Related papers: A matrix equation $X^n = aI$
We study processes with unstable particles in intermediate time-like states. It is shown that the amplitudes squared of such processes factor exactly in the framework of the model of unstable particles with continuous masses. Decay widths…
Working over an algebraically closed field $k$ of any characteristic, we determine the matrix factorizations for the --- suitably graded --- triangle singularities $f=x^a+y^b+z^c$ of domestic type, that is, we assume that $(a,b,c)$ are…
A cross matrix $X$ can have nonzero elements located only on the main diagonal and the anti-diagonal, so that the sparsity pattern has the shape of a cross. It is shown that $X$ can be factorized into products of matrices that are at most…
For given real or complex $m \times n$ data matrices $X$, $Y$, we investigate when there is a matrix $A$ such that $AX = Y$, and $A$ is invertible, Hermitian, positive (semi)definite, unitary, an orthogonal projection, a reflection, complex…
The matrix LU factorization algorithm is a fundamental algorithm in linear algebra. We propose a generalization of the LU and LEU algorithms to accommodate the case of a commutative domain and its field of quotients. This algorithm…
A recently-established necessary condition for polynomials that preserve the class of entrywise nonnegative matrices of a fixed order is shown to be necessary and sufficient for the class of nonnegative monomial matrices. Along the way, we…
We show several ways to round a real matrix to an integer one such that the rounding errors in all rows and columns as well as the whole matrix are less than one. This is a classical problem with applications in many fields, in particular,…
We establish correspondances between factorisations of finite abelian groups (direct factors, unitary factors, non isomorphic subgroup classes) and factorisations of integer matrices. We then study counting functions associated to these…
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…
A novel factorization for the sum of two single-pair matrices is established as product of lower-triangular, tridiagonal, and upper-triangular matrices, leading to semi-closed-form formulas for tridiagonal matrix inversion. Subsequent…
This paper deals with necessary and sufficient condition for consistency of the matrix equation $AXB = C$. We will be concerned with the minimal number of free parameters in Penrose's formula $X = A^(1)CB^(1) + Y - A^(1)AYBB^(1)$ for…
Given a matrix $M = (a_{i,j})$ a square is a $2 \times 2$ submatrix with entries $a_{i,j}$, $a_{i, j+s}$, $a_{i+s, j}$, $a_{i+s, j +s}$ for some $s \geq 1$, and a zero-sum square is a square where the entries sum to $0$. Recently,…
In light of recent data science trends, new interest has fallen in alternative matrix factorizations. By this, we mean various ways of factorizing particular data matrices so that the factors have special properties and reveal insights into…
Let $B$ and $C$ be square complex matrices. The differential equation \begin{equation*} x''(t)+Bx'(t)+Cx(t)=f(t) \end{equation*} is considered. A solvent is a matrix solution $X$ of the equation $X^2+BX+C=\mathbf0$. A pair of solvents $X$…
We discuss a triangulated category of graded matrix factorizations of a deformed polynomial associated to the $A_{\mu}\textrm{-}$singularity. The semi-universal deformation of the $A_{\mu}\textrm{-}$singularity is given by a certain…
In this work we consider the Takagi factorization of a matrix valued function depending on parameters. We give smoothness and genericity results and pay particular attention to the concerns caused by having either a singular value equal to…
When doubly-affine matrices such as Latin and magic squares with a single non-zero eigenvalue are powered up they become constant matrices after a few steps. The process of compounding squares of orders m and n can then be used to generate…
We give a formula for the inverse matrix to an infinite matrix with possibly noncommutative entries, generalizing the Newton interpolation formula and the Taylor formula.
Theoretical and computational properties of a vector equation $Ax-\|x\|_1x=b$ are investigated, where $A$ is an invertible $M$-matrix and $b$ is a nonnegative vector. Existence and uniqueness of a nonnegative solution is proved. Fixed-point…
We consider the following first order systems of mathematical physics. 1.The Dirac equation with scalar potential. 2.The Dirac equation with electric potential. 3.The Dirac equation with pseudoscalar potential. 4.The system describing…