# 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 $Ax=b$.

156 questions
Filter by
Sorted by
Tagged with
53 views

Let $B_1,\ldots,B_s$ be $(s\times s)$ symmetric real matrices and $x=\left(x_1,\ldots,x_s\right)^\prime$ a $(s\times 1)$ vector of unknowns. Is there a way or reference theory for studying the ...
25 views

31 views

160 views

### Solving two quadratic matrix equations [closed]

Given $10 \times 10$ matrices $A$ and $B$, I would like to find $10 \times 10$ matrix $X$ such that $$A = X B X^T \tag{1}$$ $$B = X A X^T \tag{2}$$ How can I solve the issue? if there is a way to ...
151 views

61 views

30 views

### 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. ...
515 views

### Solution for $Xa + X^Tb = c$ where $X^TX = I$? [closed]

There are three known $n\times1$ vectors: $a, b, c$, along with one unknown $n\times n$ matrix: $X$. I am only interested in the $n={2,3}$ cases. $X$ is $2\times 2$ or $3\times 3$ rotation matrix ...
88 views

186 views

### 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 ...
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 ...