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 .
4,472 questions
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How to Interpret Detection Errors and Compute Performance Metrics for an Autonomous Pedestrian Detection System? [duplicate]
I am analyzing the performance of an autonomous vehicle’s pedestrian detection system, and I want to ensure that I am interpreting the scenario correctly in terms of confusion-matrix components. This ...
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votes
1
answer
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Solving $m(x)=A m(x)+B m(-x)+C n(x)$ and its partner equation
I am working with two vector-valued functions $m(x)$ and $n(x)$ that satisfy the coupled relations
\begin{equation}
x m(x) = A\,m(x) + B\,m(-x) + C\,n(x),
\end{equation}
and
\begin{equation}
x m(...
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What is the codomain basis in the “functional view” of a system of linear equations?
Let $T : V \to W$ be a linear transformation between finite-dimensional vector spaces.
Given ordered bases $B_V = { v_1, \dots, v_n }$ of $V$ and $B_W = { w_1, \dots, w_m }$ of $W$,
the matrix ...
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votes
1
answer
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Sufficient conditions for representing the matrice $U$ as $B^\top B^{-1}$
Out of curiosity, I came across the study of matrices of the form $U = B^\top B^{-1}$ for $B \in GL(n,\mathbb{F}_2)$. In essence, U represents how matrice $B$ is asymetric.
This raised the question: ...
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2
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Can a homogenous first order matrix difference equation of this form be represented with closed form expressions in all cases?
I am dealing with a homogenous first order matrix difference equation of the following form:
$$
\begin{bmatrix}
u_{n+1} \\
u_{n} \\
\end{bmatrix}=\begin{bmatrix}
c_1 & c_2 \\
1 & 0 \\
\end{...
- 143
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58
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Eigenvalues of self-adjoint, off-diagonal, block matrix
There are many posts about the eigenvalues of
$X_2 =
\begin{pmatrix}
0_{n_1 \times n_1} & A_{n_1 \times n_2} \\
A^*_{n_2 \times n_1} & 0_{n_2 \times n_2}
\end{pmatrix}$.
Are there any ...
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2
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Matrix calculus chain rule for MSE gradient: Why the transpose?
I'm working through the derivation of the gradient for the Mean Squared Error (MSE) loss function in a standard linear regression model and have a question about the application of the chain rule in ...
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Finite Difference Discretization of $y'' + xy = 1$ with $N=4$: Forming $A$ and $b$
Problem:
Given the ODE-problem
$\frac{𝑑^2𝑦}{𝑑𝑥^2}$ + 𝑥𝑦 = 1, 𝑦(0) = 1, 𝑦(1) = 0
Discretize with the finite difference method (FDM) the problem on a grid $𝑥_𝑖 =
𝑖ℎ, 𝑖 = 0,1,2, ... , 𝑁$ ...
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1
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Transformation between curves, each having same endpoints and peak magnitude
Given are two known continuous functions where 0 <= x <= 1 and f(x1) = g(x2) = 1. Say that x1 = 0.95 and x2 = 0.5. An illustrative example:
Can a function h(x) be determined such that f(x) ...
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Determinant (eigenvalues) of the difference of a diagonal matrix and a skew-symmetric matrix.
We define $A$ to be a diagonal matrix with diagonal entries: one $p_1$ and $2n-1$ many $p_2$, here both $p_1$ and $p_2$ are real and $p_1p_2>-1$. $J$ is a skew-symmetric matrix of form
$$J=
...
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1
answer
89
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What can we say about $b(n,k,\alpha)$?
Fix positive integers $n, k$ and fix $\alpha \in [0,1]$. Let $b(n, k, \alpha)$ be the smallest integer such that for every non negative integer $n \times k$ matrix $A$, there exists a set of row ...
- 107
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Interpretation of convergence in distribution
I came across with the following situation. Let $X$ be a random vector such that $X\sim MVN(\mu_n,C_n-B_n^{\top}A^{-1}B_n^\top)$ (multivariate normal distribution), such that the elements of $\mu_n$ ...
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Does diagonalization of a matrix imply that the eigenvectors are unique to a given form of a matrix? [duplicate]
If you have a collection of n (nonzero and unique) eigenvectors, is there a way to find a general form of an n-by-n matrix that corresponds to them in such a way that 'rules out' alternative forms?
...
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2
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53
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Can you check if a collection of eigenvectors are unique to a matrix of some form? [closed]
If you have a collection of n (nonzero and unique) eigenvectors, is there a way to find a general form of an n-by-n matrix that corresponds to them? In addition, is it possible to check for some kind ...
- 117
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votes
2
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Why does $3^n-2^n=\sum_{k=0}^{\left\lfloor{n/2}\right\rfloor}(-1)^k{n-k-1\choose k}5^{n-2k-1}6^k$?
I was exploring a method for finding higher powers of $2 \times 2$ matrices without using diagonalization. I used the Cayley-Hamilton Theorem, but my question is about solving the recurrence relation. ...
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