Questions tagged [eigenvector]
The eigenvector tag has no usage guidance, but it has a tag wiki.
293 questions
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bilinear equation OR diagonal matrix search
Dear guys,
I am parametrizing a model and I face an interesting (but tough for me) problem
I have a real square $n \times n$ symmetric matrix $B$ (which consists of 2 square blocks of positive ...
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Eigenvectors and eigenvalues of a tridiagonal Toeplitz matrix
Is it possible to analytically evaluate the eigenvectors and eigenvalues of the following $n \times n$ tridiagonal matrix
$$
\mathcal{T}^{a}_n(p,q) = \begin{pmatrix}
0 & q & 0 & 0 &...
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How many iterations are required for the Lanczos algorithm to converge?
I am trying to find the n smallest eigenvalues and eigenvectors of a NxN SPD matrix using Lanczos method. What is the number of iterations usually required? I mean, does it scale as $O(N)$ or $O(\sqrt{...
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Linearization of a gradient field
Setup: Suppose we are given a smooth function $\phi$ that has a nondegenerate minimum at $x=0$. Then we can choose a coordinate system $x$ such that the gradient is given by
$$X = \mathrm{grad} \phi = ...
- 9,863
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What is the minimum of the Frobenius norm in the intersection of positive semidefinite cones?
For scalar variables $x$, we have a simple solution for the following problem.
\begin{eqnarray}
\min_x&&\alpha(x-a)^2+\beta(x-b)^2 \\\
\mathrm{s.t. }&&x\leq a\\\
&&...
- 607
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Eigenvalues of sum of an adjacent matrix and a constant
$A$ is an adjacent matrix of a network. $la$ is the largest eigenvalue of $A$ and $Va$ is its corresponding eigenvector.
I am interested in the following martix: $bA+c-dI$ ($b$, $c$, and $d$ are all ...
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583
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Efficiently computing a few localized eigenvectors
Let $H = \triangle + V(x) : \mathbb{R}^2 \rightarrow \mathbb{R}^2$. I am interested in domain decomposition for an eigenproblem involving $H$.
The lowest 1000 eigenfunctions of $H$, $ \psi_i $, can ...
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When can an eigenvector be chosen uniquely which is invariant to permutation?
Suppose $A\in\Re^{n\times n}_{sym}$ is a symmetric matrix with eigenvalues $\lambda_1,\dotsc,\lambda_n$ in decreasing order. What I seek is a way to choose an eigenvector that is invariant to ...
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673
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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
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Semiclassical expansions of eigenvalues of Schrödinger operators
Considering Schrödinger operators
$$ H(\hbar) = \hbar \Delta + V $$
where $V$ is some potential, perturbation theory tells that the eigenvalues of $H(\hbar)$ are holomorphic on some region containing ...
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Generalization of eigenvalues/vectors to modules?
What is the generalization of eigenvalues/vectors to modules?
To be specific, given a "vector" v in a module over some ring, and a linear "operator" O from the module to itself (please feel free to ...
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Stochastic Matrix: Second largest eigenvalue and second largest absolute value of eigen value
Setup
Let $A$ be a stochastic matrix.
Let the eigenvalues of $A$ be $1 = \lambda_1 \geq \lambda_2 \geq \lambda_3 ... \geq -1$.
Let $\lambda = \max_{x: x \perp 1} \frac{||Ax||}{|| x ||}$
Question:
...
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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 ...
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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 \...
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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 ...
- 9,863
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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 ...
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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 ...
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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 ...
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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 ...
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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 - ...
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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 ...
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339
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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 ...
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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?
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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-...
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517
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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 ...
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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 ...
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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 ...
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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, ...
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279
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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 $...
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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 $A(t)$ ...
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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 $...
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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 ...
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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 ...
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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 $...
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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 ...
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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,...
user1688
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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\...
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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 ...
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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 ...
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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 ...
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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 ...
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'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 ...
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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 (?) ...
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