# Questions tagged [matrix-equations]

This tag is for questions related to equations, with matrices as coefficients and unknowns. A matrix equation is an equation in which a variable stands for a matrix .

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### $ABACA = 0 \Longrightarrow BAC = 0$ if $A,B,C \ge 0$ are symmetric.

Problem. $A, B, C$ are $n \times n$ symmetric positively semi-definite matrices. Prove that $ABACA = 0 \Longrightarrow BAC = 0$ if $A,B,C \ge 0$ are symmetric. My attemp (there's mistake in it). We ...
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### Binary matrix power for a specific entry.

Consider a square $n\times n$ matrix $A$ whose entries are binary, that is, for all $i, j\in [n]$, it holds that $A_{i, j} \in \{ 0, 1\}$. I am interested in the following decision procedure: Given a ...
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### Is there any mathematical results stating when there are 0's in the inverse of a square matrix given 0's in the original matrix?

I am working with square invertible matrices. Denote the n-by-n matrix as $A \in \mathbb{R}^{n \times n}$. Say we know that there are some 0's in the matrix. For instance: $A_{ij} = 0$ for some $i,j$ ...
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### Ways to invert complicated matrix formulas

I have two somewhat complicated matrix formulas that convert the mean vector and covariance matrix for a certain variable, $\mu \in \mathbb{R}^n$ and $\Sigma \in \mathbb{R}^{n \times n}$, into the ...
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### Geometry Application for System of Linear Equations

Question : Find the intersection (if any) of the line $x=(1,0,-1)+\lambda(3,2,1)$ and the plane $x = (-1,-7,-7) + \alpha(3,5,1)+ \beta(1,-2,-5)$. My work so far: I equated both equations and ...
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### Solving a certain matrix equation

I am trying to solve the (real) matrix equation $X^{-1}AX=Y$, where the matrix $A_{n\times n}$ is given and the matrices $X_{n\times n}, Y_{n\times n}$ are unknown matrices, with the only restriction ...
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### A special property of a system of linear equations in n > 3 dimensions

The origin of the question is physics, so I have to explain the idea behind it. In three dimensions the Lorentz force $F$ of a magnetic field $B$ on a particle (with charge $q = 1$) with velocity $v$ ...
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