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 $Ax=b$.
175
questions
0
votes
0
answers
31
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, ...
- 1
0
votes
0
answers
62
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 ...
- 151
3
votes
1
answer
503
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 ...
- 151
1
vote
1
answer
135
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 ...
- 29
3
votes
0
answers
123
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 ...
6
votes
1
answer
324
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, ...
- 151
8
votes
7
answers
969
views
One observation of special type of square matrix exponentiation
I was studying the following type of matrices,
$$
A = \begin{pmatrix}
1 & x_{12} & \cdots &x_{1n}\\
0 & x_{22} & \cdots &x_{2n}\\
\vdots\\
0&\cdots&0&x_{nn}
\end{...
- 271
1
vote
0
answers
49
views
Solve linear matrix equation involving convolution
I am facing following equation:
$$
A * X + C \cdot X = D
$$
with:
$A, C, D \in \mathbb{R}^{n \times n}$ some known matrices without any particular structure,
$X \in \mathbb{R}^{n \times n}$ the ...
- 11
2
votes
0
answers
55
views
any ideas on how to solve matrix equation like this $X A_i Y = B_i$
the objective function is like
$$\operatorname*{argmin}_{X,Y} = \sum_i \|X A_i Y - B_i\|_F^2$$, and $A_i$ is a diagonal matrix
I've tried gradient-descent, but as it turns out not well, I wonder if ...
- 21
9
votes
1
answer
307
views
One question on circulant $(-1,1)$-matrices
Let $n > 13$ be a positive integer. Is there any $n\times n$ circulant $(-1,1)$-matrix $A$ satisfying the following property:
$$AA^T=(n-1)I+J$$
where $I$ is the $n\times n$ identity matrix and $J$ ...
- 220
1
vote
1
answer
107
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 ...
- 111
1
vote
0
answers
25
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, ...
- 11
0
votes
0
answers
36
views
How to minimally perturb a matrix $A$ into $\hat{A}$ so that the inexact system of equations $Ax=b$, is exact on $\hat{A}x=b$?
Given that A is over-determined and, as written in the subject, how can I minimally perturb matrix $A$ into $\hat{A}$ so that the inexact system of equations $\hat{A}x=b$ is exact? The original ...
- 101
-3
votes
1
answer
85
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
0
votes
0
answers
30
views
Matrix optimization to find ideal embedding
Basically, I am trying to find the embeddings so I can approximate $K \approx M(\vec{\phi})$. The embeddings are for each one of my samples $\vec{\phi}(x_i) \in \mathbb{R}^D$ so I thought it should ...
1
vote
1
answer
205
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$ ...
- 13
2
votes
0
answers
80
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 $, ...
- 61
2
votes
2
answers
90
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$$
Given that $X$ is ...
- 123
1
vote
0
answers
130
views
Solution that minimizes the sum of squared errors, with quadratic constraints
Given symmetric and positive definite $n \times n$ (real) matrices $A_1, \dots, A_m$ and $b_1, \dots, b_m \in {\Bbb R}^{n}$, I am trying to find the solution with the least sum of squared errors of ...
- 11
3
votes
2
answers
432
views
A truncated "geometric" matrix series
Let $A$ be an $n\times n$ matrix, $B$ be an $n\times m$ matrix, $C$ an $m \times m$ matrix, and consider the sum
$$\sum_{k = 0}^{N-1} A^k B C^k.$$
Is there any smart way to rewrite this sum in a way ...
- 503
3
votes
1
answer
266
views
Solution to a Sylvester equation with positive definite coefficients
Consider the following Sylvester equation, where each of the known coefficient matrices ($A$, $B$, $C$) is symmetric positive definite and has dimensions $n \times n$
\begin{align*}
C = A^TXA + B^TXB.
...
- 33
1
vote
0
answers
113
views
Identities for the determinant of a matrix similar to $\det(A)=\exp\circ\operatorname{tr}\circ\log(A)$ for different matrix functionals
The identity for the determinant of $A$ in the title is well know in matrix analysis and comes from the Jacobi's formula. I am interested in the existence of nontrivial formulas like this one (they do ...
- 561
3
votes
2
answers
165
views
Solving linear matrix equation
Given matrices $A, B, C' \in \Bbb R^{2 \times 6}$, where $'$ denotes matrix transposition, and matrix $L \in \Bbb R^{2 \times 2}$, how can one solve the following linear matrix equation in $X \in \Bbb ...
- 31
0
votes
1
answer
126
views
Product of matrices equal identity
I need to solve the following equation for the matrix $P \in\mathbb{R}^{r\times d}:$
$$
((PAP^\top)^{-1} P S P^\top (PAP^\top)^{-1})^2 = I_r,
$$
where $S$ is a symmetric $d\times d$ matrix, $A$ is a ...
- 235
1
vote
0
answers
82
views
In matrix product, differentiate one element with respect to another element
Background
Consider a system (roughly) along the lines of those shown in Sims, C. A. (2002). Solving linear rational expectations models, where you have
$$ AX_{t+1} = CX_t + M $$
where matrix $M$ is a ...
- 11
1
vote
2
answers
282
views
Matrix equation $P^TAP=A$
Let $A\in \mathcal{M}_{m\times m}(\mathbb R)$ , $det(A)=1$ , $A$ is positively definite. Which matrices $P$ satisfy the equation
$$P^TAP=A$$
In fact I am interested in sequences of traces $tr P^n$ of ...
- 268
9
votes
3
answers
345
views
System of quadratic equations
Let $B_1,\ldots,B_s$ be $(s\times s)$ symmetric real matrices and $x=\left(x_1,\ldots,x_s\right)^\prime$ a $(s\times 1)$ vector of unknowns. Is there a way or reference theory for studying the ...
- 111
1
vote
0
answers
129
views
Optimization problem on trace of complex matrix product
Given a complex rectangular matrix $A$ $(k \times n)$, I am interested in solving the following optimization problem over $(k\times n)$ complex matrices $x$:
$$
\mathrm{arg}\max_X \,\mathrm{trace}(X^...
- 367
1
vote
2
answers
261
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 \\
...
- 113
3
votes
1
answer
260
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 ...
- 3,939
2
votes
0
answers
169
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 ...
- 235
1
vote
1
answer
145
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
$$A_i(\alpha) = (X\Lambda(\alpha) X^...
- 235
3
votes
0
answers
41
views
Stability of matrix equation
Let $M=I+A\in \mathbb{R}^{n\times n}$ for a skew-symmetric matrix $A$ with $\|A\|<1$ in the spectral norm. Using the $LU$-decomposition of $M$, it is easy to construct a solution $L,U\in \mathbb{R}^...
- 435
1
vote
0
answers
125
views
What is the solution of the matrix equation $A X + X A' + B X B' + C = 0$ for $X$?
I know that the matrix equation $A X + X A' + C = 0$ is in the form of the time-continuous Lyapunov equation, so solving for $X$ is pretty trivial since the solution already and numeric solvers ...
- 11
0
votes
1
answer
280
views
Solving two quadratic matrix equations [closed]
Given $10 \times 10$ matrices $A$ and $B$, I would like to find $10 \times 10$ matrix $X$ such that
$$A = X B X^T \tag{1}$$
$$B = X A X^T \tag{2}$$
How can I solve the issue? if there is a way to ...
- 11
3
votes
2
answers
208
views
A problem about determinant and matrix
Suppose $a_{0},a_{1},a_{2}\in\mathbb{Q}$, such that the following determinant is zero, i.e.
$
\left |\begin{array}{cccc}\\
a_{0} &a_{1} & a_{2} \\
\\
a_{2} &a_{0}+a_{1} & a_{1}+a_{...
0
votes
1
answer
63
views
State-dependent positive definite matrix
Consider a function $f(\mathbf{x})=\mathbf{M}_\mathbf{x}$ that outputs a nonsymmetric matrix $\mathbf{M}_\mathbf{x} \in \mathbb{R}^{N \times N}$ given an input vector $\mathbf{x} \in \mathbb{R}^N$.
Is ...
- 1
1
vote
1
answer
265
views
How do you solve this quadratic matrix equation?
could you please help me solve this quadratic matrix equation? I look around, seems like there is no general solution for it..
$$-BX^2 + X - C = 0$$ for X, B and C are (3x3) matrices. B and C are ...
- 11
27
votes
1
answer
2k
views
Integer matrices which are not a power
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}$In a group $G$, an element $g$ is said to be primitive if there is no $h \in G$ and integer $n >1$ such that $g = h^n$. (For clarification, I ...
5
votes
3
answers
190
views
Determine unknown matrix function of particular form from known points
I encountered the following problem recently in a practical context.
Fix $n \ge 1$.
Suppose $f$ is an unknown function $\mathbb C ^ {n \times n} \to \mathbb C ^ {n \times n}$ of the form
$$ X \mapsto ...
3
votes
0
answers
104
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 ...
- 31
2
votes
1
answer
1k
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 ...
- 23
1
vote
0
answers
56
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 $\...
- 4,906
0
votes
1
answer
100
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\...
- 704
1
vote
1
answer
127
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 $x_0\in \mathbb{C}$ ...
- 1,398
1
vote
2
answers
104
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 ...
- 365
4
votes
3
answers
222
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 ...
- 2,662
1
vote
1
answer
102
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 $\rho_i(A(\lambda))$ ...
- 704
6
votes
2
answers
172
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 $B$ ...
- 101
8
votes
1
answer
938
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.
- 329