# Questions tagged [eigenvector]

The eigenvector tag has no usage guidance, but it has a tag wiki.

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33 views

### Exact eigendecomposition of a specific Toeplitz matrix

I am interested in diagonalizing a general $n \times n$ matrix with entries of the form
\begin{equation}
\frac{1}{|f_i-f_j|^p} \hspace{40px} 1 \le i,j \le n
\end{equation}
where $f_i,p \in \mathbb{R}$ ...

**5**

votes

**0**answers

347 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: ...

**1**

vote

**0**answers

44 views

### 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-...

**10**

votes

**2**answers

297 views

### 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(...

**1**

vote

**0**answers

73 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}
\...

**2**

votes

**0**answers

26 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,...

**1**

vote

**1**answer

54 views

### 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, ...

**1**

vote

**1**answer

81 views

### 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+...

**0**

votes

**1**answer

34 views

### 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^...

**2**

votes

**0**answers

53 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) = \...

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votes

**0**answers

25 views

### Eigenfunction of Distorted Laplacian on Smooth compact domain

Setup
Suppose that $D$ is a compact star-shaped domain in $\mathbb{R}^d$ which is diffeomorphic to a closed $d$-dimensional ball in $\mathbb{R}^d$. Let $a(t,x)>0$ be a class $\mathscr{C}^{\infty}(\...

**1**

vote

**0**answers

47 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 \...

**1**

vote

**1**answer

117 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 $...

**-2**

votes

**1**answer

100 views

### 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 $A$?

**3**

votes

**1**answer

89 views

### 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 ...

**4**

votes

**0**answers

160 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^...

**2**

votes

**1**answer

116 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 $R_s = U_s\Lambda_sU_s^*$. ...

**0**

votes

**1**answer

69 views

### 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 $(\...

**3**

votes

**1**answer

107 views

### 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}}...

**2**

votes

**1**answer

129 views

### 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 ...

**12**

votes

**3**answers

831 views

### 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 ...

**4**

votes

**2**answers

163 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 ...

**15**

votes

**2**answers

721 views

### 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,\...

**2**

votes

**0**answers

129 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 ...

**6**

votes

**0**answers

61 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 $...

**2**

votes

**0**answers

67 views

### 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&...

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votes

**1**answer

52 views

### 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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**0**answers

21 views

### Recursive computation of eigenvectors when a new column and row are added.

Assume that we have a matrix ${\bf A}={\bf X}{\bf X}^{\top}$, and we know their eigenvalues and eigenvectors. Is it possible then to compute the new eigenvalues and eigenvectors of matrix
$${\bf A}'=\...

**2**

votes

**1**answer

141 views

### 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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54 views

### Finite rank perturbations of matrices - eigenvalues and eigenvectors

Assume we know the spectral decomposition (eigenvalues and eigenvectors) of a $n\times n$ matrix $A$. Consider a finite rank perturbation of $A$ of the form $B=UV^{T}$ where $U$ and $V$ are $n\times ...

**3**

votes

**1**answer

691 views

### 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 \\
&...

**1**

vote

**1**answer

182 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}^{...

**0**

votes

**0**answers

91 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 ...

**1**

vote

**1**answer

66 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-...

**1**

vote

**1**answer

115 views

### 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-...

**0**

votes

**0**answers

112 views

### Eigenvalues and eigenvectors of “interleaved” circulant-like matrices

Let $k|n$ and $k\geq 2$ and consider $k$ matrices $$A,B,\ldots,K \in \mathbb{R}^{n\times n}.$$
Consider a $k-$interleaving of these matrices where we take every $k^{th}$ row in order from the given ...

**2**

votes

**0**answers

58 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 ...

**4**

votes

**0**answers

91 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) ...

**0**

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36 views

### Normalise radial eigenfunction - Laguerre power exponential

I am trying to normalise an eigenfunction of the form
\begin{align}
\nu(r,\theta) = r^{l-n} e^{-\frac{r^2}{2C}} L_n^{l-n}\left( \frac{r^2}{C} \right)
cos((l-n) \theta),
\end{align}
...

**2**

votes

**2**answers

182 views

### 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 ...

**2**

votes

**1**answer

179 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 ...

**1**

vote

**0**answers

186 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 ...

**6**

votes

**1**answer

227 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 $\...

**9**

votes

**1**answer

202 views

### 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&...

**1**

vote

**0**answers

124 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
$Y=U\Lambda U^T$...

**0**

votes

**0**answers

70 views

### Relaxed Condition for Rayleigh Quotient Theorem?

According to Matrix Analysis by Roger A. Horn and Charles R. Johnson, a theorem related to Rayleigh Quotient is as follows (my compact version):
[Theorem 4.2.2 (Rayleigh Quotient) ]
Let $A \in ...

**2**

votes

**0**answers

422 views

### 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 $...

**3**

votes

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117 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 &...

**2**

votes

**0**answers

25 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 ${\bf X}$ ...

**1**

vote

**1**answer

236 views

### Eigenvector of a nonnegative matrix in closed form

Consider $n\times 1$ vector $\alpha = (\alpha_{1}, ..., \alpha_{n})$, where $0<\alpha_{i}<1$, and $\sum_{i=1}^{n}\alpha_i = 1$. Construct the $n\times n$ zero-diagonal matrix $A$ with $(i,j)$-th ...