Questions tagged [matrix-equations]
This tag is for questions related to equations, with matrices as coefficients and unknowns. A matrix equation is an equation in which a variable stands for a matrix .
4,300
questions
0
votes
1
answer
39
views
Inequality for the trace of the hat matrix in Ridge regression
I was recently reviewing a research paper and came across an inequality expressed as follows:
\begin{align}
& \text{tr}\Big[\Big(\frac{1}{np}X^\top X + \rho B\Big)^{-1} \Big(\frac{1}{np}X^\top X\...
- 41
0
votes
0
answers
16
views
How can i solve this optimization problem effectively?
Recently, i met an optimization problem
$$ \arg \min_{\mathbf{x}}\Vert \mathbf {Kx} - \mathbf{y} \Vert^2_2+\frac{\eta \Vert \mathbf{Dx} -\mathbf d \Vert_2^2 }{\Vert \mathbf{Dx} \Vert_2^2} $$
from ...
1
vote
1
answer
55
views
Kernel of kronecker product of matrices
Consider a matrix $E \in \mathbb{R}^{m \times n}$ with $m\geq n$ and a nullspace $\text{ker}(E) = \{ \alpha 1_n , \, \alpha \in \mathbb{R} \}$, where $1_n$ is a column vector of ones of appropriate ...
- 168
0
votes
0
answers
15
views
How to obtain a lower triangular matrix from a vector of coefficients
I have a vector
$$
\mathbf{x} = (x_{21},x_{31},x_{32},x_{41}, x_{42}x_{51} , x_{52} , x_{53} , x_{54})^T
$$
and I need to find a way, using some matrix operations, to derive this matrix here
$$
\left(
...
- 167
-1
votes
0
answers
14
views
Using the inverse matrix, solve the matrix equation (i.e. determine the matrix X), (I – matrix unit)
exercise
A =614/3 B =3 C = 4 D = 25 E = 227 F = -73 G =22 H = 18
1
vote
1
answer
50
views
Solving $X A X^T = I$ subject to $b X = B$
I am facing the following system of matrix equations:
$$ b X = B $$
$$ X A X^T = I $$
where $X$ is the square matrix of dimension $N \times N$ to solve for, $b$ and $B$ are both row vectors of ...
- 148
0
votes
1
answer
87
views
Theorem 3.15 Linear Algebra Friedberg, Insel and Spence how is my counterexample wrong?
$\newcommand{\rank}{\operatorname{rank}}$
Part (a) of Theorem 3.15 is: let $Ax=b$ be a system of $r$ nonzero equations in $n$ unknowns. Suppose that $\rank{A}=\rank{A|b}$ and that $(A|b)$ is in ...
- 129
0
votes
0
answers
16
views
Restructuring a Discrete Equation System for Compact Representation
I have the following system of equations for $i,j=1,...,n$:
$u_{i-1,j}+u_{i+1,j}+u_{i,j-1}+u_{i,j+1}-4u_{i,j} = f_{i,j}$
The right hand side $f$ is known, as well as $u_{0,j},u_{n+1,j},u_{i,0},u_{i,n+...
- 11
0
votes
0
answers
20
views
Custom block-wise matrix
I have two block matrices $A$ and $B$ defined as:
\begin{align}
A =
\begin{pmatrix}
A_{11} & A_{12} & A_{13} & A_{14} \\
A_{21} & A_{22} & A_{23} & A_{24} \\
A_{31} & A_{...
- 33
0
votes
0
answers
27
views
Invertibility of the product of matrices when the norm is less than 1
I am reading through a book about numerical linear algebra. It is challenging but I understand the concepts after some research. However, there is one thing that I can't figure out and it's about the ...
- 25
0
votes
0
answers
53
views
Why the following objective function can be optimized?
I try to minimize an objective function as shown below
$\begin{gathered}\mathop {\min }\limits_{\bf{\beta }} {\rm{ }}L = \frac{1}{2}{\left\| {\bf{\beta }} \right\|^{\rm{2}}} + {{\bf{W}}_{\bf{S}}}\frac{...
- 1
0
votes
1
answer
55
views
Sylvester-like equation solution
I would like to solve a coupled matrix differential equation.
All are $2\times 2$ matrices. Then, I have
\begin{align}
&\dot{X}=-i (A X - X A)-\eta (B Y - Y B);\\
&\dot{Y}=-\kappa Y +(B ...
- 99
-1
votes
1
answer
84
views
Please, help me to solve this exercise. I have no idea what method or approach I should use to solve this system of equations. [closed]
Find the values of the parameter $p$, for which the given system is non-singular:
$x-2y+pz=1$
$2x-py+z=0$
$-x+y-5z=p$
- 19
1
vote
1
answer
122
views
If $A^TB$ is a symmetric matrix, then $XA=B$ has a symmetric matrix solution
Let $A,B\in\mathcal{M}_{m×n}(\mathbb{F})$ be rectangular matrices ($m\le n$) over an arbitrary field $\mathbb{F}$, such that $A$ is of full row rank. Moreover, $\require{enclose}
\enclose{...
- 1,459
0
votes
0
answers
37
views
Matrix versus component gradient descent optimization
I have 4 vectors $x_1$, $x_2$, $x_3$, and $x_4$. Let $x=[x_1^T, x_2^T, x_3^T, x_4^T]^T$. Then I have 6 matrices $A_{12}$, $A_{21}$, $A_{23}$, $A_{32}$, $A_{24}$, and $A_{42}$. Then, I define the block ...
- 33
1
vote
1
answer
41
views
Matrix commutativity, joint diagonalization, subspaces
I have two square matrices $A= U_a \Sigma_aU_a^\top $ and $B= U_b \Sigma_bU_b^\top$ of same dimension.
I have that $U_a^\top B U_a$ is a diagonal matrix. Can I conclude than that $U_a^\top U_b= I$?
...
- 848
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votes
0
answers
102
views
Calculating matrices of an endomorphism given 2 matrices
Consider the endomorphism $f: E \to E$ and the basis $B_E$ and $B'_E$ along with the matrices $A = M(f, B_E, B'_E)$ and $P = M(B_E, B'_E)$
How could i get the matrices: $B = M(f, B'_E)$, $C = M(f, B'...
- 1
2
votes
0
answers
20
views
How to solve the matrix equation $\sum_iA_i^TA_iXB_iB_i^T=\sum_iA_i^TC_iB_i^T$ efficiently?
How would I go about solving the matrix equation $\sum_iA_i^TA_iXB_iB_i^T=\sum_iA_i^TC_iB_i^T$ for $X$?
The simplest thing to do would be to, of course, consider $Y=\sum_iA_i^TC_iB_i^T$, vectorise and ...
0
votes
0
answers
47
views
Inverse of block matrix which satisfies some ODEs
Let $A(t) \in \mathbb{R}^{2n\times 2n}$, $B(t) \in \mathbb{R}^{n\times n}$ be smooth and bounded. Suppose $\Phi(t) \in \mathbb{R}^{2n\times 2n}$ solves $d\Phi(t)=A(t)\Phi(t)dt$ and $\Phi(0)=I_{2n}$ ...
- 580
0
votes
0
answers
26
views
Can this matrix expression be simplified?
Consider the following matrix function (which is related to the Kalman filter associated with a Gauss-Markov dynamical system)
$$
g(X) = X - XC^T \Gamma^T (\Gamma C X C^T \Gamma^T + R)^{-1} \Gamma C X
...
- 1,235
0
votes
0
answers
21
views
Solve VAR(2) for the n-step ahead forecast
I'm trying to find for this VARX*(2)
$$x_t=a_0+a_1t+F_1x_{t-1}+F_2x_{t-2}+\Theta_0d_t+\Theta_1d_{t-1}+\Theta_2d_{t-2}+\varepsilon_t$$
an explicit form for $x_{T+n}$, i.e. solve it as an equation for ...
- 25
1
vote
2
answers
125
views
Solve the matrix equation for $X$
Solve $X^{7}=\begin{pmatrix} 1 & 0 & 1 \\
0 & -1 & 0 \\
0 & 0 & 1 \\
\end{pmatrix}$.
We dont know the field where the entries of $X$ are. The matrix $A=X^{7}$ is not ...
- 1,191
1
vote
0
answers
58
views
Two sets of matrix conditions. Are they equivalent?
I have reasons to believe that for any four square matrices $A, B, C, D \in \mathcal{M}_{n}(\mathbb{R})$, and $\lambda \neq 0$, the following is true:
\begin{equation}
\left.\
\begin{aligned}
A^TD-C^...
- 23
3
votes
1
answer
54
views
Which matrix operation is occuring?
I am currently going through a financial book that covers some matrix multiplication but I am not certain which operation is occurring to get this result. The book provides spreadsheets so I can see ...
- 31
5
votes
2
answers
125
views
Least squares solution to underdetermined Lyapunov equation
I need to solve an underdetermined Lyapunov equation for unknown $n\times n$ matrix $X$.
$$AX + XA = B$$
The naive method is to vectorize $x=\operatorname{vec}(X)$ and use a least squares solver on ...
- 4,953
1
vote
1
answer
85
views
Solving a Nasty Matrix / Tensor Algebra Problem
I need to solve the following:
$$
\sum_{i,c,d} x_{i,a} x_{i,b} x_{i,c} x_{i,d} W_{c,d} = \sum_{i} x_{i,a} x_{i,b} y_{i}
$$
for known x and y, and W is symmetric. It is safe to assume that the elements ...
- 11
1
vote
0
answers
25
views
$(A\cdot x)\cdot (B\cdot x)=(A\cdot x)\cdot (B\cdot x)= (B\cdot x)^t * (A \cdot x)=(x^t \cdot B^t) * (A\cdot x)=x^t \cdot (B^t * A)\cdot x$ Correct?
Let $A,B$ be $m\times n$-matrices and $x\in \mathbb{R}^n$ then
$(A\cdot x)\cdot (B\cdot x)=(A\cdot x)\cdot (B\cdot x)= (B\cdot x)^t * (A \cdot x)=(x^t \cdot B^t) * (A\cdot x)=x^t \cdot (B^t * A)\cdot ...
1
vote
1
answer
59
views
Find unknown $5\times 5$ matrix
I am looking for a way to find a $5\times 5$ matrix. The matrix is used to transform colors in a photo. The matrix transforms red, green, blue, and opacity values into new values like this :
$$\begin{...
- 143
1
vote
1
answer
25
views
How to solve matrix equations with inconsistent dimensions？
I'm confused by something in one of the papers I'm reading, but I'm not sure if there's any math in it that I'm not aware of.
$\mathbf{E}=\mathbf{R}\mathbf{e}$, where $\mathbf{E}\in\mathbb{C}^{n_1\...
- 11
2
votes
0
answers
53
views
Block matrix rank inequality
I encountered an exercise (not my homework)
Let $A,B,C$ be matrices and
$$
M=\begin{pmatrix} A & C \\ 0 & B \end{pmatrix}
$$
then
$$
\mathrm{rank}~M\ge \mathrm{rank}~A+\mathrm{rank}~B
$$
...
- 407
-2
votes
0
answers
30
views
For a given natural number $n$, determine all matrices $A ∈ R^{n,n}$ such that $A·B = B·A$ for each matrix $B ∈ R^{n,n}$. [duplicate]
Don't know how to start solving this. Of course I can just check some $n$, but I don't think it is good. Any ideas?
- 25
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votes
0
answers
19
views
The sigular case of matrix logarithm and exponential for rotation matrix
This is the continuous question of how to compute Riemannian logarithm and exponential on SO(3)
According to Terse Notes on Riemannian Geometry (Example 4.7.), the matrix exponential w.r.t. $SO(3)$ is
...
- 165
0
votes
0
answers
29
views
Simplifying inversion
Issue: I have two random column vectors $a,b\in\mathbb{R}^L$ and would like to simplify the following expression:
$$
\begin{bmatrix}
\mathbb{E}[aa^\top] & \mathbb{E}[bb^\top]
\end{bmatrix}
\begin{...
- 730
0
votes
1
answer
43
views
Solve matrix differential equation around known solution
Define a matrix differential equation $\dot{X}=A(t)X(t)$ with initial condition $X(0)$, where $X=[x_1,x_2,\ldots]^T$ is a 1D vector and $A(t)$ is a complex-valued time-dependent matrix.
This system ...
- 99
1
vote
0
answers
57
views
Suppose $A \in \mathbb{R}^ {m \times n} $ has a zero null space. Show that $A$ is left invertible and $A^TA$ is positive definite.
I am stuck with the following question.:
Suppose $A \in \mathbb{R}^ {m \times n} $ has a zero null space. Show that $A$ is left invertible and $A^TA$ is positive definite.
How do I show that $A$ has ...
- 455
0
votes
1
answer
62
views
Linear Algebra THEOREM 4
I am a university student studying Linear Algebra. I read the book Linear Algebra by David C. Lay and wonder about d. of Theorem 4
Let $A$ be an $m \times n$ matrix. The following statements are ...
- 1
-1
votes
0
answers
42
views
What is the basis when we subtract out a variable from a system of equations?
For example, if we've row-reduced a matrix and converted back to equations, so that we're left with
$x_1 + x_4 = 0 \implies x_1 = -x_4 $
$x_2 - 2x_4 = 0 \implies x_2 = 2x_4 $,
what do we then say is ...
- 1
1
vote
0
answers
41
views
Spectrum of a discrete-time observability Gramian from system matrices
Suppose that $A$ is an asymmetric matrix that has all eigenvalues inside the unit circle. Let $Q$ be a symmetric, positive semidefinite matrix.
Let $W=\sum_{t=0}^{\infty} (A’)^t Q A^t$ a discrete time ...
- 162
0
votes
1
answer
25
views
Derivative of an exponential matrix that involve trace operator
I know this is a stupid question, but I got this weird doubt. Then I need to derive this expression with respect to each element of the $\Sigma$ matrix
$$
\varphi(t) = \exp\big(Tr[A(t)\Sigma] + C(t)\...
- 25
1
vote
0
answers
65
views
When $\text{exp}(A) = B$ has a solution
Let $e^A$ denote the matrix exponential, for $n \times n$ matrices over $\mathbb{C}$. I am trying to find for which $B$ there exists a solution $A$ to the equation $e^A = B$.
Clearly a necessary ...
- 374
1
vote
0
answers
47
views
relative error of p-norm of a perturbated matrix
I am working on a question related to sensitivity of linear system, and the following step is difficult to prove.
my work:
$\tilde{a}_{ij}-a_{ij}=a_{ij}\epsilon_{ij}$
then, $\tilde{A}-A=A_{\epsilon}$ ...
- 11
0
votes
1
answer
28
views
Matrix reduction for large data set to solve linear equations
I've got a large set of linear equations at work to try to solve. These could be 200-300 rows of linear equations and 100-150 variables. Some of these variables won't be solvable, ie the linear ...
- 1
0
votes
0
answers
16
views
Solving a divergence equality
Consider the equality:
$$
\mathbf{g}(\mathbf{x}, t)\mathbf{g}(\mathbf{x}, t)^\top \nabla_\mathbf{x} H(\mathbf{x}, t)^\top = - \frac{1}{2} \nabla_\mathbf{x} \cdot \left( \mathbf{G}(\mathbf{x}, t)\...
- 129
0
votes
0
answers
18
views
Solving linear equation with quadratic hadamard product
I have the following equation that I can solve iteratively, but I was wondering if it has an analytical solution that can speed up my computation.
$$ x = B(x \odot x \odot b) + a $$
where
$x, a, b \...
- 11
0
votes
0
answers
52
views
Find all solutions of the system of equations depending on the parameters $a, b \in \mathbb{C}$.
I have a system o linear equations dependent on parameters $a,b \in \mathbb{C}$. I am not quite sure what to do with it. My only intuition is, that if I were to use Gaussian elimination to get the ...
- 15
4
votes
2
answers
175
views
Given $A^2+B^2=\left(\begin{smallmatrix}1402&2022\\2022&1402\end{smallmatrix}\right)$ for which $A,B\in M_2(\mathbb{R})$, show that $AB\neq BA$
$M_2(\mathbb{R})$ is the set of all $2\times2$ matrices that their entries are in $\mathbb{R}$. Now consider $A,B\in M_2(\mathbb{R})$. We have
$$A^2+B^2=
\begin{bmatrix}1402&&2022\\
2022 &&...
- 543
0
votes
1
answer
41
views
Understanding this step in finding the transition matrix
There's such a problem about finding the transition matrix.
Let $\mathbf A=\begin{bmatrix}2&6&-15\\1&1&-5\\1&2&-6\end{bmatrix}$, find non-singular matrix $\bf P$ such that $\...
- 841
0
votes
1
answer
69
views
Partial derivatives of determinant
I have recently took and exam and one of the exercises involved finding the partial derivatives of the function $F: M_n(\mathbb{R}) \rightarrow \mathbb{R}, F(A) = \det (A)$, where $M_n(\mathbb{R})$ is ...
- 1,029
0
votes
0
answers
19
views
Question about certain integral equality
My professor proved Wigner's semi-circle law for random matrices today in class. In part of the proof, he claimed that
$$\int dA \sum_{i,j}\Big(\frac{\partial}{\partial A_{ij}}+\frac{\partial}{\...
- 1,057
3
votes
1
answer
55
views
Matrix exponentials of Jordan blocks
Let $f:\Bbb C→\Bbb C$ be an analytic function1. Let $\lambda \in\Bbb C$ and let $$
J=\begin{pmatrix}
\lambda & 1 & & &\\
&\ddots& \ddots \\
& & \ddots& 1\\
&...
- 81