# 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
Filter by
Sorted by
Tagged with
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\...
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
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 ...
15 views

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 ...
1 vote
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 vote
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^...
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 ...
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 ...
1 vote
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 ...
1 vote
25 views

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 ...
$M_2(\mathbb{R})$ 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 $$A^2+B^2= \begin{bmatrix}1402&&2022\\ 2022 &&... 0 votes 1 answer 41 views ### Understanding this step in finding the transition matrix There's such a problem about finding the transition matrix. Let \mathbf A=\begin{bmatrix}2&6&-15\\1&1&-5\\1&2&-6\end{bmatrix}, find non-singular matrix \bf P such that \... 0 votes 1 answer 69 views ### Partial derivatives of determinant I have recently took and exam and one of the exercises involved finding the partial derivatives of the function F: M_n(\mathbb{R}) \rightarrow \mathbb{R}, F(A) = \det (A), where M_n(\mathbb{R}) is ... 0 votes 0 answers 19 views ### Question about certain integral equality My professor proved Wigner's semi-circle law for random matrices today in class. In part of the proof, he claimed that$$\int dA \sum_{i,j}\Big(\frac{\partial}{\partial A_{ij}}+\frac{\partial}{\...
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\\ &...