All Questions
Tagged with eigenvector eigenvalues
133 questions
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Simple Spectrum of Jacobi matrices
I want to call a matrix a Jacobi matrix (cause there may be different notions of Jacobi matrices) if it is a tridiagonal matrix with positive off-diagonal entries. Now, I read that the spectrum of ...
- 21
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0
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184
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Oja's rule gives unit eigenvectors
Does Oja's rule for normalized Hebbian learning always result in a unit eigenvector which corresponds to the largest eigenvalue? Or are there any specific conditions or assumptions under which this is ...
- 101
3
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3
answers
3k
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Analytic perturbation of the eigenvalues/eigenvectors of non-Hermitian matrix
Consider a matrix function $A(x)$, analytically depending on single parameter $x$.
Consider the eigenvalue/eigenvector pair of $A(0)$, namely $\lambda(0)$ and $w(0)$.
The question is whether we can ...
- 175
3
votes
0
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220
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Eigenvalues and eigenvectors of nonsymmetric complex tridiagonal matrix
I wonder if it is possible to find analytically all eigenvalues and eigenvectors of the following $2n \times 2n$ non-symmetric complex tridiagonal matrix
$$M = i \begin{pmatrix}
0 & a & 0 &...
2
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0
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463
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Conditions for continuity of non-simple eigenvectors
Here, https://math.stackexchange.com/a/1146455, it is noted that eigenprojections are continuous, but eigenvectors are not. Are there any conditions where the eigenvalues are not simple, but the ...
- 37
1
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286
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Generalized eigenvalue problem with nonnegative eigenvector constraint
Consider the following problem that is known to be non-convex but can be solved as a generalized eigenvalue problem (i.e. has a global optimum solution):
$\underset{w}{\text{maximize}}\quad w^{\top}...
- 11
5
votes
2
answers
2k
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rank-one perturbation of a matrix corresponding to a specific spectrum
Let $A$ be a real symmetric matrix whose spectrum is $\lambda_1,\lambda_2,\ldots,\lambda_n$.
Let $A'$ be the matrix obtained by adding a perturbation to $A$. The requirement is that only the second ...
- 111
2
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0
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148
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Commutation relation and eigenvectors of infinite matrices [closed]
I'm given the Matrix $A$ and $A^T$:
$A =
\begin{bmatrix}
0 & 1 & 0 & 0 & \dots \\
0 & 0 & \sqrt{2} & 0 & \dots \\
0 & 0 & 0 & \sqrt{3} & \...
- 121
6
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2
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3k
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Eigenvalues of a Symmetric Positive Semi-Definite (PSD) matrix after rank one update
I have a Symmetric Positive Semi-Definite matrix $A$ which i know its eigenvalue and eigenvectors. let $v$ and $u$ be a random column vector. i want to know if it is possible to have eigenvalues of ...
- 159
3
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2
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1k
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Eigenvectors and eigenvalues of Tridiagonal matrix with varying diagonal elements
is it possible to analytically evaluate the eigenvectors and the eigenvalues of a tridiagonal $n\times n$ matrix of the form :
\begin{pmatrix}
1 & b & 0 & ... & 0 \\\
b & 2 &...
- 31
1
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0
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537
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Epsilon-net of operator norm ball around Identity
Suppose I look at the set of matrices which are invertible and satisfy
$$
\left\|A-Id\right\|_{op}<r
$$
for some $r<1$, where $Id$ is the $n\times n$ identity matrix. An $\epsilon$-net of such ...
- 205
3
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2
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4k
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Singular Value Decomposition of Noisy Matrices
I am an engineer who makes measurements of a variable over a grid
of, say, $m\times n$. Since these are actual measurements, the true
values are always corrupted by noise, and what I measure is a ...
3
votes
1
answer
561
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Reducing eigenvalues of symmetric PSD matrix towards 0: effect on ratios of original matrix elements?
Let $\boldsymbol{S}$ be $k \times k$ positive semi-definite real symmetric matrix with eigen decomposition $\boldsymbol{S} = \boldsymbol{X} \boldsymbol{\Lambda} \boldsymbol{X}'$ ($\boldsymbol{\Lambda}$...
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2
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2
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564
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Linear dynamical systems: interpretation of Frobenius eigenvector
Consider a positive linear dynamical system. $\frac{dx}{dt}=Ax$, where $A$ is a quasipositive/Metzler/essentially nonnegative matrix. By its properties, the vector $x$ will remain positive for all ...
- 143
4
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84
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Matrices with almost constant coefficient have a simple eigenvalue
As a by-product of a general result for bounded operators of a Banach space, I have the following:
A matrix $L=(\ell_{ij})_{ij}$ that has almost constant coefficients in the sense that for some $c$,...
- 14.4k
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1
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106
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Eigenvalue-related statements [closed]
(I understand this question might not be appropriate for this website, but it has been asked on MathStackexchange and did not receive any replies even with a bounty)
How can I prove that the ...
- 103
0
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2
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737
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Eigenvalues of an amplification matrix
Let $A$ and $B$ square real matrices.
I know that the matrix $A+B$ has 1 as eigenvalue of multiplicity 1 and the others eigenvalues have their modulus <1.
Can we say something about the eigenvalues ...
- 1
2
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1
answer
474
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When is there a solution to these coupled eigenvalue equations?
I am trying to find the fixed point of a dynamical system, which requires solving two coupled eigenvalue-like equations. These equations are, in general, overconstrained. I'd like to have a simple ...
- 403
1
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0
answers
270
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Eigenvalue of product of self adjoint compact operators
Suppose A is a self adjoint $m \times m$ real matrix with eigenpairs $\{e_j, \lambda_j\}$ such that $\lambda_j > \lambda_{j + 1}$. Let $B$ be another self adjoint real $m \times m$ matrix such that ...
- 157
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225
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Separating Two Groups of Data using Fisher's Linear Discriminant
I found an article (starting on page 8) that gives a neat method for finding the line/plane/hyperplane that maximizes the separation between two groups of data points in n-dimensions. It uses Fisher's ...
- 101
1
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0
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158
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Interpreting (Fiedler) spectral bisectioning
I would appreciate help on how to interpret the results of spectral bisectioning of a graph.
Given a $G=(V,E)$ with size $N$ represented by $Q$ its Laplacian matrix where the eigenvalues are ordered ...
- 355
1
vote
2
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450
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Is Rellich's function valued theorem valid for a rank defficient function valued matrix?
Theorem (Rellich). Let $\boldsymbol{A}(t) : \mathbb{R}\rightarrow\mathbb{C}^{n \times n}$ be a Hermitian matrix function that depends
on $t$ analytically.
(i) The $n$ roots of the characteristic ...
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3
votes
1
answer
264
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When is this matrix singular?
Consider matrix $A$ with $(j,k)$′th entry $A_{j,k}=\sin(\omega_j t_k+\phi_j),\,\forall j,k\in\{1,2,...,n\}$, where $\omega_j,t_k,\phi_j\in \mathbf R$.
1) For $t_k=k$, what is the condition on $\...
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1
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270
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Relationship of eigenvalue/eigenvector of hermitian matrix R and QRQ (Q is diagonal)
For a hermitian matrix R and a diagonal one Q, is there any relationship between eigenvalues/eigenvectors of R and QRQ?
To be specific, assuming the eigenvalue decomposition of R is R=VDV*, then can ...
- 101
1
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0
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136
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when can I say that $UV^T$ is a permutation matrix? [closed]
suppose we have two p.s.d matrices A and B: so we can diagonalize them like this:
A= $UΛU^T$ and $B=VΣV^T$
1: on what condition for $A$ and $B$ I can say that $UV^T$ is a permutation matrix?
2: how ...
- 59
2
votes
1
answer
80
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Is there a general way to determine the Laplacian of the eigenvalues of a real symmetric matrix? [closed]
I have a real symmetric $3\times3$ matrix $\mathbf{M}(\mathbf{r}$) which depends on $\mathbf{r} \in \mathbb{R}^3$. Each eigenvalue can be considered a scalar field $e_i(\mathbf{r})$ over $\mathbb{R}^3$...
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2
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2
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269
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Is my use of the eigendecomposition correct here?
I'm exploring different techniques to efficiently solve some matrix equations. My situation is that I have a matrix $\textbf{H} = \textbf{J}^T \textbf{J}$, where $\textbf{J}$ is a matrix with no ...
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3
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0
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125
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Eigenvalue problem
I am studying torsional Alfven waves in spicules.
In this concern I have encountered the following equation:
$
\left(1-m^2 e^{-αz}\right)y''(z)+\left(4π i m e^{-αz}-1/h\right)y'(z)+\left(\frac{1}{4h^...
1
vote
1
answer
136
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Any generic way to move a psd matrix to its neighbors?
Given a two positive matrices $A,B$. For simplicity, let's assume that $Tr A=Tr B=1$. Assume that $\|A-B\|_1\leq\varepsilon$, for some small $\varepsilon>0$, where $\|\cdot\|_1$ is the $l_1$-norm, ...
- 363
1
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0
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138
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Relationship betwen eigenvectors
Suppose that we have two matrices A and B. Matrix B is taken from A with one row and one column deleted. On the other hand A is n*n matrix and B is (n-1)*(n-1) matrix and is created by deleting last ...
- 127
2
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0
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259
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Common eigenvector
I have little experience with functional analysis beyond an undergraduate basic course, and I'm dealing with the following problem:
let $V$ be an infinite-dimensional locally convex (but not normed!) ...
- 5,407
1
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0
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191
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Eigenvectors of contraction times projection
Suppose $A$ is a real $n\times n$ matrix with real eigenvalues:
$$
1=\lambda_1>|\lambda_2|\ge \ldots\ge |\lambda_n|>0.
$$
Suppose $B$ is an involution, for simplicity let us assume that
$B$ is ...
- 650
0
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1
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106
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The influence of eigendecomposition on the periodicity of a (rank 2) Hermitian matrix (of functions)
Let $\boldsymbol{R}(u,v);~u,v\in\mathbb{R}$ be a Hermitian matrix (of Hermitian functions) with entries
\begin{equation}
r_{ij}(u,v) = 1 + Ae^{-2\pi i \phi_{ij}(ul_0 + vm_0)}; A\in\mathbb{R},l_0\in\...
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