Related papers: On the $\top$-Stein equation $X=AX^\top B+C$

We give a new elementary proof of existence and uniqueness of a solution to the Sylvester equation $AX-XB=Y$

A topology on a nonempty set $X$ specifies a natural subset of $\mathcal{P}(X)$. By identifying $\mathcal{P}(\mathcal{P}(X))$ with the totally disconnected compact Hausdorff space $2^{\mathcal{P}(X)}$, the lattice $Top(X)$ of all topologies…

Given two $k\times n$ matrices $A$ and $B$, we describe a couple of methods to solve the matrix equation $XA=BY$, where $X$ is an invertible $k\times k$ matrix, and $Y$ is an $n\times n$ permutation matrix, both of which we want to…

We describe an efficient quantum algorithm for solving the linear matrix equation AX+XB=C, where A, B, and C are given complex matrices and X is unknown. This is known as the Sylvester equation, a fundamental equation with applications in…

The higher order matching problem is the problem of determining whether a term is an instance of another in the simply typed $\lambda$-calculus, i.e. to solve the equation a = b where a and b are simply typed $\lambda$-terms and b is…

We consider the polynomial equation $$X^n + a_{n-1}\cdot X^{n-1} + \dots + a_1 \cdot X + a_0 \cdot I = O,$$ over $(2 \times 2)$-matrices $X$ with the real entries, where $I$ is the identity matrix, $O$ is the null matrix, $a_i \in \mathbb…

Systems of the form $x = (A x^s)^{1/s} + b$ arise in a range of economic, financial and control problems, where $A$ is a linear operator acting on a space of real-valued functions (or vectors) and $s$ is a nonzero real value. In these…

We study the matrix equation $XA-AX=X^p$ in $M_n(K)$ for $1< p <n$. It is shown that every matrix solution $X$ is nilpotent and that the generalized eigenspaces of $A$ are $X$-invariant. For $A$ being a full Jordan block we describe how to…

The Riesz-Kantorovich formula expresses (under certain assumptions) the supremum of two operators $S, T : X \to Y$ where $X$ and $Y$ are ordered linear spaces as $$ S \vee T (x) = \sup_{0 \leqslant y \leqslant x} [S (y) + T (x - y)]. $$ We…

The treated matrix equation $(1+ae^{-\frac{\|X\|}{b}})X=Y$ in this short note has its origin in a modelling approach to describe the nonlinear time-dependent mechanical behaviour of rubber. We classify the solvability of…

In this note we show that for any given controllable pair (A,B) and any c> 1, a gain matrix K can be chosen so that the transition matrix of the system x' = (A+BK)x decays at the exponential rate exp(-ct) and the overshoot of the transition…

We consider the convolution equation $F*X=B$, where $F\in\mathbb{R}^{3\times 3}$ and $B\in\mathbb{R}^{m\times n}$ are given, and $X\in\mathbb{R}^{m\times n}$ is to be determined. The convolution equation can be regarded as a linear system…

The Team Orienteering Problem with Service Times and Mandatory & Incompatible Nodes (TOP-ST-MIN) is a variant of the classic Team Orienteering Problem (TOP), which includes three novel features that stem from two real-world problems…

We introduce a new concept of unbounded solutions to the operator Riccati equation $A_1 X - X A_0 - X V X + V^\ast = 0$ and give a complete description of its solutions associated with the spectral graph subspaces of the block operator…

We aim to find conditions on two Hilbert space operators $A$ and $B$ under which the expression $AX-XB$ having low rank forces the operator $X$ itself to admit a good low rank approximation. It is known that this can be achieved when $A$…

The supporting vectors of a matrix A are the solutions of max || x ||_2 =1 {||Ax||_2^2}. The generalized supporting vectors of matrices A_1 , . . . , A_k are the solutions of max || x ||_2 =1 {||A_1x||_2^2 + ||A_2x||_2^2 + ... +…

Given two metastable states A and B of a biomolecular system, the problem is to calculate the likely paths of the transition from A to B. Such a calculation is more informative and more manageable if done for a reduced set of collective…

In this paper, we study a matrix equation $X^n = aI$. We factorize $X^n - aI$ based upon the factorization of $x^n - a$ and then give a necessary and sufficient condition for one of the factors to be the zero matrix.

One well-known necessary and sufficient condition for equality in the Frobenius rank inequality due to Tian and Styan is that the matrix equation B = BCX + YAB be solvable for X and Y. We develop an algorithm to construct the matrices X and…

Based on the conjecture for the exact eigenvalue of the transfer matrix of the higher half-integer spin XXZ chain at the Razumov-Stroganov point, we evaluate the corresponding Baxter's Q operator in closed form by solving the TQ equation.…