# Questions tagged [matrix-equations]

Equations whose unknown is a matrix, such as, for instance, algebraic Riccati equations $XAX+XB+CX+D=0$ or matrix differential equations (e.g. $\dot X(t)=AX(t)$. This tag is *not* meant for general systems of linear equations $Ax=b$.

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### Example for diagonal Lyapunov equation with orthogonal matrix

This question is related to previous posts here and here. Here, I'm asking for an example. Given the orthonormal matrix $U$ of size $n \times n$, i.e., $U U^T = I$, and $Q$ is non-singular diagonal. ...
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### Reversing system of linear equation [closed]

I have a problem that I cannot handle. I have a system of polynomial equations, for example: \begin{eqnarray} a_0=x_{0,0}b_0+x_{0,1}b_1+x_{0,2}b_2+x_{0,3}b_3 \\ a_1=x_{1,0}b_0+x_{1,1}b_1+x_{1,2}b_2+...
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### A number related to rank

Given $n$ a natural number we know minimum $m$ over all choices of $v_i=\begin{bmatrix}a_{i1}\\\vdots\\a_{in}\end{bmatrix}\in\mathbb R^n$ such that $$I=\sum_{i=1}^mv_iv_i'$$ holds where $'$ is ...
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### Conditions for a certain matrix equation to have a full rank solution

Assume that we have the following equation to solve $$\sum_{\ell=1}^L A_\ell X_{\ell} B_{\ell} =0$$ over complex matrices where each $A_{\ell}$ is a given $m\times n$ matrix, each $B_{\ell}$ is a ...
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### Distance between two algebraic sets

We are in $M_n(\mathbb{R})$ equipped with the Frobenius norm $||A||^2=tr(AA^T)$. Let $Z=\{(A,B)\in M_n(\mathbb{R})^2;A^2-AB-B^2=0\}$ and $T=O(n)^2$. It is easy to see that $Z\cap T=\emptyset$ and ...
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### Solving $AXB + X\odot C = D$

I need to solve the following equation for $X$ with $d$-by-$d$ matrices $A,B,C,D$ and Hadamard product $\odot$ $$AXB + X\odot C = D$$ Vectorizing all terms gives a solution with $O(d^6)$ complexity, ...
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### Low-rank solution of generalized Sylvester equation

Let $n \in \mathbb{N}$. Let $A,B,C,D$ be non-singular $n \times n$ matrices. If the matrix pencils $A-\lambda C$ and $B-\lambda D$ are regular and have disjoint spectra, then $$AXB-CXD = 0$$ has a ...
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### A vanishing sum of symmetric matrices

Let $\{G_i\}_{i=1}^N\in\mathbb{R}^{n\times m}$ be a set of full column rank matrices (i.e., $\mathrm{rank}(G_i)=m$ for all $i$) and $\{P_i\}_{i=1}^N\in\mathbb{R}^{m\times m}$ be a set of positive ...
Let $A,C\in\mathbb{R}^{m\times n}$, $n\ge m$, $B\in\mathbb{R}^{n\times m}$, and $P$ be a real positive definite $m\times m$ matrix. Denote by $\mathcal{S}^n$ the space of $n\times n$ real symmetric ...
### A conjugation matrix $X\in \mathbb{C}^{ n\times p}$ where $p< n$
Given a Hermitian positive definite matrix $A\in \mathbb{C}^{n \times n}$ and a Hermitian matrix $B\in\mathbb{C}^{ p\times p},$ find the matrix $X$ so that $X^HAX=B$ holds where $X^H$ denotes ...