Questions tagged [eigenvector]
The eigenvector tag has no usage guidance, but it has a tag wiki.
100 questions with no upvoted or accepted answers
10
votes
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Eigenfunctions of the integral kernel $1/(x^2 + x'^2)$
My question seems elementary, yet I could not find the solution after working on and searching for several days...
I'd like to find the eigenfunctions of a simple integral kernel:
\begin{equation}
\...
- 101
8
votes
0
answers
221
views
Density of odd and even eigenstates of an integral operator
Consider an integral operator $(Kf)(x)=\int_{-1}^{1}K(x-y)f(y)dy$, where the kernel $K(-x)=K(x)$ is an even function.
Let $\lambda_n$ be the ordered eigenvalues of $K$ and $f_n(x)$ the ...
- 81
7
votes
0
answers
384
views
Concept of eigenvector restricted to nonnegative entries
Let $X\in \mathbb{R}^{n\times n}$ be a positive semidefinite matrix. The leading eigenvector $v\in \mathbb{R}^n$ of $X$ is the solution to the problem
$\arg \max_{v:\lVert v\rVert_2=1} \lambda\quad$ ...
- 373
6
votes
0
answers
188
views
Measurability of eigenvalues-eigenvectors of a positive compact operator
Let $H$ be a separable Hilbert space over $\mathbb{R}$. Let ${A} = \{a\colon H\to H\,|\,a\text{ is a positive, compact linear operator}\}$.
By the spectral theorem, given $a \in A$, there are scalars $...
- 163
6
votes
0
answers
96
views
Finding the maximal component of a vector in sublinear time
Given a vector $u \in \Bbb R^n$, finding the value of the largest component of $u$ needs linear time in $n$. However, what if we additionally know that $u$ lies in some linear subspace $U \subset \Bbb ...
- 13.6k
6
votes
0
answers
88
views
Density of squares of radial eigenfunctions
The eigenfunctions of the Laplace operator on the disc can be written in polar coordinates as $f(r,\theta)=R_{nk}(r)e^{ik\theta}$, where $k\in\mathbb Z$ and $n\in\mathbb N$ and the radial function is $...
- 8,086
6
votes
0
answers
565
views
What are the eigenvectors of the Lagrange interpolation matrix?
Let $F$ be a field.
Let $x_1,\ldots,x_k,y_1,\ldots,y_k\in F$ be distinct elements in the field.
Consider the $k\times k$ matrix that in position $i$, $j$ has the element
$\frac{\prod_{l\neq i}(y_i - ...
- 179
5
votes
0
answers
208
views
Perturbation of Neumann Laplacian
Consider the $N \times N$ matrix
$$A_{\alpha}=\begin{pmatrix} \lambda_1 & -1 & -\alpha & 0 & \cdots & 0\\
-1 & \lambda_2 & -1 & -\alpha & \cdots & 0\\
-\alpha &...
- 73
5
votes
0
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409
views
Spectral theory without topology
How much of spectral theory can be developed just working with vector spaces (finite or infinite dimensional) without referring to a choice of topology ?
Something along these lines, for example: ...
- 51
5
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0
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392
views
Preconditioner for finding the smallest eigenpairs of a large, but structured, matrix
I'm trying to find the eigenvector corresponding to the second smallest eigenvalue of a large $(4,000,000 \times 4,000,000)$ matrix $L$. $L$ is a graph Laplacian, with the following structure: $L = D -...
- 500
5
votes
0
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282
views
Linearization of a gradient field
Setup: Suppose we are given a smooth function $\phi$ that has a nondegenerate minimum at $x=0$. Then we can choose a coordinate system $x$ such that the gradient is given by
$$X = \mathrm{grad} \phi = ...
- 9,863
4
votes
0
answers
70
views
Local energy estimate in a semiclassical regime
Let us consider $h_n=(2n+1)^{-1/2}\to 0$ as $n\to \infty$ be a small parameter, which we just write as $h$ for convenience, and $u_h : \mathbb{R} \to \mathbb{R}$ be functions satisfying $Pu_h=0$ (I ...
- 489
4
votes
0
answers
447
views
How to find eigenvalues of following block matrices?
Is there a procedure to find the eigenvalues of A?
$$A=\begin{bmatrix}X & I &&&&&&&&& 0\\I & 0 & P &&&&&&&&\\& P^t ...
- 41
4
votes
0
answers
298
views
Stationary distribution of mixture of Markov Chain with "complete" Markov Chain
I already asked this question in StackExchange, but found little attention. So I'm just going to copy-paste my original question here.
Let $P$ be a stochastic matrix (of an irreducible Markov Chain) ...
- 153
4
votes
0
answers
2k
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What is the time complexity of the largest singular value and its vectors?
Full zero-error SVD on an $m \times n$ matrix $A$ would cost $O(\min(m^2n,mn^2))$. What is the time complexity if we need only the largest singular value and its corresponding vectors? I think it is $...
- 123
4
votes
0
answers
84
views
Matrices with almost constant coefficient have a simple eigenvalue
As a by-product of a general result for bounded operators of a Banach space, I have the following:
A matrix $L=(\ell_{ij})_{ij}$ that has almost constant coefficients in the sense that for some $c$,...
- 14.4k
4
votes
0
answers
463
views
The distribution of the elements of an eigenvector of random matrices
Suppose a random matrix $A$ with its elements following Gaussian distribution with non-zero mean. We know that the eigenvalues of $A$ have two patches: one is at the real axis that is far away from ...
- 39
4
votes
0
answers
137
views
What do we know about the generalized eigenvalue problem involving a projector?
Consider a matrix $A\in\mathbb{R}^{n\times n}$ and a projector $P\in\mathbb{R}^{n\times n}$.
Are there results regarding the generalized eigenpairs $(v,\lambda)$ of the generalized eigenproblem
$$...
- 325
3
votes
0
answers
115
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Recovering the matrix when the Schur decomposition of its blocks are known
Let E be a real symmetric matrix in $M_n(\mathbb{R})$ where $ n=2m$ and
$$E=\left(\begin{array}{cc}
G & X \\
X^t & H
\end{array}\right)$$
where $G,H,X$ are $m\times m$ matrices.
Suppose that $...
- 4,058
3
votes
0
answers
47
views
What is the supremum of 1-dim Hausdorff measure of the nodal set of Neumann eigenfunction $u$ for planar convex domain
All descriptions of this question are limited to 2-dimension for simplicity. Recently, I read some papers on the nodal set of Laplacian eigenfunctions. Denote $\Sigma=\{u(x)=0\}$ be the nodal set of a ...
- 131
3
votes
0
answers
921
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Principal eigenvector of non-negative symmetric block matrix is approximated by a linear combination of the principal eigenvectors of the blocks
Let $
M \in \mathbb{R}^{n \times n} =
\begin{bmatrix}
A & B \\
B^T & C
\end{bmatrix}
$ for some nonnegative $A \in \mathbb{R}^{k \times k}, B \in \mathbb{R}^{k \times n-k}, C \in \mathbb{R}^{n-...
- 123
3
votes
0
answers
220
views
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
$$M = i \begin{pmatrix}
0 & a & 0 &...
3
votes
0
answers
125
views
Eigenvalue problem
I am studying torsional Alfven waves in spicules.
In this concern I have encountered the following equation:
$
\left(1-m^2 e^{-αz}\right)y''(z)+\left(4π i m e^{-αz}-1/h\right)y'(z)+\left(\frac{1}{4h^...
3
votes
0
answers
2k
views
Relation between the eigenspace of a covariance matrix and eigenspace of correlation matrix
I was discussing applying Principal Component Analysis to a covariance matrix versus applying PCA to the corresponding correlation matrix with a collegue. This led me to think about the following ...
- 131
3
votes
0
answers
1k
views
How many iterations are required for the Lanczos algorithm to converge?
I am trying to find the n smallest eigenvalues and eigenvectors of a NxN SPD matrix using Lanczos method. What is the number of iterations usually required? I mean, does it scale as $O(N)$ or $O(\sqrt{...
- 31
2
votes
0
answers
86
views
Smallest eigenvalue of certain PD matrix decreases under sparse perturbation
Let $\omega_1<\dots<\omega_n\in\mathbb{R}$. Then, define $G\in\mathbb{C}^{n\times n}$ such that $G_{k\ell}=\frac{1}{1-i(\omega_\ell-\omega_k)}$. For example, if $n=3$ we obtain $$ G=\begin{...
- 83
2
votes
0
answers
143
views
Convergence of eigenfunctions
In their 1999 paper "Sturm-Liouville operators with singular potentials", Savchuk and Shkalikov prove the uniform resolvent convergence of an operator $L_\varepsilon \rightarrow L$ for $\...
2
votes
0
answers
251
views
Infinite positive matrices with probability eigenvector
Let $A$ be an infinite non-negative matrix with integer entries ($a_{ij} \geq 0, \forall i,j \in \mathbb N$).
Suppose that $A$ is irreducible, aperiodic, and recurrent. So that it satisfies the ...
- 351
2
votes
0
answers
43
views
References on discrete Sturm-Liouville eigenvectors convergence
Let $ L : u_n \mapsto a_n u_{n + 1} + b_n u_n + a_{n - 1} u_{n -1} = \nabla ( a_n \Delta u_n ) + (b_n + a_n + a_{n - 1}) u_n $ be a discrete Sturm-Liouville operator, with $ \nabla u_n := u_{n + 1} - ...
- 593
2
votes
0
answers
114
views
Is $A$ is small on bounded functions, is there a large subdomain on which $A$ is small?
Let $A$ be a symmetric linear operator of norm $\leq 1$ on the space of functions $f:S\to \mathbb{R}$, where $S$ is a set with $N$ elements. Define the inner product $\langle \cdot,\cdot\rangle$ by ...
- 20.2k
2
votes
0
answers
301
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Eigenvector of Hadamard matrix functions
Let $X\in\mathbb{R}^{n\times n}$ with SVD $X=UDV^T$. Are there known results regarding the eigenvectors of $Y=X^{\odot g}$? I am mainly interested in simple functions such as $g(z)=z^2$, i.e. $Y_{ij}=...
- 21
2
votes
0
answers
245
views
Relationship between eigenvectors of projected and original matrix
Let $A = \mathrm{Diag}(\lambda_1, \dots, \lambda_n)$ where $\lambda_1 \le \lambda_2 \dots \le \lambda_n$. Let $P = I - ww^T$ be a projection operator on an arbitrary $n$-dimensional hyperplane. Let $B ...
- 231
2
votes
0
answers
117
views
Approximate Simultaneous Diagonalization of Non-Hermitian Matrices
Let $A_1,A_2$ two $n\times n$ complex matrices. $A_1$ and $A_2$ are also non-normal, especially, non-hermitian and do not commute. I would like to find an invertible matrix $V$ such that
$$
\sum_{i=1,...
- 909
2
votes
0
answers
340
views
Determinant of a rank r perturbation
In the following paper:
Restricted Rank Modification of the Symmetric Eigenvalue Problem: Theoretical Considerations
on page 79, Golub et al. have the following set of equations:
$f(\lambda) = \...
- 121
2
votes
0
answers
171
views
List of analytically known eigensystems?
In condensed matter physics, we often come across matrices that are multi-diagonal or banded. For example, I may have a matrix with three tridiagonal bands, or a tridiagonal band and two/four ...
2
votes
0
answers
79
views
Conditions on a $n\times n$ Hermitian matrix such that its extremal eigenvectors have equal magnitude entries
Is it possible to find (necessary and sufficient) conditions on a general $n\times n$ Hermitian matrix $A$, such that its extremal eigenvectors (the eigenvectors corresponding to the maximum and ...
- 31
2
votes
0
answers
701
views
Simple random walk on a discrete torus - the eigensystem, reference
My problem concerns finding a reference in which the formulae for the eigenvalues and the corresponding eigenvectors ($n$ linearly independent eigenvectors!) for the transition matrix of a simple ...
user83658
2
votes
0
answers
479
views
Eigendecomposition of the Hadamard product of a rank one symmetric matrix and a positive definite symmetric matrix
Is it possible to say anything about the eigenvalues and eigenvectors of a matrix
$X = Y \circ xx^T$
where $Y$ is a positive definite symmetric matrix with known eigen-decomposition
$Y=U\Lambda U^T$...
- 71
2
votes
0
answers
28
views
Comparison of principal diagonals of two positive definite matrix
Let us consider a matrix with positive elements: ${\bf X}^{k\times2}=[X_1:\ldots:X_k]'$ with $X_i=(1,X_{1i})',\;i=1,\ldots,k$. Also consider ${\bf X}^{(-1)}$ as the Moore-Penrose inverse of ${\bf X}$ ...
- 213
2
votes
0
answers
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
2
votes
0
answers
259
views
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
2
votes
0
answers
673
views
How to compute the stationary distribution?
What is the most computationally efficient way to compute the stationary distribution of a reversible finite-state continuous-time Markov process?
Assume that the process is irreducible and the rate ...
- 266
2
votes
0
answers
339
views
Sparse Eigenvectors for the Discrete Fourier Transform matrix
There are many ways to choose eigenbasis for the Discrete Fourier Transform matrix since it has only $4$ distinct eigenvalues taken from $\{\pm 1,\pm i\}$.
Has there been any refereed work that ...
- 800
2
votes
0
answers
279
views
Eigenvectors of convolution with a normal distribution over a restricted interval
Suppose I have a random variable $X_0$ with a p.d.f $f_0$ supported on the real interval $[a_0, b_0]$. $X_1$ is the restriction to $[a_1, b_1]$ of the sum $X_0 + g$, where $g$ is normally distributed $...
- 1,902
1
vote
0
answers
68
views
Breakdown of the fourier series identity $e_n e_m = e_{n+m}$ on a perturbed torus
I would like to apologize for leaving things a bit vague. I think my question could be stated much more precisely but right now that is difficult for me to do. I nevertheless think it is an ...
- 243
1
vote
0
answers
64
views
Eigenvalues and eigenvectors of the path Laplacian
Consider the Laplacian matrix of the path graph:
$$
L = \begin{bmatrix}
1 & -1 & 0 & \cdots & 0 & 0\\
-1 & 2 & -1 & \cdots & 0 & 0\\
0 & -1 & 2 & \...
- 11
1
vote
0
answers
80
views
Inequality involving random vectors and absolute values
Let $\mathbb{X}, \mathbb{Y} \subset \mathbb{R}^d$ be finite sets. Suppose random vectors $X \in \mathbb{X}$ and $Y \in \mathbb{Y}$ are sampled according to a joint distribution $\mathbb{P}_{XY}$. ...
1
vote
0
answers
48
views
Random matrices: Relation between leading eigenvector and a vector in culumn space
Let $X$ be a $n\times n$ symmetric matrix with iid zero-mean random entries on and above the diagonal. Denote by $v$ the eigenvector corresponding to the largest eigenvalue of $X$. Let $a$ be a fixed $...
- 31
1
vote
0
answers
183
views
Connection of eigenspace of finite Hilbert matrix and its continuous operator counterpart
I am trying to understand the connection between the eigenspace of the continuous operator
$$
H(x,y) = \frac{1}{x+y}
$$
which is nothing but the square of the Laplace operator, and its discrete ...
- 33
1
vote
0
answers
120
views
Formula for the kernel of an operator
Let $\mathcal H$ be a Hilbert space and let $O$ be an operator. Obviously $M=O^\dagger O$ is a semi-positive definite operator and $v\in\ker M$ if and only if $v\in\ker O$. Therefore it seems to me ...
- 521