Questions tagged [eigenvector]
The eigenvector tag has no usage guidance, but it has a tag wiki.
293 questions
5
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Can I assign the term "is eigenvector" and "is eigenmatrix" of matrix $P$ in my specific (infinite-size) case?
Remark: I asked this in MSE, the question got views and votes but seemingly no one had an answer so far.
Background: I'm rereading a couple of my exploratory (surely not research-level) math-essays ...
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8
votes
1
answer
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Eigenvectors of Kronecker Product [closed]
Conjecture If $A$ and $B$ are two complex square matrices, then every eigenvector of $A\otimes B$ is of the form $x\otimes y$, where $x$ is an eigenvector of $A$ and $y$ is an eigenvector of $B$.
...
12
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2
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What is known about the eigenvectors of the $2^n \times 2^n$ Hadamard matrix?
What is known about the eigenvectors of the $2^n \times 2^n$ Hadamard matrix defined recursively by $H_1=(1)$ and $$ H_N=\begin{pmatrix}H_{N/2} & H_{N/2} \\ H_{N/2} & -H_{N/2}\end{pmatrix}, $$ ...
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1
vote
0
answers
439
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Derivative of singular value decomposition of $I + \alpha X$
For an application in physics I would like to estimate the effect of a small pertubation on an ideal system. For that, I require the change to the eigenvectors and eigenvalues of an diagonal matrix ...
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2
votes
1
answer
661
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Leading eigenvector value problem as an optimisation problem for asymmetric matrices
As noted in 1806.05647, given a symmetric matrix $A$, the leading eigenvector value problem (LEVP)
$$Av = \lambda v,$$
where $A = A^T \in \mathbb{R}^{n \times n}$, $\lambda$ is the largest ...
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0
votes
1
answer
820
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What are the eigenvalues and eigenvectors of a composition of two arbitrary linear transformations? [closed]
Let $S$ and $T$ be two linear transformations on $\mathbb{R}^n$. We can find the eigenvalues and eigenvectors of $S$ and $T$. I am trying to find out the relation(s) among the eigenvalues and ...
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3
votes
2
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Eigenvectors of graph Laplacian for spectral clustering
I have the following questions regarding the graph Laplacian for spectral clustering:
What is the intuition behind projecting the Laplacian (D-A, where D is the degree matrix and A is the affinity ...
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6
votes
0
answers
96
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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
2
votes
1
answer
508
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Jordan decomposition of a block matrix
Assume $A$ is a block matrix of the form:
$$A=\left[\begin{array}{cccc}
A_{11}&A_{12}&\ldots&A_{1n}\\
A_{21}&A_{22}&\ldots&A_{2n}\\
\vdots&\vdots&\ddots&\vdots\\
...
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1
vote
0
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86
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Solution of a manipulated equation vs the maximum eigenvalue and eigenvector of a non-negative matrix
Lets assume we have the following equation:
$AU=\lambda U \Rightarrow\left[
\begin{array}{c|c|c}
0 &A_{12}&A_{13}\\
\hline
A_{21}& 0& A_{23}\\
\hline
A_{31}&A_{32}&0
\end{...
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0
votes
1
answer
123
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Probability eigenvectors of discrete random matrix are orthogonal to discrete random vector
Let $W=(w_{ij})_{1 \leq i, j \leq N}$ and $\textbf{v}=(v_j)_{1 \leq j \leq N}$ be a random $N\times N$ matrix and N-vector, respectively, where all $w_{ij}$ are jointly independent and have discrete ...
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1
vote
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An inequality concerning the eigenvalues and eigenvectors of an SPD matrix
Let $Ax_i=\lambda_ix_i, \ (i=1,\cdots,n)$ be an eigensystem of the symmetric positive-definite diagonally-dominant matrix $A=\{a_{ij}\}$. Let
$$b_{jk}=\sum_{i=1}^{n}{\frac{(x_i(j)-x_i(k))^2}{\...
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409
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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: ...
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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
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2
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373
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Can a vector-function $v:\mathbb{R}^n\to \mathbb{R}^n$ be an eigenvector of its own Jacobian matrix?
Good morning,
I've came across this question, which has been puzzling me for some days. Suppose we are given a vector-valued function $v:\mathbb{R}^n\to \mathbb{R}^n$, $v(x)=\left( v_1(x),\dots, v_n(...
- 311
1
vote
0
answers
171
views
Eigenvalues of non-negative block matrices
$B$ is a non-negative irreducible block matrix as follows:
$$B=
\left[
\begin{array}{c|c|c}
0 &B_{12}&B_{13}\\
\hline
B_{21}& 0& B_{23}\\
\hline
B_{31}& B_{32}&0
\end{array}
\...
- 21
2
votes
0
answers
117
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Approximate Simultaneous Diagonalization of Non-Hermitian Matrices
Let $A_1,A_2$ two $n\times n$ complex matrices. $A_1$ and $A_2$ are also non-normal, especially, non-hermitian and do not commute. I would like to find an invertible matrix $V$ such that
$$
\sum_{i=1,...
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1
vote
1
answer
77
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Efficient way to compute eigenvalue decomposition for following problem
I have an optimization problem
$$\begin{array}{ll} \text{minimize} & Tr(X^TAX) \\ \text{subject to} & X^TX=I
\end{array}$$
where $A\in R^{n \times n}$ and it is symmetric positive definite, ...
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1
vote
1
answer
96
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Can eigenvectors determine their original matrices given the basis of matrices space is small?
Let $A_1$, $A_2$, and $A_3$ be three different mutually orthogonal random hermitian operators, say dimension of $30\times 30$. For three random real numbers $a_1$, $a_2$, $a_3$, we have
$$
(a_1A_1+...
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0
votes
1
answer
79
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Will truncated SVD ever flip the sign of any element of the matrix?
For a symmetric p.s.d matrix $A \in \mathcal{R}^{n\times n}$, we can calculate its SVD as $A=USV^T$, then we can use the truncated SVD to approximate it with a low-rank matrix $\tilde{A} = \sum_i^...
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2
votes
0
answers
340
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Determinant of a rank r perturbation
In the following paper:
Restricted Rank Modification of the Symmetric Eigenvalue Problem: Theoretical Considerations
on page 79, Golub et al. have the following set of equations:
$f(\lambda) = \...
- 121
1
vote
0
answers
71
views
Nodal domains on a surface
What is it known about the topology of nodal domains of eigenfunctions of self-adjoint operators?
In particular I'm interested in self-adjoint operators on a complete, non-compact, surface $\Sigma \...
- 823
2
votes
1
answer
3k
views
Lanczos algorithm for finding $k$ smallest eigenvector
I am trying (and have been recommended) to use the Lanczos algorithm to find the $k$ smallest eigenvectors. However, all of the literature seems to talk about this algorithm as a way to estimate the $...
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votes
1
answer
970
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What can we say about the rank of the sum of a multiple of the identity matrix and a symmetric rank-$1$ matrix? [closed]
Suppose we have the following symmetric matrix.
$$A = \sigma^2 I + u u^T$$
What can we say about the eigendecomposition of $A$?
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1
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473
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Statistical independence of eigenvectors of real symmetric Gaussian random matrices
What is known about the statistical independence of the eigenvectors of a real symmetric matrix with independent Gaussian entries with zero mean, and finite variance? The matrix elements are not ...
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4
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0
answers
447
views
How to find eigenvalues of following block matrices?
Is there a procedure to find the eigenvalues of A?
$$A=\begin{bmatrix}X & I &&&&&&&&& 0\\I & 0 & P &&&&&&&&\\& P^t ...
- 41
3
votes
1
answer
145
views
Connections between eigenvectors after matrix multiplication
Suppose we have an M$\times$N complex matrix $H$ and its singular value decomposition $H=U\Lambda V^*$ and an N$\times$N covariance matrix $R_s$ with its eigendecomposition $R_s = U_s\Lambda_sU_s^*$. ...
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1
vote
1
answer
496
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Convergence rate of eigenvectors
Let us suppose that $A,A_1,A_2,\ldots$ are non-negative definite self-adjoint bounded linear operators in $L(\mathbb H)$, where $\mathbb H$ is a separable Hilbert space. $(v_j)_{j\ge1}$ and $(\...
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4
votes
1
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The existence of $v\in A\otimes_{\mathbb{K}}A$ such that $(a\otimes_{\mathbb{K}}1)v=(1\otimes_{\mathbb{K}}a)v$
If $A$ is a finite dimensional commutative, associative, unital algebra over a field $\mathbb{K}$ then does there exist a non-zero vector $v\in A\otimes_{\mathbb{K}}A$ such that $(a\otimes_{\mathbb{K}}...
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4
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1
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380
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Rotatable matrix, its eigenvalues and eigenvectors
We say that a real matrix is rotatable iff after turning it clockwise on $90^{\circ}$ it doesn't change.
I'm interesting about eigenvalues and eigenvectors (belonging to non-zero eigenvalues) of such ...
13
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3
answers
3k
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Differentiability of Eigenvalues - Perturbation Theory
first, I have a general question. In perturbation theory, I saw perturbations in eigenvalues and eigenvectors of square, non-symmetric matrices and the calculations were all right but no one ever ...
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2
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748
views
Lower bound on the entries of the Perron vector
Let $A$ be a matrix that satisfies all the conditions of Perron- Frobenius theorem. From the theorem it is known that the entries of the eigenvector corresponding to the largest eigenvalue will be ...
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2
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Constructive proof of a rational version of Perron-Frobenius?
In the following, we work with vectors and matrices whose entries are rational numbers. Inequalities between such vectors are understood to be coordinatewise: e.g., two vectors $a = \left(a_1,a_2,\...
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2
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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 ...
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votes
0
answers
88
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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
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0
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406
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Asymptotic behavior of the Dirichlet-Laplacian eigenvalues [closed]
I found in a math book http://www.cambridge.org/dz/academic/subjects/mathematics/differential-and-integral-equations-dynamical-systems-and-co/introduction-partial-differential-equations?format=PB&...
- 617
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1
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Stability of eigenvectors for diagonal perturbations
In a previous question I asked about the stability of eigenvalues with respect to diagonal perturbations. Following results from the book Matrix Analysis (by Roger A. Horn & Charles R. Johnson) ...
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2
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1
answer
154
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Finding a similarities and differences of sent of matrices
Suppose we have a set of rank deficient covariance matrices. How can I know the similarities and differences between those set of matrices?
Regards,
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1
answer
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Eigenvalues and eigenvectors of tridiagonal matrices
What can I say about the eigenvalues and eigenvectors of the tridiagonal matrix $T$ given as
$T = \begin{pmatrix}
a_1 & b_1 \\
c_1 & a_2 & b_2 \\
& c_2 & \ddots & \ddots \\
&...
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1
vote
1
answer
2k
views
Gradients of the Dominant Eigenvalue and Eigenvector
How can I compute the partial derivatives of the dominant eigenvalue and eigenvectors of a real symmetric matrix $\mathbf{A}$?
In particular, given
$ \mathbf{v}^* = \arg\max_{\mathbf{v}} \mathbf{v}^{...
- 137
0
votes
0
answers
184
views
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 ...
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1
vote
1
answer
193
views
Is there any relation between weights in the eigenvector (corresponding to least eigenvalue) and the columns of a correlation matrix?
This problem is motivated from one of my pattern mining research projects. Any helpful suggestions will be highly appreciated.
Consider an $n \times n$ correlation matrix A such that all the off-...
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1
vote
1
answer
179
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Connection between weights in the last eigenvector (corresponding to least eigenvalue) and the corresponding column of a correlation matrix
This problem is motivated from one of my pattern mining research projects. Any helpful suggestions will be highly appreciated.
Consider an $n \times n$ correlation matrix A such that all the off-...
- 141
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 ...
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4
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0
answers
298
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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
2
votes
2
answers
446
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Entrywise modulus matrix and the largest eigenvector
Disclaimer. This is a cross-post from math.SE where I asked a variant of this question two days ago which has been positively received but not has not received any answers.
Let $A$ be a complex ...
- 623
3
votes
1
answer
932
views
Integer eigenvectors
Is there a known way or software to find integer eigenvectors for an integer matrix with integer eigenvalues?
In particular, I have a large real symmetric matrix with only a small number of ...
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
7
votes
1
answer
464
views
"Unimodality" of the positive eigenvector of a non-negative irreducible matrix?
Consider an eigenvalue / eigenvector problem for a matrix $A$ that is known to be non-negative and irreducible (so the Perron-Frobenius theorem applies):
$$\sum_j A_{ij} x_j = \lambda x_i$$
Here $\...
- 884
9
votes
1
answer
436
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Eigenvalues of a matrix with binomial entries
I am trying to determine the eigenvalues and eigenvectors of the following matrix:
$$M_{ij} = 4^{-j}\binom{2j}{i}$$
where it is understood that the binomial coefficient $\binom{m}{k}$ is zero if $k&...
- 884