Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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

0
votes
1answer
28 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
0answers
51 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) = \...
0
votes
0answers
24 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
0answers
43 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 \...
0
votes
0answers
30 views

Eigenvalue sensitivity matrix calculation for perturbation.

I have a matrix $J(x)=J_o+J_d(\Delta x)$. I got the expression of eigenvalue sensitivity matrix by partial differentiating the relation of eigenvalue and eigenvectors: $$\frac {\partial\lambda}{\...
0
votes
0answers
89 views

Eigenvectors of convex combinations of stochastic matrices

Suppose we have $k$ square ($n$ by $n$) stochastic/probability matrices, $M_1, M_2,\dots ,M_k$ (so for each matrix the entries are non-negative and all column sums are one). Suppose we have a ...
1
vote
1answer
69 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
1answer
98 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
1answer
86 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
0answers
156 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
1answer
115 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
1answer
58 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
1answer
105 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
1answer
126 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
3answers
780 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
2answers
123 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 ...
14
votes
2answers
703 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
0answers
124 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
0answers
60 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
0answers
66 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
1answer
45 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) ...
0
votes
0answers
19 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
1answer
138 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,
0
votes
0answers
49 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
1answer
519 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
1answer
143 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
0answers
90 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 ...
0
votes
0answers
40 views

Optimal measurement by projectors for density operators!

Theorem: Let $\{\rho_i,1\leq i\leq m\}$ be a quantum state ensemble consisting of linearly independent density operators $\rho_i$ with prior probabilities $p_i$. Then the optimal measurement is a von ...
1
vote
1answer
65 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
1answer
107 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
0answers
103 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
0answers
56 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
0answers
72 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
votes
0answers
33 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
2answers
178 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
1answer
162 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
0answers
168 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
1answer
222 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
1answer
199 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
0answers
109 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
0answers
66 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 ...
1
vote
0answers
351 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
0answers
113 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
0answers
24 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
1answer
224 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 ...
0
votes
0answers
172 views

Continuity/differentiability of eigenvectors corresponding to semisimple eigenvalues

From my reading and intuition, I'm pretty sure that the following is true: The eigenvectors corresponding to semisimple eigenvalues (i.e. algebraic multiplicity = geometric multiplicity) of the ...
2
votes
2answers
120 views

Solving linear system when one eigenvalue is known

I have a huge sparse linear system $Ax = b$ where I know that an eigenvalue/eigenvector pair is $1$ and a vector of all $1$'s. Can this knowledge help me in solving the linear system at all? It seems ...
0
votes
0answers
130 views

product of eigenvalue vs. eigenvalue of product of matrices

Is there any relationship between the product of eigenvalues and eigenvalues of product of two matrices? eig(A)eig(B) vs. eig(AB)
5
votes
3answers
764 views

Proving that a certain non-symmetric matrix has an eigenvalue with positive real part

Suppose that $X$ is the $n \times n$ matrix of all ones $Y$ is an arbitrary $n \times n$ matrix with zeroes on the diagonal and all other entries equal to $0$ or $1$ $0 < \delta < 1$ Let $Z = ...
2
votes
0answers
66 views

Commutation relation and eigenvectors of infinite matrices [closed]

I'm given the Matrix $A$ and $A^T$: $A = \begin{bmatrix} 0 & 1 & 0 & 0 & \dots \\ 0 & 0 & \sqrt{2} & 0 & \dots \\ 0 & 0 & 0 & \sqrt{3} & \...