# Questions tagged [eigenvector]

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

199
questions

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80 views

### Eigenvalues of adjacency matrix of a k-regular graph

If $A_G$ is the adjacency matrix of a k-regular graph, let $B = J+xA_G$, where J is the matrix whose elements are all 1s and $x\in R$ is a scalar. If $\lambda_1\geq\lambda_2\geq \dots \geq \lambda_n$ ...

**0**

votes

**0**answers

28 views

### Perturbation analysis and sensitivity of eigenvector matrix product with specific perturbation

In my research in applied linear algebra and probability (Wiener filtering) I have come across this rather interesting problem:
For a matrix $ U $ we denote by $ U_k $ the matrix formed by taking ...

**2**

votes

**1**answer

99 views

### Common eigenvalues for two Sturm-Liouville problem

Does exist in literature any results concerning the common eigenvalues for the two eigenvalue problems of the form
$$y''(x)=\lambda^2 y(x)+\lambda a(x)y(x), \ x\in(0,1), $$$$z''(x)=\lambda^2 z(x)-\...

**1**

vote

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41 views

### Spectral theorems for generalized Hermitian matrices

Let $k$ be a field, and let $\sigma$ be a nontrivial involutory automorphism of $k$. Let $A$ be a square matrix with entries in $k$, such that $(A^{\sigma})^T = A$; here $A^\sigma$ means the matrix $(...

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54 views

### Eigenvector of Hadamard matrix functions

Let $X\in\mathbb{R}^{n\times n}$ with SVD $X=UDV^T$. Are there known results regarding the eigenvectors of $Y=X^{\odot g}$? I am mainly interested in simple functions such as $g(z)=z^2$, i.e. $Y_{ij}=...

**2**

votes

**0**answers

87 views

### Relationship between eigenvectors of projected and original matrix

Let $A = \mathrm{Diag}(\lambda_1, \dots, \lambda_n)$ where $\lambda_1 \le \lambda_2 \dots \le \lambda_n$. Let $P = I - ww^T$ be a projection operator on an arbitrary $n$-dimensional hyperplane. Let $B ...

**4**

votes

**1**answer

275 views

### 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-...

**5**

votes

**1**answer

534 views

### 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 $B$.
...

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votes

**2**answers

797 views

### 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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105 views

### 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 ...

**2**

votes

**1**answer

127 views

### 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 ...

**0**

votes

**1**answer

203 views

### 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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votes

**0**answers

71 views

### Difference between second largest and smallest eigenvalue

The question is related to spectral graph theory. Wrt Fiedler
number and algebraic connectivity of graphs, sometimes academic literature uses second largest eigenvalue, sometimes second smallest. For ...

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vote

**1**answer

205 views

### 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 ...

**6**

votes

**0**answers

78 views

### 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 ...

**2**

votes

**1**answer

195 views

### 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\\
...

**1**

vote

**0**answers

80 views

### 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{...

**0**

votes

**1**answer

104 views

### 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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71 views

### 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}{\...

**5**

votes

**0**answers

376 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: ...

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134 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

308 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(...

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**0**answers

97 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

53 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

63 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

85 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

45 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**

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**0**answers

98 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) = \...

**1**

vote

**0**answers

65 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

594 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

105 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 $A$?

**4**

votes

**1**answer

122 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**

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**0**answers

221 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^...

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votes

**1**answer

119 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

128 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 $(\...

**4**

votes

**1**answer

118 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

174 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

1k 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 ...

**6**

votes

**2**answers

388 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

824 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,\...

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**0**answers

142 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

72 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 $...

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**0**answers

104 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&...

**0**

votes

**1**answer

130 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) ...

**2**

votes

**1**answer

144 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,

**3**

votes

**1**answer

2k 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

598 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

102 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

84 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

140 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-...