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 .

Definition: Let $~v_1,~v_2,~\cdots~,v_n~ $ and $~b~$ be vectors in $~\mathbb{R^n}~$. Consider the vector equation $$x_1~v_1+x_2~v_2+~\cdots~+x_n~v_n=b~$$This is equivalent to the matrix equation$$~Ax=b~$$

where $~~A=\begin{pmatrix} \cdot & \cdot & \cdots & \cdot \\ v_1 & v_2 & \cdots & v_n \\ \cdot & \cdot & \cdots & \cdot \\ \end{pmatrix};~~ x=\begin{pmatrix} x_1 \\ x_2\\ \cdots\\ x_n \end{pmatrix} ~~\text{and}~~ b=\begin{pmatrix} b_1 \\ b_2\\ \cdots\\ b_n \end{pmatrix}$

Since a matrix equation $ ~AX=B~$ (where $ ~X~$ is a column vector of variables) is equivalent to a system of linear equations, we can use the same methods we have used on systems of linear equations to solve matrix equations. Namely:

$(1.)~~$ Write down the augmented matrix $ ~A \vdots B$.

$(2.)~~$ Row-reduce to a new augmented matrix $~ \overline A \vdots \overline B~$ in row echelon form.

$(3.)~~$ Use this new matrix to write a matrix equation equivalent to the original one.

$(4.)~~$ Use this new, equivalent matrix equation to find the solutions to the original equation.

In mathematics, matrix equation (which is a system of linear equations) is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A system of non-linear equations can often be approximated by a linear system (see linearization), a helpful technique when making a mathematical model or computer simulation of a relatively complex system.

Very often, the coefficients of the equations are real or complex numbers and the solutions are searched in the same set of numbers, but the theory and the algorithms apply for coefficients and solutions in any field. For solutions in an integral domain like the ring of the integers, or in other algebraic structures, other theories have been developed, see Linear equation over a ring. Integer linear programming is a collection of methods for finding the "best" integer solution (when there are many). Gröbner basis theory provides algorithms when coefficients and unknowns are polynomials. Also tropical geometry is an example of linear algebra in a more exotic structure.

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Why $x^TLx = \sum_{(u,v)\in E} w_{uv}(x(u)-x(v))^2$ for Laplacian L?

Question from paper "Graph Sparsification by Effective Resistances" by Daniel A. Spielman and Nikhil Srivastava again. I was trying to find some notes/lectures on this topic and for instance I found this: The link for some notes regarding…
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determinant of $4\times 4$ matrix by elimination

I am trying to find the determinant of this $4\times 4$ matrix. I got the wrong answer but I can't find the mistake The answer is supposed to be $-44$ but I got $-176$ the matrix $$ \begin{bmatrix} -2 & 2 & 3& -3\\ -3 & 3 & -1 & 2\\ …
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Solve $x \sqrt{x^T A x} + c =0$ for x in terms of $A$ and $c$ only

Question: Solve $x \sqrt{x^T A x} + c =0$ for x in terms of $A$ and $c$ only. Here $x \in \mathbb{R}^n$, $c \in \mathbb{R}^n$, and $A \in \mathbb{R}^{n\times n}$. Lastly, $A$ is symmetric and PSD, not sure if that matters. This seems like it should…
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Pseudoinverses giving weird results

Suppose H and E are two $n\times m$ matrices with $n>m$, now I have an equation: $$\begin{equation}H=E\rho \end{equation}$$ where $\rho$ is $m\times m$ since $n>m$, we have moore-penrose invereses $H^{+}$ and $E^{+}$ such that $H^{+}H=I$ and…
Kutsit
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If $A^2 = - I$, is $A$ invertible?

If $A^2 = - I$, is $A$ invertible? My attempt: We know $A^4 = I$. So, $A(A^3) = I$, and thus, the inverse of $A$ is $A^3$. Is that correct? It is a really basic question but it's a new subject for me.
DarkLeader
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Solve this "quasi diagonalization" matrix equation

I would like to know if it is possible to solve for matrix T any matrix equation in the form of T^-1 * F * T = G where F and G are nxn known matrix. How can I solve equation and find such matrix T? In my example, with Matlab representation of…
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Is there a matrix $X$ possible such that $AXB=O$?

Let $A$ be a $9\times4$ matrix and $B$ be a $7\times3$ matrix. Is there a $4\times 7$ matrix $X$ possible such that $X\neq O$ and $AXB=O$? My approach: I tried to prove that there is a nonzero solution to $AX=O$ by proving that the number of…
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Characterizing the unique solution to a matrix equation

Setup Let $\Pi$ be an $n\times n$ matrix where each row sums to one. Let $\mathcal Y$ be an eigenvector of $\Pi'$ such that \begin{equation*} \Pi' \mathcal Y=\mathcal Y \end{equation*} Denote $\Psi$ as the diagonal matrix whose diagonal is the…
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Solution of a linear matrix equation

i'm trying to rewrite a linear matrix equation $$ X=A_1XA_2+A_3X^TA_4+B $$ where $ X\in\mathbb{R}^{m\times n} $ and $ A_1\text{, }A_2\text{, }A_3\text{, }A_4\text{, }B $ are matrices of appropriate sizes. So far i got to express the equation…
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Solutions to $X^3 = I_2$

Show that there are an infinite number of solutions to $X^3 = I_2$ in $M_2(\mathbb{Q})$. $\operatorname{det} (X) = 1$ because $X^3 = I_2$ and $\operatorname{det} (X) \in \mathbb{Q}$ $X^2 = X^{-1}$ and, using Cayley-Hamilton $X^2 =…
LIR
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Adding the column space of a matrix with another matrix?

Supposed I have these matrices: \begin{gather} M = \begin{bmatrix} 0 & 1 & 1 \\ 0 & 1 & 0 \\ 0 & 0 &1 \end{bmatrix} \hspace{1.5em} N := ℝ^3 \end{gather} \begin{gather} A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ 0 & 1…
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How can the Poisson equation be numerically solved when its matrix is ​ill-conditioned in nature?

In the numerical solution of Poisson partial differential equation PPD by the method of finite differences FDM, the linear system of algebraic equations that is usually met: Ax = b. If the PPDE matrix A itself is inherently ill-conditioned, the…
user737980
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Finding the symmetric square roots of diagonal matrices

Let $D=\text{diag}(d_1,\dots,d_n)$ be a real diagonal matrix, where $0\le d_1 \le d_2 \le \dots \le d_n$. Let $a_1 < a_2 < \dots < a_m$ be its distinct eigenvalues (counted without multiplicities). Now, let $A$ be a real symmetric $n \times n$…
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solve matrix equation $2AX-X^TA=B$

Given matrices $A=\begin{pmatrix} 2 & 4\\ 3 & 6 \end{pmatrix},B=\begin{pmatrix} 3 & -6\\ 7&-4 \end{pmatrix}$ Find all matrices $X\in M_2(\mathbb{R})$ such that $$2AX-X^TA=B$$ I have no idea how to do it.
Hrackadont
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Matrix inequality that follows from other inequality

I have the following inequality: $$ f(x)^T(f(x) - A x) \leq 0 $$ The $A$ is a positive definite matrix of dimension $a \times a$, $x$ is a $a \times 1$ vector and $f$ is a vector valued function that has $a$ dimensions (everything real valued). Why…
user3137490
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