Related papers: On the matrix equation XA-AX=X^p

In these notes, we consider the problem of finding the logarithm or the square root of a real matrix. It is known that for every real n x n matrix, A, if no real eigenvalue of A is negative or zero, then A has a real logarithm, that is,…

We give a full analytic solution to a particular case of the algebraic Riccati equation $XWW^*WX=W^*$ for any matrix $W$ (possibly non-square or non-symmetric) in using the Schur method, terms of the SVD decomposition of $W$. In particular,…

We propose and study a generalized version of the Lipman-Zariski conjecture: let $(x \in X)$ be an $n$-dimensional singularity such that for some integer $1 \le p \le n - 1$, the sheaf $\Omega_X^{[p]}$ of reflexive differential $p$-forms is…

It is well-known that a nilpotent n by n matrix B is determined up to conjugacy by a partition of n formed by the sizes of the Jordan blocks of B. We call this partition the Jordan type of B. We obtain partial results on the following…

We prove regularity results for solutions of the equation \[div(< AXu,X u>^{(p-2)/2} AX u) = 0,\] $1<p<\infty$, where $X=(X_1,...,X_m)$ is a family of vector fields satisfying H\"ormander's ellipticity condition, $A$ is an $m\times m$…

The condition of nilpotency is studied in the general linear Lie algebra $\mathfrak{gl}_{n}(\mathbb{K})$ and the symplectic Lie algebra $\mathfrak{sp}_{2m}(\mathbb{K})$ over an algebraically closed field of characteristic 0. In particular,…

For a complex nilpotent finite dimensional Lie algebra of matrices,and a Jordan-H\"older basis of it, we prove a spectral radius formula which extends a well-known result for commuting matrices.

In the paper, we introduce a matrix method to constructively determine spaces of polynomial solutions (in general, multiplied by exponentials) to a system of constant coefficient linear PDE's with polynomial (multiplied by exponentials)…

In this paper we completely characterize all possible pairs of Jordan canonical forms for mutually annihilating nilpotent pairs, i.e. pairs $(A,B)$ of nilpotent matrices such that $AB=BA=0$.

Properties of solutions of the RPA equation is reanalyzed mathematically, which is defined as a generalized eigenvalue problem of the stability matrix $\mathsf{S}$ with the norm matrix $\mathsf{N}=\mathrm{diag.}(1,-1)$. As well as physical…

We prove that an integral Jacobson radical ring is always nil, which extends a well known result from algebras over fields to rings. As a consequence we show that if every element x of a ring R is a zero of some polynomial p_x with integer…

This note is concerned with the linear matrix equation $X = AX^\top B + C$, where the operator $(\cdot)^\top$ denotes the transpose ($\top$) of a matrix. The first part of this paper set forth the necessary and sufficient conditions for the…

We review a family of algorithms for Lyapunov- and Riccati-type equations which are all related to each other by the idea of \emph{doubling}: they construct the iterate $Q_k = X_{2^k}$ of another naturally-arising fixed-point iteration…

Let $M_n(\mathbb{F})$ denote the algebra of $n \times n$ matrices over an algebraically closed field $\mathbb{F}$ of characteristic different from $2$. For $n \ge 2$, we classify all maps $\phi : M_n(\mathbb{F}) \to M_n(\mathbb{F})$…

Given linear diophantine equation Ax=b, rank A=m. Let d be the maximum of absolute values of the mxm minors of the matrix (A | b). It is shown that if M={x : Ax=b, x nonnegative and integer} is nonempty, then there exists x=(x1,...,xn) in…

Consider the $n$th degree polynomial equation, $X^n+A_{n-1}X^{n-1}+...+A_1X+A_0=0$ over the ring of 2 by 2 complex matrices. If this equation has more than ${2n \choose 2}$ solutions, then it has infinitely many solutions. We show here that…

W.E. Roth (1952) proved that the matrix equation $AX-XB=C$ has a solution if and only if the matrices $\left[\begin{matrix}A&C\\0&B\end{matrix}\right]$ and $\left[\begin{matrix}A&0\\0&B\end{matrix}\right]$ are similar. A. Dmytryshyn and B.…

Matrix functions extend scalar function concepts to linear operators, offering a unified framework with broad applications in mathematics, science, and engineering. Classical definitions--via power series, spectral calculus, or Jordan…

Consider the nonlinear matrix equation X-sum_{i=1}^{m}A_{i}^{*}X^{-1}A_{i}=Q. This paper shows that there exists a unique positive definite solution to the equation without any restriction on A_{i}. Three perturbation bounds for the unique…

Let $k$ be an algebraically closed field of characteristic $p >0$. We consider the variety of nilpotent pairs $(A,B)$ with $[A,B]=\lambda I$, namely the set of pairs $ X = \{ (A,B) \in M_n(k) \times M_n(k) \mid A,B \text{ nilpotent},…