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 $Ax=b$.
187
questions
-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 ...