Questions tagged [eigenvector]

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Independence of rotated spherical harmonics

Hi, Consider a spherical harmonic of degree $l$, denoted by $y_l^m$. I rotate this harmonic using $2l+1$ different rotations. The set of functions I get is not an orthogonal set, but the functions ...
Cyril Soler's user avatar
1 vote
1 answer
527 views

Relating eigenvectors of two self-adjoints operators

Suppose I have a self-adjoint operator $\mathbf{L}$ which I seperate in two parts which are themselves self-adjoint. I write this in terms of their eigenvalues/eigenvectors: $\mathbf{v} \Lambda \...
Bramiozo's user avatar
3 votes
3 answers
3k views

Countability of eigenvalues of a linear operator

Is it true that every closed operator on a separable Hilbert H space only has countably many eigenvalues? Or put the other way around, if I want to ensure that a (not necessarily bounded) linear ...
Matthias Ludewig's user avatar
2 votes
2 answers
455 views

Eigenvalues in the semiclassical limit

Consider the Schrödinger operator $H_\hbar = -\hbar^2\Delta + V$ on $M=\mathbb{R}^n$, where $V$ is a potential that behaves well in a certain sense ($C^\infty$, bounded from below, going to infinity ...
Matthias Ludewig's user avatar
1 vote
0 answers
249 views

Operator eigenvalues and eigenvalues of pointwise evaluation matrix

Let $D\subset\mathbb{R}$ be a bounded interval and $f: D\times D \rightarrow \mathbb{R}$ a real-valued analytic function of two variables such that $f\in L_2(D\times D)$. Suppose we have upper bounds ...
alext87's user avatar
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7 votes
2 answers
5k views

the complexity of Lanczos method

Hi, all I am working on an algorithm which uses Lanczos method to compute K smallest eigenvalue(and their eigenvectos) of a sparse matrix, just want some information or links about the complexity of ...
rechardchen's user avatar
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1k views

Eigenvalues of anti-circulant matrices

Is there any theorem to find the eigenvalues of any anti-circulant matrix using the equivalent (with the same first row) circulant matrix. I found out that, for any anti-circulant matrix, the ...
Udara's user avatar
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6 votes
0 answers
549 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 - ...
user17119's user avatar
  • 179
1 vote
1 answer
269 views

Spectral analysis of sparse symmetric integer matrices

Hi all, A project I'm currently working on requires me to compute the eigenvectors / eigenvalues of sparse symmetric integer matrices. This is needed in the context of Principal Component Analysis. I ...
guidupuy's user avatar
  • 113
2 votes
0 answers
320 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 ...
user16007's user avatar
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10 votes
1 answer
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Eigendecomposition after multiplying by diagonal matrix

Hello, If we possess the eigendecomposition of a positive definite matrix: $X = U \Sigma U^T$, is there an efficient way to compute the eigendecomposition of $D X D$ where $D$ is a diagonal matrix?
Martin McCormick's user avatar
7 votes
3 answers
6k views

Minimize trace of inverse of convex combination of matrices.

Hello! (First question--please forgive me if its unclear.) I am interested in efficient/approximate optimization techniques for minimizing a norm of a convex combination of symmetric, positive semi-...
jvdillon's user avatar
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1 vote
2 answers
438 views

Optimizing directly on the eigenspectrum of a matrix

I have an application where I want to the eigenvalues of the graph to be involved in the objective and constraints in a flexible way (moreso than just the nuclear or frobenius norm). Whats a good ...
DoubleJay's user avatar
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-1 votes
1 answer
419 views

linear versus non-linear integral equations

I'm having trouble solving an integral equation. It appears to me to be a homogenous fredholm equation of the second kind. However, I'm being told that this can't be a fredholm equation, because it ...
adhanlon's user avatar
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5 votes
2 answers
2k views

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 ...
Vedarun's user avatar
  • 111
0 votes
1 answer
30k views

Difference between Principal Component Analysis(PCA) and Singular Value Decomposition(SVD)? [closed]

I am confused between PCA and SVD. The wikipedia page for PCA has this line. "PCA can be done by eigenvalue decomposition of a data covariance matrix or singular value decomposition of a data matrix, ...
user15019's user avatar
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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 $...
Arthur B's user avatar
  • 1,882
17 votes
5 answers
2k views

Can always a family of symmetric real matrices depending smoothly on a real parameter be diagonalized by smooth similarity transformations?

This question is related to another question, but it is definitely not the same. Is it always possible to diagonalize (at least locally around each point) a family of symmetric real matrices $A(t)$ ...
Cristi Stoica's user avatar
6 votes
3 answers
1k views

Analytical solution to a Linear advection-reaction PDE

I am looking for an analytical solution for the linear PDE $(1)\qquad\qquad \qquad f_t+ A f_x + B f = 0, $ Where $A$ and $B$ are constant matrices and $f=f(x,t)$ is a vector. Clearly each one of $...
Yossi Farjoun's user avatar
6 votes
1 answer
822 views

Dominant eigenvector of a real symmetric tridiagonal matrix

What is the most efficient way to calculate the dominant eigenvector of a real symmetric tridiagonal matrix? What's the corresponding time complexity bound? Could someone give me a reference for ...
tom's user avatar
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4 votes
1 answer
1k views

dominant eigenvector

Hi, everyone! Is there any efficient way to simplify the following tensor product $X \otimes X + X^T \otimes X^T$, where $X$ is a square $n \times n$ matrix. My goal is to efficiently compute the ...
person's user avatar
  • 41
0 votes
3 answers
587 views

Modified Eigen Problem

Any reference on how to solve the problem $Ax + c = \lambda Bx$ , where $A$, $B$ are full rank matrices, $c$ and $x$ are vectors and $\lambda$ is an unknown constant. I want to solve for both $x$ and $...
user12463's user avatar
2 votes
1 answer
203 views

Statistical estimation of singular values and vectors

My question is about the well known and well studied singular value decomposition (SVD). What I am working on right now requires performing an SVD repeatedly on a slowly varying matrix. Since I don't ...
Bernard's user avatar
  • 111
11 votes
1 answer
800 views

Algebraicity of Eigenvectors in a Hilbert space

Let $(e_j)_{j\in\mathbb N}$ be an orthonormal basis of a Hilbert space $V$. Let $T:V\to V$ be continuous, selfadjoint linear operator. Assume that for all $i,j\in\mathbb N$ the number $\langle Te_i,...
user avatar
19 votes
2 answers
8k views

Conditions for smooth dependence of the eigenvalues and eigenvectors of a matrix on a set of parameters

Let $A\in\mathcal M_n$ be an $n\times n$ real [symmetric] matrix which depends smoothly on a [finite] set of parameters, $A=A(\xi_1,\ldots,\xi_k)$. We can view it as a smooth function $A:\mathbb R^k\...
Cristi Stoica's user avatar
3 votes
10 answers
37k views

practical applications of eigenvalues and eigenvectors [closed]

We're making a video presentation on the topic of eigenvectors and eigenvalues. Unfortunately we have only reached the theoretical part of the discussion. Any comments on practical applications would ...
user9621's user avatar
  • 147
7 votes
1 answer
6k views

The difference between Principal Components Analysis (PCA) and Factor Analysis (FA)

I am trying to understand the difference between PCA and FA. Through google research, I have come to understand that PCA accounts for all variance, while FA accounts for only common variance and ...
Spencer_K's user avatar
7 votes
3 answers
2k views

What is the right citation for the power iteration method to find eigenvalues?

What is the right citation for the power iteration method to find eigenvalues, if I want to cite the method in a paper? I've seen some Google PageRank references in this context. But Brin and Page ...
user9162's user avatar
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0 votes
1 answer
3k views

Eigenvector centrality

I was wondering if you can calculate eigenvector centrality with undirected graphs and if you can, what is the best means of doing so. I understand how to calculate the adjacency matrix and how to ...
Spencer's user avatar
13 votes
2 answers
697 views

'Eigenvectors' of evolute operation

The evolute of a curve is the locus of its centers of curvature. The evolute of some plane curves is a scaled, or scaled and reflected/rotated, version of that curve. For example, the evolute of a ...
Joseph O'Rourke's user avatar
0 votes
3 answers
1k views

Intuitions/connections/examples for "eigen-*"

There are many concepts in mathematics that begin with the German word "eigen": eigenvector, eigenvalue, eigenspace, eigenstate, eigenfunction, eigensystem etc. (to name just the most important (?) ...
vonjd's user avatar
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