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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16 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. ...
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0answers
46 views

Reversing system of linear equation [closed]

I have a problem that I cannot handle. I have a system of polynomial equations, for example: \begin{eqnarray} a_0=x_{0,0}b_0+x_{0,1}b_1+x_{0,2}b_2+x_{0,3}b_3 \\ a_1=x_{1,0}b_0+x_{1,1}b_1+x_{1,2}b_2+...
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0answers
68 views

A number related to rank

Given $n$ a natural number we know minimum $m$ over all choices of $v_i=\begin{bmatrix}a_{i1}\\\vdots\\a_{in}\end{bmatrix}\in\mathbb R^n$ such that $$I=\sum_{i=1}^mv_iv_i'$$ holds where $'$ is ...
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0answers
58 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
36 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
163 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 ...
12
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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
259 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
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1answer
41 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
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1answer
158 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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101 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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30 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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0answers
90 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
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1answer
88 views

How to solve a quadratic matrix equation with positive semidefinite constaint

I have the following quadratic matrix equation: $ XAX+X = B $ where $A$ and $B$ are all positive definite matrix. The constraint here is that $X$ is actually a covariance matrix and hence should be ...
1
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1answer
98 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
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1answer
91 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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0answers
92 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}&\...
10
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1answer
401 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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95 views

Existence of solution for system of quadratic equations

I have a system of quadratic equations $${\bf{x}}_m {\bf{A}}_{k} {\bf{y}}_k^{\mathrm{H}}= Z_{m,k}, \quad 1\leq m \leq M,\;\; 1\leq k \leq K,\;\; M>K $$ where ${\bf{x}}_m$ is $1\times N$, and ${\bf{...
5
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2answers
192 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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0answers
78 views

Matrix logarithm for d-dimensional cyclic permutation matrix

I want to find the matrix $\hat{H}_d$ which, when exponentiated, leads to a d-dimensional cyclic permutation transformation matrix. I have solutions for d=2: $$ \hat{U}_2 =\left( \begin{matrix} ...
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0answers
45 views

Coexistence of different solutions in a nonlinear matrix differential equation

I've faced a system of first-order nonlinear matrix differential equation, and I have tried to use perturbation method to approach the solutions. The differential equation has the form: \begin{...
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0answers
56 views

How to solve a quadratic matrix equation?

\begin{equation} \boldsymbol{\omega^H} \boldsymbol{G} \boldsymbol{\Theta^H} \boldsymbol{h_r} \boldsymbol{h_r^H} \boldsymbol{\Theta} \boldsymbol{G^H} \boldsymbol{\omega}=a\\ \boldsymbol{\omega^H} \...
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0answers
91 views

Solving a nonlinear matrix equation

Consider the following nonlinear matrix equation: $B=PX^{−1}AX$ where $B$ and $P$ are a $1\times n$ row vector and $A$ is a $n\times n$ matrix which are all strictly positive, and $X=diag(x_1,...,...
3
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0answers
127 views

What is topology of all Square matrices such that matrix times it transpose is diagonal

What is the topology of the subspace $X_n\subset M_n(\mathbb R)$ consisting of all non-zero $n\times n$ matrices $A$ such that $A^t A$ is diagonal ? For example $X_2$ is the product of a torus and an ...
2
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1answer
54 views

Equalities between transforms of matrices that are extremely different

I have two $2N\times 2N$ matrices, defined by blocks: $$ A = \begin{bmatrix} a & 0 \\ 0 & 0 \end{bmatrix} $$ $$ B = \begin{bmatrix} 0 & 0 \\ 0 & b \end{bmatrix} $$ where $a$ and $b$ ...
16
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2answers
431 views

The number of 0-1 normal matrices

Let $A\in\{0,1\}^{n\times n}$ be a $n\times n $ matrix with entries in the discrete set $\{0,1\}$. My question: What is the number of matrices in $\{0,1\}^{n\times n}$ that are normal, that is, ...
6
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2answers
272 views

Representation over matrices $A_i^3=I$, $A_0A_1^\dagger+A_1A_2^\dagger+A_2A_0^\dagger=0$, $A_0^\dagger A_1+A_1^\dagger A_2+A_2^\dagger A_0=0$

I would like to know what all the possible finite-dimensional representations of the following relations are. $$A_0^3 = A_1^3 = A_2^3 = I \tag{1}$$ $$A_0 A_1^\dagger + A_1 A_2^\dagger + A_2 A_0^\...
7
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0answers
162 views

A limiting sequence of positive definite matrices

Let $A\in\mathbb{R}^{n\times n}$ be a matrix with eigenvalues having (strictly) negative real part. Let $X\in\mathbb{R}^{n\times n}$, $X\succ 0$, be a positive definite matrix and let $P\succ 0$ be ...
3
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0answers
200 views

On the existence of fixed points of a matrix iteration

Let $A\in\mathbb{R}^{n\times n}$ be a Hurwitz stable matrix (i.e. all eigenvalues of $A$ have negative real part). Let $\succeq$ denote the standard partial order in the cone of positive semidefinite ...
0
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1answer
98 views

Matrix equation of the form $C A C^\intercal = D$

Consider the following square matrix \begin{align} A = \left(\matrix{d & 0 & -\frac12 & 0 & 0 & 0 & 0 & 0 \\ 0 & d & -d+1 & -\frac12 & 0 & ...
5
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2answers
309 views

On the existence of integer square root of a $3 \times 3$ positive definite matrix

As far as I know, a real square matrix $M$ has a real square root if $M$ is positive semidefinite, i.e., if all eigenvalues are nonnegative. And, in fact, its square root is unique. I have read some ...
3
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0answers
131 views

On a matrix inequality based on the Schur-Horn theorem

Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and (strictly) positive eigenvalues. (Notice that $A$ is not required to be symmetric.) Let $A_s$ denote the symmetric part of $A$...
2
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0answers
155 views

Space of change of basis matrices between two similar matrices - how to reduce it with additional tests?

Assume we have two real symmetric $n\times n$ matrices: $A, B$. We can easily test their similarity: $\textrm{Tr}(A^k)=\textrm{Tr}(B^k)$ for $k=1..n$. In this case both can be rotated to the same ...
9
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2answers
432 views

A trace-constrained maximization problem in the cone of positive definite matrices

Let $A\in\mathbb{R}^{n\times n}$ be a matrix having eigenvalues with strictly negative real part (in other words, $A$ is supposed to be Hurwitz stable). Let $\mathrm{tr}(\cdot)$ denote the trace ...
4
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3answers
1k views

Non linear matrix equation

I want to solve the following non linear matrix equation for $X\in\mathbb{R}^{N\times N}$: \begin{equation} XX^{\top}+ABX^{\top}-A=0 \qquad (1) \end{equation} For a given matrices $A\in\mathbb{R}^{...
0
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1answer
79 views

How to derive the solution of Tikhonov Regularization via SVD [closed]

The solution to Tikhonov Regularization is $$x=(A^HA+\sigma^2_{min}I)^{-1}A^Hb$$ where $\sigma^2_{min}$ is the minimum of the singular values of $A$. Then we apply $SVD$ to $A$ such that, $$A=U\Sigma ...
5
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1answer
674 views

Solving system of bilinear equations

Consider a collection of $m$ matrices $A_i$ of size $n\times n$, and a vector $b$ of size $m$. I want to solve the bilinear system $\{x^T A_i y=b_i:i=1,\dots,m\}$ in variables $x,y$. Is there an ...
6
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2answers
372 views

Is there any efficient solution of the matrix equation AXB + (AXB)' + cX = D

I am trying to find the symmetric solution $X\in \mathbb{R}^{p\times p}$ of following matrix equation: $AXB + (AXB)^T + cX = D$ where $A,B\in \mathbb{R}^{p\times p}$ are symmetric positive ...
1
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1answer
210 views

On a condition for a matrix sum to be zero

Let $\{Y_i\}_{i=1}^N\in\mathbb{R}^{n\times m}$ be a set of full column rank matrices ($\mathrm{rank}(Y_i)=m$ for all $i$) and $\{P_i\}_{i=1}^N\in\mathbb{R}^{m\times m}$ be a set of positive definite ...
2
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1answer
203 views

Solving a “reversed” Stein equation

Let $P$ and $Q$ be positive definite matrices. Consider the following matrix equation $$\label{star}\tag{$\star$} XPX^\top - P = -Q, \quad X\in\mathbb{R}^{n\times n}. $$ My question. Is it true ...
0
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0answers
30 views

Solutions to this equation of the form $A(t_1,t_2)x = b(t_2)$

Given two symmetric positive definite matrices $L_1, L_2\in\mathbb{R}^{n\times n}$ and a vector $w\in\mathbb{R}^n$, under what conditions does the following system of equations of the form $M_1(t_1,...
1
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0answers
35 views

Lower bounds on eigenvalues of Lyapunov solutions

Let $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$ and let $X\in\mathbb{R}^{n\times n}$, $X=X^\top>0$ be the solution of the following Lyapunov algebraic equation $$ AX+XA^\top=-BB^\top....
1
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2answers
133 views

Low-rank solution of generalized Sylvester equation

Let $n \in \mathbb{N}$. Let $A,B,C,D$ be non-singular $n \times n$ matrices. If the matrix pencils $A-\lambda C$ and $B-\lambda D$ are regular and have disjoint spectra, then $$AXB-CXD = 0$$ has a ...
3
votes
4answers
199 views

Coupled Sylvester equations

Let $n \in \mathbb{N}$. Let $A,B,C$ real matrices of size $n \times n$. Let $\alpha,\beta,\gamma,\delta \in \mathbb{R}^{4}$ such that $\alpha,\beta$ are non-zero. I an looking for two matrices $...
3
votes
1answer
133 views

A matrix monotonicity question

Let $X\in\mathbb{R}^{n\times n}$ be a positive semi-definite matrix and $A\in\mathbb{R}^{n\times n}$ be a stable matrix, i.e. a matrix whose eigenvalues are strictly inside the left-half complex plane....
2
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1answer
125 views

Quadratic matrix equation for $X\in \mathbb{C}^{ n\times p}$ with Hermitian parameters

Let $A\in \mathbb{C}^{ n\times n}$ and $B \in \mathbb{C}^{p \times p}$ be Hermitian matrices with $p < n$. Find matrix $X$ such that $X^*AX=B.$ Solution in the case of positive definite $A$ and $...
1
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0answers
164 views

A vanishing sum of symmetric matrices

Let $\{G_i\}_{i=1}^N\in\mathbb{R}^{n\times m}$ be a set of full column rank matrices (i.e., $\mathrm{rank}(G_i)=m$ for all $i$) and $\{P_i\}_{i=1}^N\in\mathbb{R}^{m\times m}$ be a set of positive ...
3
votes
1answer
642 views

Completing the square of a matrix expression

Let $A,C\in\mathbb{R}^{m\times n}$, $n\ge m$, $B\in\mathbb{R}^{n\times m}$, and $P$ be a real positive definite $m\times m$ matrix. Denote by $\mathcal{S}^n$ the space of $n\times n$ real symmetric ...
0
votes
1answer
106 views

A conjugation matrix $X\in \mathbb{C}^{ n\times p}$ where $p< n$

Given a Hermitian positive definite matrix $A\in \mathbb{C}^{n \times n}$ and a Hermitian matrix $B\in\mathbb{C}^{ p\times p},$ find the matrix $X$ so that $X^HAX=B$ holds where $X^H$ denotes ...