrete valuation rings can be descent…
We examine the possibility that a certain class of neutrino mass matrices, namely those with two independent vanishing minors in the flavor basis, regardless of being invertible or not, is sufficient to describe current data. We compute…
The combined matrix is a very useful concept for many applications. Almost strictly sign regular (ASSR) matrices form an important structured class of matrices with two possible zero patterns, which are either type-I staircase or type-II…
A banded matrix is a real square matrix where nonzero entries appear around the main diagonal. In this article, we consider linear complementarity properties of (variants) of banded matrices. Focusing on triangular matrices and the newly…
We exhibit a Cayley-Hamilton trace identity for $2\times2$ matrices with entries in a ring $R$ satisfying $[[x,y],[x,z]]=0$ and 1/2 \in R$.
Let $A$ be a $C^{*}$ algebra and $T: A\rightarrow A$ be a linear map which satisfies the functional equation $\begin{cases}T(x)T(y)=T^{2}(xy)\\T(x^{*})=T(x)^{*} \end{cases}$ We prove that under each of the following conditions, $T$ must be…
We find a class of algebras A satisfying the following property: for every nontrivial noncommutative polynomial, the linear span of all its values in A equals A. This class includes the algebras of all bounded and all compact operators on…
The noncommutative (or mixed) trace algebra $T_{nd}$ is generated by $d$ generic $n\times n$ matrices and by the algebra $C_{nd}$ generated by all traces of products of generic matrices, $n,d\geq 2$. It is known that over a field of…
This paper examines with elementary proofs some interesting properties of numbers in the binary quadratic form $a^2+ab+b^2$, where $a$ and $b$ are non-negative integers. Key findings of this paper are (i) a prime number $p$ can be…
Let $\mathcal{M}$ be a square matrix over a commutative ring and let $\mathcal{A}$ be a principal submatrix. We give relations between the determinants of $\mathcal{M}$ and $\mathcal{A}$ based on an annihilating polynomial for one of them.…
We prove an explicit formula for the number of $n \times n$ upper triangular matrices, over $GF(q)$, whose square is the zero matrix. This formula was recently conjectured by Sasha Kirillov and Anna Melnikov[KM].
A cycle is algebraically trivial if it can be exhibited as the difference of two fibers in a family of cycles parameterized by a smooth scheme. Over an algebraically closed field, it is a result of Weil that it suffices to consider families…