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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5
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0answers
53 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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25 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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2answers
234 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 \\ ...
2
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1answer
109 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 ...
2
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0answers
141 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 ...
1
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1answer
92 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^...
3
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31 views

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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85 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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25 views

Norm of vector components optimization of linear matrix combination

Given complex matrices $A_1, A_2, \dots, A_k\in\mathbb{C}^{m \times n}$, $B \in\mathbb{R}^{m \times n}$, the objective is to find a vector $x \in \mathbb{C}^k$ such that: $\max {||x_i||}$ , $i\in 1,2.....
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1answer
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 ...
3
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2answers
151 views

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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1answer
49 views

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 ...
1
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1answer
71 views

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 ...
25
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1answer
1k 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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0answers
36 views

The relation between unit quaternions and the virtual rotation of an angular velocity vector?

I am working with a calculus employed in multi rigid body dynamics problems introduced by professor E. Haug (c.f. this book). Let's define $\boldsymbol{\mathrm{e}} \in \mathcal{R}^4$ as a set of unit ...
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0answers
39 views

Solving a Lyapunov inequality with a fixed block

Let $A\in\mathbb{R}^{n\times n}$ be the following partitioned matrix $$ A = \begin{bmatrix}A_{11} & A_{12} \\ A_{21} & A_{22}\end{bmatrix} $$ where $A_{11}\in\mathbb{R}^{n_1\times n_1}$, $A_{...
1
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1answer
61 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 ...
2
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0answers
54 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 ...
2
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1answer
235 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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0answers
50 views

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 $\...
0
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1answer
81 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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97 views

Multiplication of matrix-represented polynomials

Suppose we have a multivariate polynomial $f(\mathbf{x})$ represented by some matrix $A$, i.e. $f(\mathbf{x}) = b^T A b$, where $b = (1, x_1, x_2, \dots, x_{m-1}x_m^{n-1}, x_m^n)^T$ is a monomial ...
1
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1answer
78 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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2answers
91 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 ...
2
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0answers
73 views

How to find the similarity transformation of a matrix operator?

Consider the matrix operator $$ \mathcal{L}= \begin{pmatrix}{} -A\cdot \partial_x^2-(3\varphi^2-1)-2(\varphi',\partial_x\cdot)_{L^2}\varphi'' & c\partial_x\\ -c\partial_x & 1 \end{pmatrix} $$...
4
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3answers
192 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 ...
1
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1answer
80 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 $\rho_i(A(\lambda))$ ...
6
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1answer
129 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 $B$ ...
7
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1answer
263 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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2answers
57 views

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 $c_{ij}(\lambda) = O(\frac{1}{\lambda})$ $A+C$ ...
0
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1answer
91 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?
4
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1answer
309 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 ...
2
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3answers
210 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 ...
0
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0answers
45 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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0answers
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. ...
1
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5answers
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 ...
1
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0answers
88 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 = ...
1
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0answers
37 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 ...
4
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2answers
182 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 ...
13
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1answer
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 ...
3
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2answers
277 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 ...
2
votes
1answer
57 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 \...
3
votes
1answer
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 ...
2
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0answers
106 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 ...
2
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0answers
31 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_{...
1
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0answers
104 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 ...
2
votes
1answer
162 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 ...
1
vote
1answer
111 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 ...
5
votes
1answer
119 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 ...
1
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0answers
96 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}&\...