Questions tagged [eigenvalues-eigenvectors]
Eigenvalues are numbers associated to a linear operator from a vector space $V$ to itself: $\lambda$ is an eigenvalue of $T\colon V\to V$ if the map $x\mapsto \lambda x-Tx$ is not injective. An eigenvector corresponding to $\lambda$ is a non-trivial solution to $\lambda x - Tx = 0$.
14,651 questions
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How to use Exercise 2.1 to solve Exercise 2.5(a)? (Numerical Linear Algebra by Lloyd N. Trefethen and David Bau, III.)
I am reading Numerical Linear Algebra by Lloyd N. Trefethen and David Bau, III.
On p.16 in this book:
Exercise 2.5. Let $S\in\mathbb{C}^{m\times m}$ be skew-hermitian, i.e., $S^*=-S$.
(a) Show by ...
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Factorization of matrices
Given
$A\in \mathbb{R}^{n\times n}$ a diagonal matrix where all diagonal elements are positive
$B\in \mathbb{R}^{m\times n}$ a matrix with $m<n$,
$I_n \in \mathbb{R}^{n\times n}$ an identity ...
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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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Discrepancy in inverse calculated using GHEP and HEP
Say we have a matrix $A = L + \beta^{2} M$, where $\beta$ is a real scalar. The matrices $L$ and $M$ are symmetric positive semi-definite and symmetric positive definite respectively. I am interested ...
- 31
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Uniqueness of radial solution on the perturbed cylinder
Let $T>0$, $v\in C^{2,\alpha}(\mathbb{R}/T\mathbb{Z})$ whose norm is small enough, where $\mathbb{R}/T\mathbb{Z}$ denotes the circle of perimeter $T$. Define the perturbed cylinder $C_{1+v}^T$ by
$$...
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An alternative expression for $\frac{\left[zI-A\right]^{-1}}{z- \lambda }$
I am following a discrete controls theory course and one of the professor's theory slides states that if $I$ is the identity matrix and $\lambda$ is not an eigenvalue of matrix $A$, it can be shown ...
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how to construct a volume element of a coordinate system warped by a kernel function
Take a grid in an arbitrary number of dimensions. I construct a graph kernel to define the connectivity of the grid, and apply that kernel to the grid to create a weighted digraph.
I Construct the ...
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2
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Eigenvalues of a traceless matrix
It is well known that the eigenvalues $(\lambda_i)_i$ of a traceless square $n\times n$ matrix $M$ (with no other assumptions) check :
\begin{equation}
\sum_{i=1}^n \lambda_i = 0
\end{equation}
For $2\...
- 201
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Find Hermitian matrix from another matrix, preserving the real eigenvalues
Given a square matrix $M\in\mathbb{C}^{n\times n}$, I am looking for another matrix $P$ hermitian, i.e. $P=P^{\dagger}$, which has the same eigenvalues, or at least :
\begin{equation}
\lambda\in\...
- 201
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Assessing Positive Semi-Definiteness of Covariance Matrices in Random Cylindrical Shell Generation
I am working on generating random cylindrical shells using a multivariate Gaussian distribution. To construct the covariance matrix, I am employing the following function:
$$
K \left( \theta_L, z_L \...
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Proving that a matrix and its transpose have the same characteristic polynomial [duplicate]
I'm working on a linear algebra problem about characteristic polynomials and would appreciate some guidance.
For any square matrix $A$, prove that $A$ and $A^T$ have the same
characteristic ...
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I do not understand the rewriting of the scalar helmoltz equation into an eigenvalue problem done by this researcher using finite differences
I am trying to do what is done in the following article : A "“poor man’s approach” to modelling of micro-structured optical fibres". At some point in the article they consider the following ...
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Can LP calculate maximum/minimum eigenvalue of a matrix
It's not hard to show that the maximum eigenvalue of a matrix $A\in \mathbb{S}^n$ can be calculated through the following SDP:
\begin{align*}
\max&\ Tr(AX) \\
\text{subject to}& \ Tr(X) = 1\\
...
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Efficiently updating eigenpairs when bordering a symmetric matrix
Let $A\in\mathbb{R}^{n\times n}$ be a symmetric matrix. Consider the bordered matrix
$$
B(x, v) \;=\; \begin{bmatrix}
A & v\\[2pt]
v^{\top} & x
\end{bmatrix}\in\mathbb{R}^{(n+1)\times(n+1)}.
$$...
- 163
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Minimal eigenvalue of Gram matrix generated by orthonormal basis
Consider an orthonormal basis $(\phi_i)_{i\in\mathbb N}\subset L^2(\mathscr X)$, where $\mathscr X\subset \mathbb R^n$ is some compact Euclidean space; for simplicity, we may just take $\mathscr X=[0,...
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