Questions tagged [eigenvector]

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3
votes
1answer
245 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 ...
0
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0answers
32 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 ...
9
votes
2answers
337 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, \...
2
votes
1answer
59 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\...
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0answers
40 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 ...
1
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1answer
122 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 ...
2
votes
1answer
155 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 ...
1
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0answers
64 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 ...
3
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1answer
69 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} ...
0
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1answer
65 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 ...
13
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1answer
336 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 ...
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0answers
84 views

Geometric interpretation of generalized eigenvalue problem

I'm trying to shed some light on a recurrent problem I find while studying control systems. In many of the systems I work with, their stability depends on the eigenvalues of a matrix $B = U^{-1}A$, ...
1
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1answer
89 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{...
2
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0answers
23 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} - ...
2
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1answer
144 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 \...
4
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1answer
220 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 ...
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0answers
48 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 ...
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0answers
50 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 ...
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0answers
41 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, \...
10
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1answer
627 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 $A$ ...
2
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0answers
109 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 ...
1
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2answers
142 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 ...
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0answers
224 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,...
6
votes
1answer
192 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 \...
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0answers
21 views

Eigenvector of $U |y \rangle \equiv |xy (mod N) \rangle$, when $ x \le N$ and $x, N$ coprimes

In the book quantum computing and quantum computation of Nielsen and Chuang, in the chapter relating to order-finding, they say that via a simple calculation, an eigenvector of $U |y \rangle \equiv |...
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0answers
88 views

Lower bound for eigenvalue problem with single linear constraint

Consider the problem $$\max_x x^\top B x$$ subject to $\|x\|=1$ and $b^\top x = a$, where $b$ is a unit vector but not necessarily an eigenvector of $b$. Suppose that $B$ is symmetric and positive ...
1
vote
1answer
168 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 $V$ ...
1
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1answer
277 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 $\lambda_1\geq\lambda_2\geq \dots \geq \lambda_n$ ...
2
votes
1answer
108 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)-\...
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0answers
45 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 $(...
1
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0answers
95 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
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0answers
93 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 ...
5
votes
1answer
450 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 ...
6
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1answer
2k 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 $B$. ...
11
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2answers
1k 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 $$ H_N=\begin{pmatrix}H_{N/2} & H_{N/2} \\ H_{N/2} & -H_{N/2}\end{pmatrix}, $$ ...
1
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0answers
202 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
1answer
145 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
1answer
359 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 ...
1
vote
1answer
336 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
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0answers
80 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
1answer
243 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
0answers
81 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
1answer
109 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 ...
1
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0answers
83 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
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0answers
390 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
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0answers
307 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
2answers
313 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
0answers
139 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
0answers
64 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
1answer
68 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, ...

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