Questions tagged [eigenvector]
The eigenvector tag has no usage guidance, but it has a tag wiki.
282
questions
4
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1
answer
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complexity of computing the singular vector corresponding to the smallest singular value
It is known that the singular value decomposition of an $m \times n$ matrix $A$ is in general of complexity of the order $m n^2$, assuming that $m \ge n$. But what if we only want to compute say the ...
0
votes
1
answer
806
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Do these matrices have the same null space?
Let $[\theta_1,\theta_2, \dots, \theta_N]^\mathrm{T} \, \in \mathbb{R}^N$. The angles are not all identical (on the circle), i.e. $[\theta_1,\theta_2, \dots, \theta_N] \not \equiv c [1,1,\dots, 1]^\...
7
votes
1
answer
251
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Eigenvectors of a matrix with entries involving combinatorics
In the question Eigenvalues of a matrix with entries involving combinatorics No_way asked about eigenvectors of $n\times n$ matrix $M$ with entries \begin{eqnarray*}
M_{ij}=(-1)^{i+j}F(n, l, i, j),
\...
1
vote
0
answers
280
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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}...
5
votes
4
answers
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Differentiability of eigenvalue and eigenvector on the non-simple case
Let $h:\mathbb{R}^n\to\mathbb{R}^m, n>1$ be a twice continuously differentiable function and $J_h:\mathbb{R}^n\to\mathbb{R}^{m\times n}$ be its jacobian matrix. Let us consider the functions $A(x):=...
4
votes
0
answers
83
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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 $c$,...
-2
votes
1
answer
676
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The spherical harmonics are the EIGENVECTORS of Beltrami operator [closed]
In the well-known book "THE PRINCETON COMPANION TO MATHEMATICS" page 296, it is indicated that the spherical harmonics are the EIGENVECTORS of the Beltrami operator. In the document Spectral Geometry ...
8
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3
answers
526
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Why $M_1 \subset M_2 \not \Rightarrow N_{M_1} (\lambda) \leq N_{M_2} (\lambda)$ for eigenvalue problem? (EDIT)
We know that for a direct problem with Dirichlet Boundary Condition (with Laplacian operator) that if two domains $M_1$ and $M_2$ are such that $M_1 \subset M_2$, then $\lambda(M_2) \leq \lambda(M_1)$,...
2
votes
1
answer
857
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How can I efficiently approximate the stationary distribution of an infinite CTMC with a sparse rate matrix?
I am looking for methods to approximate the stationary distribution of an infinite CTMC with a sparse rate matrix. Each row and column of the rate matrix has a finite number of non-zero elements. ...
2
votes
1
answer
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How to retrieve eigenvectors from shifted QR algorithm?
I understand that the key to retrieve eigenvectors in the non-shifted QR algorithm is to accumulate the transformations at each steps in the following way:
$Q = \Pi_i Q_i$
Can we accumulate the ...
2
votes
1
answer
935
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Eigenvectors of symmetric positive semidefinite matrices as measurable functions
I'm currently interested in how discontinuous can get the eigenprojections of a continuous function taking values in a particular subspace of symmetric matrices.
I've been searching everywhere for an ...
2
votes
0
answers
452
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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 ...
11
votes
1
answer
892
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Exact eigenvalues of a specific tridiagonal matrix
I'm studying the following tri-diagonal matrix
$$
X = \begin{pmatrix}
0 & x_0 & 0 & 0 &\cdots & 0 & 0 & 0 \\\
x_0 & 0 & x_1 & 0 &\cdots & 0 & ...
0
votes
1
answer
105
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Eigenvalue-related statements [closed]
(I understand this question might not be appropriate for this website, but it has been asked on MathStackexchange and did not receive any replies even with a bounty)
How can I prove that the ...
10
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2
answers
9k
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Derivative of eigenvectors of a matrix with respect to its components
Suppose that $B$ is a real, positive-definitive symmetric ($3\times3$) matrix (more accurately, $B$ is a tensor) with distinct eigenvalues, and that we can write it as
$$
B= \sum_{i=1}^3 \lambda_{i}(...
0
votes
1
answer
339
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Almost sure convergence of smallest eigenvector of diagonal matrix
I have that a sequence of random matrices, $M_n$, converges almost surely to a diagonal matrix, $D$, with finite real entries on its diagonal. During convergence, the off-diagonals are not necessarily ...
7
votes
1
answer
403
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Sum of the absolute eigenvalues of A>=B
Kindly help me to prove/disprove the following statement.
Let $A$ be a symmetric matrix of order $n \times n$ with all the diagonal entry equal to $0$, and other non-diagonal entry equal to $k$ (...
3
votes
3
answers
344
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Can a block matrix with at least 3 zero blocks of different size on the diagonal and 1's everywhere else have only integer eigenvalues?
Let $M=\begin{pmatrix}
\begin{array}{cccccccc}
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
1 & 1 & 0 & 0 & ...
4
votes
2
answers
460
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Non-asympototic version of Gelfand's formula
Let $A$ be a $n\times n$ matrix. Let $\|A\|$ be the spectral norm of $A$, and $\rho(A)$ be the spectral radius. I am wondering whether the following statement is true.
There exists universal ...
4
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0
answers
452
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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 ...
1
vote
0
answers
120
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Algorithm for finding eigenfunctions
I have an $ L^2(\mathbb{R}) $ operator that looks like
$$
\Omega = \int \partial\phi(a, b)\ \ |b, a\rangle \langle b, a |,
$$
where $ \langle x | a, b \rangle = f_a(x - b) e^{x^2/2} $ and $ f_a \in L^...
2
votes
1
answer
310
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Is there an algorithm to test whether a vector is an eigenvector of a power of a matrix?
Given a square matrix $A\in k^{n\times n}$ and a vector $x\in k^n$ over some field $k$, is there an algorithm to test whether there are $s\in\mathbb{N}$ and $\lambda\in k$ such that $A^sx=\lambda x$? ...
1
vote
0
answers
156
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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 ...
4
votes
2
answers
498
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XOR circulant matrices?
Take a function $f: Z_N\rightarrow R$. Construct an $N \times N$ matrix where the $(i,j)$th element of the matrix is $f(i-j)$, where $i-j$ is interpreted mod $Z_N$. The resulting matrices are ...
2
votes
1
answer
157
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Shared maximum eigenvector
Let us consider two arbitrary Hermitian square matrices $\mathbf{A,B}$ with the same dimension. Given $\mathbf{v}$ the eigenvector associated to the maximum eigenvalue of $\mathbf{A}$:
Are there ...
3
votes
1
answer
222
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Concentration and Correlation for Magnitudes of Gaussian Vectors
Suppose we have a large collection of standard normal random variables $a_i\in\mathbb{R}^n$. We know by standard concentration results that if we take $m \geq C\left(t/\epsilon\right)^2n$ samples, ...
1
vote
1
answer
3k
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Hadamard Product and Eigendecomposition
I just found this related question in here Q1.
Given a positive definite matrix $\mathbf{A}$, consider its eigendecomposition $(\mathbf{A}\mathbf{V} = \mathbf{V}\mathbf{D})$. Consider an arbitrary ...
1
vote
0
answers
510
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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 ...
6
votes
1
answer
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Eigenvectors as continuous functions of matrix - diagonal perturbations
The general question has been treated here, and the response was negative. My question is about more particular perturbations. The counterexamples given in the previous question have variations not ...
2
votes
1
answer
78
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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 $\mathbb{R}^3$...
3
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2
answers
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Derivative of an eigenvector with respect to his own 3x3 real symmetric matrix
$\mathbf{C}$ is a real, positive-definitive 3x3 symmetric matrix (I am thinking about the right Cauchy-Green tensor in solid mechanics). We perform eigendecomposition and get:
$$\mathbf{C} = \sum_{i=...
3
votes
1
answer
544
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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}$...
4
votes
1
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Why are 1 and -1 eigenvalues of this matrix?
This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix $\mathbf{A}$.
First, let's define two matrices:
...
2
votes
0
answers
723
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Reference: Continuity of Eigenvectors [closed]
I am looking for an appropriate reference for the following fact. I already posted on math.stackexchange, but got no answer.
For each $X \in \mathbb{R}^{n \times n}_{\text{sym}}$ (symmetric matrix),...
1
vote
0
answers
263
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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 ...
0
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0
answers
224
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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 ...
1
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3
answers
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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 ...
11
votes
3
answers
8k
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Eigenvectors of a symmetric positive definite Toeplitz matrix
I wish to efficiently compute the eigenvectors of an n x n symmetric positive definite Toeplitz matrix K. A full eigendecomposition would be even better.
Although I assumed this would be a well ...
1
vote
0
answers
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 ...
3
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0
answers
124
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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^...
7
votes
1
answer
196
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Compute only selected components of an eigenvector
I am wondering whether it is possible to compute portions of the eigenvectors of a given (possibly very big) matrix. More formally, consider the eigenvalue problem $\mathbf{Ax} = \lambda \mathbf{x}$, ...
4
votes
1
answer
725
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Helmholtz equation Poynting vector integral
The Maxwell's equation for harmonic time dependent field in vacuum is
\begin{align}
\nabla \times B + i\omega E &= 0\\
\nabla \times E - i\omega B &= 0 \\
\nabla \cdot B &= 0 \\
\nabla \...
1
vote
1
answer
175
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Linear map with two "incompatible" representations
Let $K$ be a field and let $V$ be the set of sequences $\{v_1,v_2,\dots\}$ of elements of $K$. If $A=\{a_1,a_2,\dots\}$ is also a sequence of elements of $K$, then it defines an endomorphism of $V$ $$...
1
vote
1
answer
214
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True or false: if a set of 2D points has valid symmetry axes, then at least one of them is equal to a principal component vector
I posted this question on math.stackexchange but got no answer, so I decided to post it here instead. Sorry about the impreciness, not professional mathematician here.
Let's assume we have a set of ...
0
votes
1
answer
630
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Kneser graphs eigenvalues
Basically, I want to prove that, in the Kneser graph (wikipedia has a good definition),$K_{n, m}$, if $n_{-}(A(G)) $ and $n_{+}(A(G))$ denote the number of negative and positive eigenvalues of A(G) ...
1
vote
0
answers
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 ...
1
vote
1
answer
134
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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, ...
1
vote
0
answers
143
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Bounding Rayleigh quotient for stochastic matrix
Suppose you have an irreducible, stochastic matrix $A$ with left Perron-Frobenius eigenvector $v$ (corresponding to the eigenvalue $1$), and suppose the next largest eigenvalue for $A$ is $\lambda$. ...
6
votes
1
answer
486
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Recovering Spherical Harmonics from Discrete Samples
Consider a collection of $N$ points on the 2-sphere chosen uniformly at random. Let's say that there's an edge between two such vertices if their geodesic distance is less than $r_N$. The resulting ...
1
vote
0
answers
130
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interpretation of generalized eigenvalue/vectors in spectral graph theory [closed]
Let us say I have a symmetric graph adjacency matrix A, a degree matrix D, a laplacian L (D-A). I have a generalized eigenvalue equation $Av=\lambda Lv$. Does the eigenvalue/vectors produced in this ...