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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Solving linear matrix equation

Given matrices $A, B, C' \in \Bbb R^{2 \times 6}$, where $'$ denotes matrix transposition, and matrix $L \in \Bbb R^{2 \times 2}$, how can one solve the following linear matrix equation in $X \in \Bbb ...
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1 answer
123 views

Product of matrices equal identity

I need to solve the following equation for the matrix $P \in\mathbb{R}^{r\times d}:$ $$ ((PAP^\top)^{-1} P S P^\top (PAP^\top)^{-1})^2 = I_r, $$ where $S$ is a symmetric $d\times d$ matrix, $A$ is a ...
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1 vote
0 answers
80 views

In matrix product, differentiate one element with respect to another element

Background Consider a system (roughly) along the lines of those shown in Sims, C. A. (2002). Solving linear rational expectations models, where you have $$ AX_{t+1} = CX_t + M $$ where matrix $M$ is a ...
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1 vote
2 answers
195 views

Matrix equation $P^TAP=A$

Let $A\in \mathcal{M}_{m\times m}(\mathbb R)$ , $det(A)=1$ , $A$ is positively definite. Which matrices $P$ satisfy the equation $$P^TAP=A$$ In fact I am interested in sequences of traces $tr P^n$ of ...
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8 votes
3 answers
246 views

System of quadratic equations

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 ...
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  • 101
1 vote
0 answers
78 views

Optimization problem on trace of complex matrix product

Given a complex rectangular matrix $A$ $(k \times n)$, I am interested in solving the following optimization problem over $(k\times n)$ complex matrices $x$: $$ \mathrm{arg}\max_X \,\mathrm{trace}(X^...
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  • 327
1 vote
2 answers
253 views

Proving that a system of polynomial matrix equations over $\mathbb{F_2}$ has no solution

I am working on a problem involving nilpotent matrices over $\mathbb{F}_2$ and I was able to reduce it to proving that the system \begin{equation} \begin{cases} A^2+ BC+ BCA+ ABC+A = I_4 \\ ...
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3 votes
1 answer
171 views

Is the solution to the Fredholm integral equation of the first kind a continuous analogue of Cramer's rule for matrix equations?

When we have a system of of $n$ linear equations represented by $$A \vec{x} = \vec{b} $$ with $\vec{x} = (x_{1}, x_{2}, \dots, x_{n})^{\intercal} $, we can solve for each component of this vector by ...
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2 votes
0 answers
161 views

System of matrix equations

Problem definition: Let $x_i \in \mathbb{R}^d$ and $a_i \in [0,1]$, for all $i = 1,\dots, k$ (with $k\geq d$). Define $M(a) = \sum_{i = 1}^k a_i x_ix_i^T,$ and assume $M(a) \succ 0.$ Question: Is ...
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1 vote
1 answer
126 views

Matrix equation involving quadratic form

Let $X,Y\in\mathbb{R}^{n\times k}$, $\Lambda(\alpha) = \text{diag}(\alpha)$, with $\alpha\in\mathbb{R}^k$, and let $c,d\in\mathbb{R}^+$ be positive constants. Let $$A_i(\alpha) = (X\Lambda(\alpha) X^...
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3 votes
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Stability of matrix equation

Let $M=I+A\in \mathbb{R}^{n\times n}$ for a skew-symmetric matrix $A$ with $\|A\|<1$ in the spectral norm. Using the $LU$-decomposition of $M$, it is easy to construct a solution $L,U\in \mathbb{R}^...
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0 answers
117 views

What is the solution of the matrix equation $A X + X A' + B X B' + C = 0$ for $X$?

I know that the matrix equation $A X + X A' + C = 0$ is in the form of the time-continuous Lyapunov equation, so solving for $X$ is pretty trivial since the solution already and numeric solvers ...
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1 answer
221 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 ...
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2 answers
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A problem about determinant and matrix

Suppose $a_{0},a_{1},a_{2}\in\mathbb{Q}$, such that the following determinant is zero, i.e. $ \left |\begin{array}{cccc}\\ a_{0} &a_{1} & a_{2} \\ \\ a_{2} &a_{0}+a_{1} & a_{1}+a_{...
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0 votes
1 answer
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State-dependent positive definite matrix

Consider a function $f(\mathbf{x})=\mathbf{M}_\mathbf{x}$ that outputs a nonsymmetric matrix $\mathbf{M}_\mathbf{x} \in \mathbb{R}^{N \times N}$ given an input vector $\mathbf{x} \in \mathbb{R}^N$. Is ...
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1 vote
1 answer
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How do you solve this quadratic matrix equation?

could you please help me solve this quadratic matrix equation? I look around, seems like there is no general solution for it.. $$-BX^2 + X - C = 0$$ for X, B and C are (3x3) matrices. B and C are ...
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27 votes
1 answer
2k views

Integer matrices which are not a power

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}$In a group $G$, an element $g$ is said to be primitive if there is no $h \in G$ and integer $n >1$ such that $g = h^n$. (For clarification, I ...
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2 votes
1 answer
79 views

Determine unknown matrix function of particular form from known points

I encountered the following problem recently in a practical context. Fix $n \ge 1$. Suppose $f$ is an unknown function $\mathbb C ^ {n \times n} \to \mathbb C ^ {n \times n}$ of the form $$ X \mapsto ...
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2 votes
0 answers
73 views

Is there a method to solve a non-linear quadratic matrix equation?

I am interested in solving the following quadratic equation: $$x^{\top} A x = \sqrt{x^{\top} B x}$$ Here, $x \in \mathbb{R^q}$ is an unknown vector, and A and B are two q$\times$q-dimensional ...
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2 votes
1 answer
628 views

Number of 5x5 matrix permutations without repetitions in rows or columns

Context In the boardgame Azul, your goal is to complete as much as possible of a $5\times5$ board by placing 25 tiles of 5 different colours (5 tiles of each colour) so that no colour appears twice in ...
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1 vote
0 answers
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Approximation bounds for matrix multiplication

$\DeclareMathOperator{\op}{\mathrm{op}}$Since matrix multiplication is continuous, I expect that if $A_n\to A\in \mathrm{Mat}_{d\times d}(\mathbb{R})$ for the operator-norm and if $x_n\to x$ in $\...
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0 votes
1 answer
90 views

An otherwise linear matrix equation with the presence of a signum function : reference request

Consider the equation $$\pmb{c}+\text{sign}(G\pmb{c}) = L$$ $\pmb{c}$ is a $n\times1$ matrix. $G$ is a $n\times n$ matrix which is also positive definite. matrices $G$ and $c$ are real. $L$ is a $n\...
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1 vote
1 answer
101 views

Global polynomial basis for the kernel of a matrix polynomial

Let $M(x)$ be an $m$ by $n$ matrix with entries in $\mathbb{C}[x]$. Suppose that for all $x\in \mathbb{C}$ the rank of $M(x)$ is constant and equal to $r<n$. Therefore, for any $x_0\in \mathbb{C}$ ...
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1 vote
2 answers
98 views

Rank of a linear combination of linear operators

I asked this question a few days ago in MathExchange and received no satisfatory answer. I hope it is well suited for MathOverflow. Suppose I have two linear operators $X,\,Y$ on $\mathbb{C}^n$. Now ...
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  • 355
4 votes
3 answers
208 views

Properties of matrix $X=\left[\frac{1}{1-\bar\alpha_i \alpha_j}\right]_{ij}$

Let $\{\alpha_i\}_{i=1}^n$ be complex numbers such that $|\alpha_i|<1$, and consider the following $n\times n$ structured matrix $$ X=\left[\frac{1}{1-\bar\alpha_i \alpha_j}\right]_{ij}. $$ Such ...
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1 vote
1 answer
95 views

Asymptotic behavior of a matrix equation and its eigenvalues

We have a matrix valued function $A:\mathbb{R}_+\to \mathbb{R}^{m\times m}$. It is known that $A(\lambda)$ is a positive definite matrix for all $\lambda\in\mathbb{R}_+$ Denoting $\rho_i(A(\lambda))$ ...
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6 votes
2 answers
163 views

How to obtain matrix from summation inverse equation

I have a set of square matrices $\{A_i\}_{i \in \{1,..., n\}}$ and another square matrix of equal size $K$. Under the assumption that such a matrix exists and is unique, I want to find the unique $B$ ...
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7 votes
1 answer
578 views

Closed form solution for $XAX^{T}=B$

Let $d \times d$ matrices $A, B$ be positive definite. Is there a closed form solution for the following quadratic equation in $X$? $$X A X^{T} = B$$ Thank you.
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0 votes
2 answers
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Asymptotic expansion involving a matrix equation

Let $b = [b_1,b_2,b_3,...b_n]^T$ $A = [a_{ij}]_{n \times n}$ such that $a_{i,j} = 1\forall 1\le i,j \le n$ $C = [c_{ij}(\lambda)]_{n \times n}$ such that $c_{ij}(\lambda) = O(\frac{1}{\lambda})$ $A+C$ ...
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0 votes
1 answer
99 views

Solution for $AX+XA^T+XBX=C $ where $X$, $B$ and $C$ are symmetric

Is there a solution for $AX+XA^T+XBX=C$ where $X$, $B$ and $C$ are symmetric?
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4 votes
1 answer
354 views

Solving equation of matrix valued functions

Given $n\times n$ matrices with entire functions entries (holomorphic on all of the complex plane $\mathbb{C}$) $A(z)=[a_{ij}(z)],B(z)=[b_{ij}(z)]$, i.e., $a_{ij}(z),b_{ij}(z)$ are entire functions ...
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2 votes
3 answers
242 views

Efficient algorithm for matrix equation $AXB + BXA = F$

For $n\in\mathbb{N}$, let $A\in\mathbb{R}^{n\times n}$ and $B\in\mathbb{R}^{n\times n}$ be two symmetric positive definite matrices and let $F\in\mathbb{R}^{n\times n}$ be arbitrary. Is there any ...
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0 votes
0 answers
61 views

Solution to the matrix equation AX=XB in the form of generalized inverses

It is known that the solution (suppose it exists) of Sylvester matrix equation $AX-YB=C$ can be given in the form of generalized inverses as $$X=A^{-}C + A^{-}ZB + (I-A^{-}A)W,$$ and $$Y = - (I-AA^{-})...
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1 vote
5 answers
547 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 ...
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1 vote
0 answers
106 views

A real system of bilinear equations with $2n$ unknown and equations

I have the following system of $2n$ bilinear equations, for a square invertible matrix $A \in \mathbb{R}_{n \times n}$, and $2n$ unknowns organized in vectors $x,y \in \mathbb{R}^n$: $$ diag(y) A x = ...
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  • 550
1 vote
0 answers
39 views

Nonlinear fixed-point equation with linear solutions?

Let $S$ be an $N\times N$ row-stochastic matrix and let $w'$ be the left Perron eigenvector of $S$ (i.e., $w$ is the stationary distribution of the Markov chain represented by $S$). Let $T$ be the ...
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4 votes
2 answers
193 views

Is there any general reference on matrix quadratic equations?

I am studying a problem where a quadratic matrix equation emerges. The equation is as follow (all capital letters are n by n matrices) $(I-X^{\prime}L)X=D$ where $L$ and $D$ are both symmetric and ...
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  • 41
13 votes
1 answer
1k views

Necessary conditions for the existence of solution of Sylvester equation AX=XB

Let's consider square matrices $A_{n \times n}$, $B_{n \times n}$ and $X_{n \times n}$ with elements from $\mathbb{R}$. Could you tell me please, what would be the necessary conditions for the ...
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3 votes
2 answers
300 views

Diagonal Lyapunov equation with rank 1

Given the discrete-time Lyapunov equation (1): $$ A^T P A - P = bb^T $$ such that $P$ shall be diagonal and positive definite and $b$ is a column vector. How to characterize $A$ and $b$, where ...
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2 votes
1 answer
91 views

Matrix factorization for dimensional reduction similar to spectral decomposition/SVD

I have a graph clustering problem I'm working on and it basically involves finding a factorization of the adjacency matrix $A$ such that the following equations are (approximately) satisfied: $$ A \...
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  • 123
3 votes
1 answer
241 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 ...
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2 votes
0 answers
112 views

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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  • 2,850
2 votes
0 answers
36 views

Solving the inverse of a matrix under a uniform distribution

I am looking to solve the following equation: $$\left(\begin{array}{cc}{ g_{11}} & { g_{12}} \\ { g_{21}} & { g_{22}}\end{array}\right)=\int_{a}^{b} \frac{1}{b-a}\left(\begin{array}{cc}{- g_{...
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  • 125
1 vote
0 answers
136 views

solving a non-linear Matrix equation

I am working on a problem of optimization of wireless sensor networks in order to achieve the optimal power allocation in the network. I find an equality matrix problem that can be written as ...
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  • 327
2 votes
1 answer
242 views

How to solve a quadratic matrix equation with positive semidefinite constraint?

I have the following quadratic matrix equation: $$ XAX+X = B $$ where both $A$ and $B$ are given positive definite matrices, and $X$ is a covariance matrix and, hence, positive definite. When there is ...
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1 vote
1 answer
124 views

find a PSD matrix that that verify matrices sum of equality

$A $, $ C$ $(n,n)$ are symmetric PSD matrices, $B$ is PD symmetric matrix, and $H_i$ $\; $ $(i=[1,m])$ represent $ m $ complex matrices. $H_i$ are all one rank matrix Our objectif is to find ...
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  • 327
5 votes
1 answer
177 views

How can I solve an orthogonal-constrained Sylvester equation?

I am currently facing a Sylvester equation $AX+XB = C$ where $A$, $B$, $C$ are all symmetric and a special constraint here is that $X$ should be orthogonal. The Sylvester equation itself may not ...
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  • 91
1 vote
0 answers
99 views

Controlling the rank of a Matrix product

Let $\bf{Q}$ be an $(m-k)\times m$ matrix satisfying $\bf{QQ}^H=\bf{I}$, $\bf{W}$ an $m\times m\ell$ matrix of the form $$\bf{}W=\left[\begin{array}{ccccc}{\bf{w}}_1 &\bf{0} &\bf{0}&\...
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10 votes
1 answer
509 views

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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5 votes
2 answers
530 views

Symmetric linear least-squares solution

Given tall matrices $A$ and $Y$ and the following overdetermined linear system in square matrix $X$ $$AX=Y$$ is there an explicit formula for the least-squares solution if $X$ is constrained to be ...
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