Questions tagged [eigenvalues]
eigenvalues of matrices or operators
831
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What is the natural module?
Lemma 2.9 of [1]:
Let $\operatorname{char}(K) \neq 2 $ and let $G$ be $\operatorname{Spin}(m,K)$, $n=\operatorname{rank} G$, and let $V$ be the natural $m$-dimensional module. Suppose $f\in G$ and $f^...
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Is there a differential form which corresponds to an eigenvalue of the homomorphism in cohomology?
Let $M$ be a closed manifold and $f:M\to M$ be a diffeomorphism. Suppose the homomorphism $f^*:H^k(M;\mathbb R)\to H^k(M;\mathbb R)$ has an eigenvalue $\lambda\in\mathbb{R}$. Note that $\lambda$ is ...
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Maximal eigenvalue of a real symmetric Toeplitz matrix
The $n×n$ matrix $A_n$ is defined by the elements $a_{ij}=n−|i−j|$.
\begin{bmatrix}
n & n-1 & n-2 & \cdots & 1\\
n-1 & n & n-1 & \cdots & 2\\
n-2 & n-1 & n &...
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Eigenvalues of N×N correlation matrices as N tends to infinity
I want to find a 𝑁×𝑁 positive definite correlation matrix, which ensures that as 𝑁 goes to infinity, only a finite number of eigenvalues remain non-zero, while the rest eigenvalues approach zero.
...
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How to obtain the eigenvalues of the matrix $\left(\cos(i-j)\frac{\pi}{n}\right)_{i,j=1}^n$?
The eigenvalues of the skew-circulant matrix $A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^{\frac{n-1}{2}}$ have been successfully computed enter link description here.
Q:How to obtain the ...
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17
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Why does the random shift in the QR eigenvalue algorithm work in the non-symmetric case over the complex field
I tried to implement the QR algorithm for non-symmetric matrices with complex entries to show to my students. The main part of the implementation was standard: the Householder reduction to the ...
- 60.9k
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An example of a matrix whose eigenvalues fullfill 'No-resonance' condition
No-resonance for a matrix is defined in terms of its eigenvalue as (last para page-3 in ref.):
$$\lambda_i \neq \sum_{j=1}^N m_j\lambda_j;\ \forall m_j\in \mathbb{Z}\ \ and\ m_j\geq 0$$
$$such\ that\ \...
- 101
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Second largest eigenvalue of graph
Let $G$ be a connected $d$-regular n-vertex graph and let $k:= k(n)\in \mathbb{N}.$ Given a Non-empty set of vertices $\phi\neq B\subseteq V(G),$ how can I prove that all but at most $\frac{\lambda_2|...
- 177
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Function of eigenvalues of Laplacian matrix
Let $G$ be a simple $n$-vertex graph and let $\mu_n\geq\mu_{n-1}\geq\dots\geq\mu_1$ be the eigenvalues of its Laplacian matrix, how can I find a function $$f(\mu_1,\mu_2,\dots\mu_n) \text{ such that } ...
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3
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Low rank perturbation of non-Hermitian $A$, where all eigenvalues are real
Suppose $A,E$ are Hermitian $(n \times n)$-matrices and $E$ is of low rank. There are well known results bounding the difference in spectra of $A$ and $A+E$. For example the Eigenvalue Interlacing ...
- 1,253
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Reference request for non-banded Toeplitz matrix
I want to know references that treat the property of eigenvalues and eigenstates of the non-banded Toeplitz matrix.
I mean for example, the Toeplitz matrix $A$ whose matrix element is given by $A_{ij}=...
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2
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79
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Smallest eigenvalue of certain PD matrix decreases under sparse perturbation
Let $\omega_1<\dots<\omega_n\in\mathbb{R}$. Then, define $G\in\mathbb{C}^{n\times n}$ such that $G_{k\ell}=\frac{1}{1-i(\omega_\ell-\omega_k)}$. For example, if $n=3$ we obtain $$ G=\begin{...
- 83
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Initial guess in shifted QR algorithm
I'm comparing timings of two implementations of algorithms for the computation of Gauss-Legendre nodes.
1 - The first is a Newton algorithm to find the roots of the Gauss-Legendre polynomials. Quite ...
- 249
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Eliminating nullity for enhanced non-singularity
If we have an
$n\times n$ matrix $A$ with entries either $0$ or $1$, where all diagonal entries are $0$ and the rank is $k<n$, can we reach full rank by changing exactly $n-k$ zero off-diagonal ...
- 4,030
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How to show the following matrix has eigenvalues $-d,-d+1,...,d$?
Consider the following $(2d+1)\times (2d+1)$ matrix:
$$
A = \begin{pmatrix}
0 &\frac{2d}{2} & 0 &0 & \cdots &0 & 0 \\
\frac{1}{2} & 0 & \frac{2d-1}{2} &0& \...
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