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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-2 votes
0 answers
128 views

Is this regression problem solvable? [duplicate]

I have a random vector $\pmb{x}=(X_1,...,X_p)^T\in \mathbb{R}^p$, a symmetric matrix $$\Theta = \left(\begin{matrix}0 & \theta_{12} & \theta_{13} & \cdots & \theta_{1p}\\ \theta_{12} &...
-1 votes
0 answers
77 views

How to find two sets of vectors which satisfy a set of matrix equations [migrated]

In my trial to solve a system of matrix equations, I wish to find two sets of non-zero vectors of $\mathbb{R}^3$ (which may be not unique) $\{ A_i \}$ and $\{ B_i \}$ where $i \in I$ (an index set, ...
4 votes
2 answers
121 views

What does the matrix-valued solution of $X = A X A^T + \operatorname{Id}$ look like?

Let $A \in \mathbb{R}^{n \times n}$ be an invertible contraction, i.e. all singular values are in $(0,1)$. By reformulating the equation \begin{align*} & X = A X A^T + \operatorname{Id} \tag{1} \...
9 votes
1 answer
598 views

One question on circulant $\pm1$ matrices

Let $n > 13$ be a positive integer. Is there any $n \times n$ circulant $\pm1$ matrix $A$ satisfying the following property $$AA^T=(n-1)I+J$$ where $I$ is the $n \times n$ identity matrix and $J$ ...
1 vote
1 answer
332 views

Solvability of $A X B=C$ with $X=X^\mathrm{T}$

I am studying symmetric solutions to the complex matrix equation \begin{equation} A X B=C, \end{equation} where $A$, $B$, and $C$ are $m\times n$, $n \times k$, and $m \times k$ complex matrices, ...
5 votes
3 answers
228 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 ...
6 votes
1 answer
175 views

Efficiently solve the Sylvester equation $AX+XA = C$ where $X$ is skew-symmetric

Is there a way (more efficient than the standard vectorization) to solve the following Sylvester equation in the skew-symmetric matrix $X$ $$AX+XA = C$$ where the matrix $A$ is symmetric positive ...
2 votes
0 answers
248 views

Two questions about three circulant matrices

Consider the following matrix equation in $n \times n$ circulant $\pm 1$ matrices $A$, $B$, $C$ $$2AA^T+BB^T+CC^T=(4n+4)I-4J$$ where $I$ is the $n \times n$ identity matrix and $J$ is the $n×n$ matrix ...
0 votes
0 answers
118 views

On a matrix equation with Kronecker product

Is there any work on the matrix equation in unknowns $X, Y \in {\Bbb C}^{n \times n}$ $$(X \otimes Y + Y \otimes X) \operatorname{vec}(A)=0$$ where $\otimes$ is the Kronecker product? Or, in general, ...
3 votes
1 answer
141 views

Does this matrix equation always have a solution?

Let $\{A_i, i\ge 3\}$ be the matrices whose columns represent numbers from $0$ to $2^i-1$ in binary form. For example, $A_3 = \begin{bmatrix} 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \...
1 vote
1 answer
232 views

Solve permutation matrix equations of the form: $X^T A X = B_1$ and $X A X^T = B_2$

I have a hard time solving the following two matrix equations for unknown permutation matrix $X \in \mathbb{R}^{n \times n}$: $$X^T A X = B_1$$ $$X A X^T = B_2$$ where, $A$, $B_1$ and $B_2$ are all $n ...
2 votes
1 answer
117 views

Is an almost-solvable linear equation with integer coefficients solvable?

Let $M$ be a fixed $m \times n$ rectangular matrix ($m > n$) with non-negative integer coefficients. Does there exist a pair $(R, \epsilon)$ with the following properties: If $b$ is a $m \times 1$ ...
1 vote
1 answer
151 views

Symmetric linear least-squares solution ${\bf X} {\bf A} = {\bf B}$

Given the wide matrices ${\bf A} \in {\Bbb R}^{n \times m}$ and ${\bf B} \in {\Bbb R}^{p \times m} $, where $m > n > p$, form an overdetermined linear system in ${\bf X} \in {\Bbb R}^{p \times n}...
6 votes
2 answers
631 views

Sprinkling signs in unitary matrices

Let $A$, $B$ be $n\times n$ unitary complex matrices, such that for all indices $i,j$ we have $|a_{ij}|=|b_{ij}|$. Does there then exist diagonal unitary matrices $D,D’$ such that $DAD’=B$? This can ...
1 vote
0 answers
142 views

Constrained trace optimization with relavance to optimal asset selection

Let $D$ and $Q$ be two real $m\times m$ diagonal matrices given $$ D=\left(\begin{array}{cccc} d_1 & 0 & \cdots & 0\\ 0 & d_2 & \cdots & 0\\ \vdots & \vdots & \ddots &...
0 votes
1 answer
97 views

Least square error problem ill conditioning

I am trying to understand why I am getting an almost singular matrix in a problem I have. The problem is a simple as $$ \min_{X \in \mathbb{R}^{m,n}} \left\lVert AX - B \right\rVert_F^2 $$ Obvioulsy ...
0 votes
0 answers
36 views

Does this recurent matrix sequence admit an explicit writing?

I have sequence defined by : 𝐏(n+1)=(𝐈−(Ф.𝐏(n).Ф′+𝐐).𝐇′.(𝐇.(Ф.𝐏(n).Ф′+𝐐).𝐇′+𝐑)^(−𝟏).𝐇).(Ф.𝐏(n).Ф′ +𝐐) Where : P(n), Q, R are square, NxN, symmetric, positive semidefinite. R is square, ...
0 votes
0 answers
70 views

Follow-up question regarding real singular matrices with additional details

After my question whose answer turned out to be false, I re-examined the course of my proof, which is actually seperate from the one in my question, and found out that there's another condition, at ...
3 votes
1 answer
517 views

Is the set of real matrices with at least one real logarithm closed under multiplication?

Let $S$ be the set of real matrices with at least one real logarithm. For some couple of its elements, for example those with at least (one real logarithm each with submultiplicative norm smaller than ...
3 votes
0 answers
138 views

Solvability of a matrix exponential equation - generalized matrix logarithm

For a given invertible real matrix $G\in \mathrm{GL}_d$ with $\det G>0$, we ask for a solution $B$ of the matrix exponential equation $$ G = \exp(B) \exp\bigl(\tfrac{1}{2}(B^T-B)\bigr) . $$ Basic ...
1 vote
1 answer
157 views

Solution to commutator equation in semisimple algebraic group

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Let $K$ be a field of characteristic zero, and $\SL_n$ and $\GL_n$ the special and general linear groups over $K$. Let $\Phi \in \GL_n(K), H \in ...
7 votes
1 answer
459 views

Is the set of purely real square matrices, that are complex-diagonalisable, dense in the set of real matrices?

A quick search for "diagonalisable matrix" on Wikipedia immediately gives the result that the set of real-diagonalisable matrices is not dense in the set of real matrices. I need, however, ...
8 votes
7 answers
1k views

One observation of special type of square matrix exponentiation

I was studying the following type of matrices, $$ A = \begin{pmatrix} 1 & x_{12} & \cdots &x_{1n}\\ 0 & x_{22} & \cdots &x_{2n}\\ \vdots\\ 0&\cdots&0&x_{nn} \end{...
1 vote
0 answers
61 views

Solve linear matrix equation involving convolution

I am facing following equation: $$ A * X + C \cdot X = D $$ with: $A, C, D \in \mathbb{R}^{n \times n}$ some known matrices without any particular structure, $X \in \mathbb{R}^{n \times n}$ the ...
2 votes
0 answers
56 views

any ideas on how to solve matrix equation like this $X A_i Y = B_i$

the objective function is like $$\operatorname*{argmin}_{X,Y} = \sum_i \|X A_i Y - B_i\|_F^2$$, and $A_i$ is a diagonal matrix I've tried gradient-descent, but as it turns out not well, I wonder if ...
6 votes
1 answer
1k views

Solve equation with matrix variable

I want to solve a matrix $\Omega$ from a equation $\sum_k (\Omega + \Theta_k)^{-1} = Q$. The $Q$ and $\Theta, \forall k=1,\ldots,K$ are known, and are positive definite matrices. $\Omega$ also has to ...
1 vote
1 answer
116 views

Is it possible to simplify the coefficient matrix for large values of $x$?

If I have a system of $8$ linear equations for the eight variables $\{\alpha ,\beta ,\gamma ,\delta ,\eta ,\lambda ,\xi ,\rho \}$ and with the three parameters $\{x,y,z\}$ reals and $x>0$. I want ...
1 vote
0 answers
28 views

Finding variance-minimizing weights [closed]

I'm trying to solve the following matrix calculus problem: $\text{argmin}_{v \in R_+^K}(v'\Sigma v) \hspace{0.5pc} \text{subject to} \hspace{0.5pc} 1'v=1$ where $\Sigma$ is a well-behaved (symmetric, ...
-3 votes
1 answer
414 views

Multiplication of a symmetric matrices [closed]

I'm wonder if the next claim is true or not: If A,B is a symmetric matrices over the real numbers, and A is PSD , B is PD implies than AB is PSD. PD - positive definite PSD - positive semidefinite If ...
1 vote
1 answer
228 views

Two unknowns: one vector, one scalar, one equation

I would like to know if this equation is solvable for $a$ and $\alpha$: \begin{equation} \Sigma = \Gamma + a \left( \alpha 1^\top + 1\alpha^\top \right) +a^2 b \end{equation} $\Sigma$ & $\Gamma$ ...
1 vote
1 answer
304 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} ...
4 votes
3 answers
4k 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}^{N\...
2 votes
0 answers
97 views

Gradient of QZ decomposition

Let $A$ and $B$ be an $m \times n$ matrix of rank $ k_1 \le \min(m,n) $ and $ k_2 \le \min(m,n) $. Then the QZ decomposition or the generalized Schur decomposition is $A = USV^T$ and $B = UTV^T $, ...
2 votes
2 answers
146 views

Orthonormal solution of overdetermined linear equations

I have a two matrices $A$ and $B$ in $\mathbb{R}^{m \times n }$ ($m \gg $ n) such that there exists an orthonormal matrix $X \in \mathbb{R}^{n \times n }$, such that: $$AX = B$$ Given that $X$ is ...
8 votes
1 answer
1k 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.
1 vote
0 answers
196 views

Solution that minimizes the sum of squared errors, with quadratic constraints

Given symmetric and positive definite $n \times n$ (real) matrices $A_1, \dots, A_m$ and $b_1, \dots, b_m \in {\Bbb R}^{n}$, I am trying to find the solution with the least sum of squared errors of ...
3 votes
2 answers
621 views

A truncated "geometric" matrix series

Let $A$ be an $n\times n$ matrix, $B$ be an $n\times m$ matrix, $C$ an $m \times m$ matrix, and consider the sum $$\sum_{k = 0}^{N-1} A^k B C^k.$$ Is there any smart way to rewrite this sum in a way ...
3 votes
1 answer
321 views

Solution to a Sylvester equation with positive definite coefficients

Consider the following Sylvester equation, where each of the known coefficient matrices ($A$, $B$, $C$) is symmetric positive definite and has dimensions $n \times n$ \begin{align*} C = A^TXA + B^TXB. ...
1 vote
0 answers
132 views

Identities for the determinant of a matrix similar to $\det(A)=\exp\circ\operatorname{tr}\circ\log(A)$ for different matrix functionals

The identity for the determinant of $A$ in the title is well know in matrix analysis and comes from the Jacobi's formula. I am interested in the existence of nontrivial formulas like this one (they do ...
3 votes
2 answers
325 views

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 ...
6 votes
2 answers
175 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 $B$ ...
0 votes
1 answer
134 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 ...
10 votes
3 answers
548 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 ...
1 vote
0 answers
84 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 ...
1 vote
2 answers
331 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 ...
16 votes
2 answers
9k views

Solving a quadratic matrix equation

This might be a well-known problem but I am having trouble to find this. For square matrices $X, A, B,$ how to obtain the general solution for $X$, for the quadratic matrix equation $X A X^{T} = B$ ? ...
1 vote
0 answers
149 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^...
1 vote
2 answers
269 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 \\ ...
1 vote
5 answers
611 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 ...
3 votes
1 answer
308 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 ...