# 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 $Ax=b$.

186
questions

-1
votes

0
answers

23
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### Correspondence of forms of the same solution of a system of linear equations [migrated]

I am currently reading Mathematics for Machine Learning book (p.27-28, freely available). And I am really confused with how two different forms of a solution of a system of linear equations correspond ...

4
votes

2
answers

125
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}
\...

1
vote

1
answer

333
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, ...

6
votes

1
answer

184
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

250
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

119
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

143
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

238
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 ...

1
vote

1
answer

152
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

633
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 ...

2
votes

1
answer

119
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

0
answers

143
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

101
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

37
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 ...

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 ...

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 ...

7
votes

1
answer

474
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

63
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 ...

9
votes

1
answer

600
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

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

482
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

229
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}
$\Sigma$ & $\Gamma$ ...

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

157
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 ...

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

633
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

325
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

133
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

338
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 ...

0
votes

1
answer

136
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 ...

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

336
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 ...

10
votes

3
answers

566
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

151
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 \\
...

3
votes

1
answer

313
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
votes

0
answers

173
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
vote

1
answer

195
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
votes

0
answers

47
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}^...

1
vote

0
answers

173
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 ...

0
votes

1
answer

312
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
votes

2
answers

218
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_{...

0
votes

1
answer

71
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
vote

1
answer

357
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 ...

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 ...