Questions tagged [eigenvector]

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Diagonalizing a symmetric block matrix

Let us consider the matrix $$ A = \begin{pmatrix} a & c+ib \\ c-ib& a \end{pmatrix},$$ then this matrix has eigenvalues $a\pm \sqrt{c^2+b^2}.$ Now, let us consider a block matrix $$ A = \begin{...
Guido's user avatar
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5 votes
0 answers
206 views

Perturbation of Neumann Laplacian

Consider the $N \times N$ matrix $$A_{\alpha}=\begin{pmatrix} \lambda_1 & -1 & -\alpha & 0 & \cdots & 0\\ -1 & \lambda_2 & -1 & -\alpha & \cdots & 0\\ -\alpha &...
Guido Li's user avatar
4 votes
2 answers
361 views

How close to uniform are Perron-Frobenius eigenvectors?

Let $A=(a_{i,j})$ be a square matrix with non-negative entries. (Assume $A$ is symmetric, if it helps.) Let $v$ be a Perron-Frobenius eigenvector. What do we need to assume about $A$ in order to have ...
H A Helfgott's user avatar
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When must an eigenvector have only non-negative entries?

What would be a reasonable sufficient condition on a real symmetric matrix that would force its eigenvector with largest eigenvalue (or one of its eigenvectors with maximal eigenvalue) to have only ...
H A Helfgott's user avatar
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1 vote
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146 views

Eigenvalues and eigenvectors of non-symmetric elliptic operators

We know that the operator $A=\Delta$ with domain $D(A)=\{u\in W^{2, 2}(\Omega): u=0 \ \ \text{on } \partial\Omega\}$ (say $\Omega$ is a bounded nice domain) has eigenvalues $\lambda_1>\lambda_2\ge \...
Y Wu's user avatar
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1 vote
1 answer
384 views

Trace minimization for generalized eigenvalue problem

In [1], it is shown in theorem 1.2 that for symmetric $n \times n$ matrices $A$, $B$, we have $$ \min_{Y \in Y^*} \text{tr}(Y^TAY) = \text{tr}(X^TAX) = \sum_{i=1}^p \lambda_i, $$ with $$ \text{ $X^...
drommedaris's user avatar
0 votes
0 answers
261 views

Eigenvector to zero eigenvalue of general Laplacian

I was wondering what we can say about the eigenvectors of a matrix $A$ fullfilling $Ax =0$ where $A$ is symmetric with a diagonal equal to one and every row sums up to 0. Obviously this is a ...
RZA Chris's user avatar
0 votes
1 answer
645 views

Centrality measures in a network with negative correlations

I have a bidirectional network where the weights of edges are based on partial correlation matrix. I have both positive and negative values as weights. Now, I want to compute centrality measures as ...
statwoman's user avatar
  • 109
1 vote
1 answer
152 views

Derivative of eigenpair with respect to matrix

Suppose that $A$ is real and symmetric matrix (or tensor) of dimension $3 \times 3$, with its spectral decomposition $$A = \sum_{i=1}^3 \lambda_i\ n_i\otimes n_i$$ where $\lambda_i$, $n_i$ and $\...
TARS's user avatar
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3 votes
1 answer
277 views

Eigenvectors of a tensor in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$

I want to find the critical point of tensor $f=a_0b_0c_0 + a_1b_1c_1$ in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$, and I followed this construction: First, I take the following partial ...
B.K-Theory's user avatar
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67 views

Fast decay of eigenvector elements

Let A be a set of similar (symmetric) matrices, sharing the same eigenvalues. I understand that their eigenvectors would be different. Let us focus on one eigenvector (e.g. corresponding to the lowest ...
twofiveone's user avatar
10 votes
2 answers
580 views

Lower eigenvectors of nonnegative matrices with zero trace

Let $A$ be an $N\times N$ nonnegative matrix with all diagonal entries equal to zero and such that there is $n_0$ such that all entries of $A^{n_0}$ are strictly positive. Let $\lambda_1,\ldots, \...
Serguei Popov's user avatar
2 votes
1 answer
515 views

Distribution of eigenvectors of random matrices and link with the components of the matrix

Let $M$ be a real symmetric matrix of size $N$ with its components $M_{ij}$ following a normal distribution centered around 0. Let $x\in\mathbb{R}^N$ be an eigenvector of $M$ with eigenvalue $\lambda\...
Matt's user avatar
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0 answers
66 views

Conjugate gradient and the eigenvectors corresponding to the large eigenvalues [closed]

I am working on an optimization problem (for example, conjugate gradient) to solve $Ax=b$, where $A$ is a symmetric positive definite matrix. I can understand that the CG (conjugate gradient) has ...
Joshuashiing's user avatar
2 votes
1 answer
1k views

Derivative of eigenvectors of an Hermitian matrix

In the question "Derivative of eigenvectors of a matrix with respect to its components", Liviu Nicolaescu has provided an answer valid for a real matrix. As outlined in the following, the ...
S.B.'s user avatar
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2 votes
1 answer
266 views

Find a way to apply the MLE on Fisher or Covariance matrix to make cross-correlations

I have 2 Fisher matrixes which represent information for the same variables (I mean columns/rows represent the same parameters in the 2 matrixes). Now I would like to make the cross-correlations ...
youpilat13's user avatar
2 votes
0 answers
194 views

Infinite positive matrices with probability eigenvector

Let $A$ be an infinite non-negative matrix with integer entries ($a_{ij} \geq 0, \forall i,j \in \mathbb N$). Suppose that $A$ is irreducible, aperiodic, and recurrent. So that it satisfies the ...
SIB's user avatar
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4 votes
1 answer
611 views

Condition number for matrix of eigenvectors of a diagonalizable matrix

Let $A$ be a diagonalizable matrix, i.e., $A=SDS^{−1}$. For any matrix $A$, condition number is defined as $\kappa(A)=\|A\| \|A\|^{-1}$. For $A$ being a diagonalizable matrix, define $G_A=\{{S: S^{-1} ...
Das Dipayan's user avatar
0 votes
1 answer
92 views

If there are eigenvectors with largest components $i$ resp. $j$, then is there an eigenvector with two largest components $i$ and $j$?

Let $G=(V,E)$ be a connected (finite simple) graph with vertex set $V=\{1,...,n\}$ and let $\theta_2\in\Bbb R$ be the second-largest eigenvalue of its adjacency matrix. I wonder about the following ...
M. Winter's user avatar
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13 votes
1 answer
647 views

$\ell^1$-norm of eigenvectors of Erdős-Renyi Graphs

Setting. Let $G(n,p)$ denote the usual Erdős-Renyi (random) graphs. For each such graph there is an associated Laplacian matrix $L = D - A$ where $D$ collects the degrees on the diagonal and $A$ is ...
Stefan Steinerberger's user avatar
1 vote
1 answer
133 views

Requirements for finite backward derivatives of degenerate eigenvectors

A matrix, $\mathbf{A}(\theta)\in\mathbb{R}^{n\times n}$, has elements that depend on a parameter $\theta$. The $j$-th eigenvalues and eigenvectors of the matrix are denoted as $\lambda_j$ and $\mathbf{...
Firman's user avatar
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2 votes
0 answers
41 views

References on discrete Sturm-Liouville eigenvectors convergence

Let $ L : u_n \mapsto a_n u_{n + 1} + b_n u_n + a_{n - 1} u_{n -1} = \nabla ( a_n \Delta u_n ) + (b_n + a_n + a_{n - 1}) u_n $ be a discrete Sturm-Liouville operator, with $ \nabla u_n := u_{n + 1} - ...
Synia's user avatar
  • 549
2 votes
1 answer
842 views

Calculating second derivatives of eigenvectors of a matrix with some degenerate eigenvalues

Given real symmetric matrix $\mathbf{M}$ with eigenvalues $\lambda_i$ and eigenvectors $\mathbf{v}_i$, the derivative of an eigenvector is $$\dot{\mathbf{v}}_i = \sum_{j \ne i} \frac{\mathbf{v}_j \...
ehermes's user avatar
  • 23
5 votes
1 answer
863 views

Real eigenvectors of complex matrices

Let $A$ be a nonsingular complex $(3 \times 3)$-matrix (that is, an element of $\mathrm{GL}_3(\mathbb{C})$). Then what are some of the best-known criteria which guarantee $A$ to have real eigenvectors ...
THC's user avatar
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0 votes
0 answers
148 views

Minimize a vector from a matrix operation

I want to minimize a certain vector that results from a matrix operation with some constraints and i don't exactly know how to tackle this problem. Lets say we have $$ (L+A)*s = v $$ L is the ...
leo_bouts's user avatar
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0 answers
77 views

Eigendecomposition of $A=I+BDB^H$

Suppose that we have $$A = I_m + BDB^H$$ where matrix $A$ is $m \times m$, matrix $B$ is $m \times k$, $BB^H \neq I_m$ and $D$ is a $k \times k$ diagonal matrix. Can we obtain the eigendecomposition ...
user164237's user avatar
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50 views

What transformation is required to find a unique solution of this problem instead of multiple solutions?

$$ \max\limits_{\mathbf{f},\ \|\mathbf f\|=1 } \log_2\left(\prod^K_{i=1} \ \frac{ \mathbf{f}^H {\mathbf E} (\mathbf{W}_i, \Theta, \tau_i) \mathbf{f}} { \mathbf{f}^H \mathbf{G}_i ( \mathbf{W}_i, \...
syam's user avatar
  • 1
11 votes
1 answer
2k views

Eigenvalues of the complement of a graph

Let $A$ and $\widetilde A$ be the adjacency matrices of a graph $G$ and of its complement, respectively. Is there any relation between the eigenvalues of $A + \widetilde A$ and the eigenvalues of $A$ ...
GA316's user avatar
  • 1,219
2 votes
0 answers
114 views

Is $A$ is small on bounded functions, is there a large subdomain on which $A$ is small?

Let $A$ be a symmetric linear operator of norm $\leq 1$ on the space of functions $f:S\to \mathbb{R}$, where $S$ is a set with $N$ elements. Define the inner product $\langle \cdot,\cdot\rangle$ by ...
H A Helfgott's user avatar
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1 vote
2 answers
150 views

From one eigenvector to many, in a very local graph?

Let $\Gamma$ be an undirected graph of bounded degree $d$ with $V = \{1,2,\dotsc,N\}$ as its set of vertices, and edges only between vertices that are at a distance $\leq M$ apart (where $M$ is much ...
H A Helfgott's user avatar
  • 19.3k
0 votes
0 answers
234 views

Eigenvectors of a matrix

Let $M$ be a square matrix of order $n\times d$. Let $\xi_{1},\dots,\xi_{n\times d}$ be an orthonormal basis of $\mathbb{R}^{n\times d}$ formed of eigenvectors of $M$. We have $$\xi_{i}=(\lambda_1, 0,...
yassine yassine's user avatar
6 votes
1 answer
284 views

Continuity of eigenvectors

Let $\mathbb{C} \ni z \mapsto M(z)$ be a square matrix depending holomorphically on a parameter $z$ with the property that $\operatorname{dim}\ker(M(z)))=1$ for $z $ away from a discrete set $D \...
Sascha's user avatar
  • 506
1 vote
1 answer
481 views

Simultaneous diagonalization in Matlab [closed]

Crossposted from StackOverflow. The generalised diagonalization of two matrices $A$ and $B$ can be done in Matlab via [V,D] = eig(A,B); where the columns of $V$ ...
MathMax's user avatar
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1 vote
1 answer
1k 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 $\lambda_1\geq\lambda_2\geq \dots \geq \lambda_n$ ...
RayyyyySun's user avatar
2 votes
1 answer
132 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)-\...
Gustave's user avatar
  • 545
1 vote
0 answers
62 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 $(...
THC's user avatar
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2 votes
0 answers
254 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}=...
JZRedw's user avatar
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2 votes
0 answers
211 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 ...
Dmitry's user avatar
  • 221
5 votes
1 answer
543 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-level) math-essays ...
Gottfried Helms's user avatar
8 votes
1 answer
5k 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 $B$. ...
Henrique de Oliveira's user avatar
12 votes
2 answers
2k 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 $$ H_N=\begin{pmatrix}H_{N/2} & H_{N/2} \\ H_{N/2} & -H_{N/2}\end{pmatrix}, $$ ...
MCH's user avatar
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1 vote
0 answers
384 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 ...
Dirk B.'s user avatar
  • 111
2 votes
1 answer
472 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 ...
user144910's user avatar
0 votes
1 answer
796 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 ...
MAS's user avatar
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1 vote
2 answers
915 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 ...
YtroS's user avatar
  • 11
6 votes
0 answers
93 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 ...
M. Winter's user avatar
  • 12.5k
2 votes
1 answer
437 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\\ ...
user293017's user avatar
1 vote
0 answers
86 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{...
afra's user avatar
  • 21
0 votes
1 answer
122 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 ...
dwood04's user avatar
1 vote
0 answers
92 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}{\...
shuai's user avatar
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