Related papers: Sums and products of square-zero matrices
We present a simple proof of the factorization of (complex) symmetric matrices into a product of a square matrix and its transpose, and discuss its application in establishing a uniqueness property of certain antilinear operators.
Matrices are very popular and widely used in mathematics and other fields of science. Every mathematician has known the properties of finite-sized matrices since the time of study. In this paper, we consider the basic theory of infnite…
We consider two seemingly unrelated questions: the relationship between nonnegative polynomials and sums of squares on real varieties, and sparse semidefinite programming. This connection is natural when a real variety $X$ is defined by a…
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 study algebraic properties of matrices whose rows are mutual neighbours, and are also neigbours of 0 ("neighbour" in the sense of a certain nilpotency condition). The intended application is in synthetic differential geometry. For a…
The concepts of differentiation and integration for matrices were introduced for studying zeros and critical points of complex polynomials. Any matrix is differentiable, however not all matrices are integrable. The purpose of this paper is…
Several characterizations are given for a square matrix that can be written as the product of two positive (semidefinite) projections. Based on one of these characterizations, and the theory of alternating projections, a Matlab program is…
The aim of this note is to show that certain number theoretic inequalities due to Nesbitt and Shapiro have noncommutative counterparts involving positive definite matrices.
This text is a survey on symmetric matrices. It serves as a script for a module to be taught at university.
The article is devoted to the investigation of transformation groups of polynomials over Cayley-Dickson algebras and their manifolds of zeros. The problems about expressibility of zeros with the help of roots and decomposibility of…
In this work, we show how the eigenstructures of summands are related to that of the sum. In particular, we show that the sum of two positive semidefinite matrices can be written as the inner product of two block matrices $\mathbf{C} =…
It is known that singular values of idempotent matrices are either zero or larger or equal to one \cite{HouC63}. We state exactly how many singular values greater than one, equal to one, and equal to zero there are. Moreover, we derive a…
On the set of mappings of the given set, we define the product of mappings. If A is associative algebra, then we consider the set of matrices, whose elements are linear mappings of algebra A. In algebra of matrices of linear mappings we…
Sums-of-squares formulas over the integers have been studied extensively using their equivalence to consistently signed intercalate matrices. This representation, combined with combinatorial arguments, has been used to produce…
This work studies the distribution of the nonsymmetric matrix $\mathbf{E}^{-1}\mathbf{H}$. This random product is of fundamental interest under the general multivariate linear hypothesis setting. Specifically when $\mathbf{H}$ and…
Recently, Bapat and Kurata [\textit{Linear Algebra Appl.}, 562(2019), 135-153] defined the Cartesian product of two square matrices $A$ and $B$ as $A\oslash B=A\otimes \J+\J\otimes B$, where $\J$ is the all one matrix of appropriate order…
We study multiplicative nested sums, which are generalizations of harmonic sums, and provide a calculation through multiplication of index matrices. Special cases interpret the index matrices as stochastic transition matrices of random…
The product of a Hermitian matrix and a positive semidefinite matrix has only real eigenvalues. We present bounds for sums of eigenvalues of such a product.
The notion of a capped tensor product, introduced by G. Gr\"{a}tzer and the author, provides a convenient framework for the study of tensor products of lattices that makes it possible to extend many results from the finite case to the…
Square matrices of the form $\widetilde{\mathbf{A}} =\mathbf{A} + \mathbf{e}D \mathbf{f}^*$ are considered. An explicit expression for the inverse is given, provided $\widetilde{\mathbf{A}}$ and $D$ are invertible with…