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-5
votes
0answers
35 views

prove that, for every markov matrix M, there exists a vector x such that Mx = x [closed]

(A Markov matrix is a square matrix whose columns are probability vectors) This proof I found seems simple at first but I can't understand the last step. If you have another proof, it would be ...
1
vote
1answer
57 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 $...
0
votes
0answers
46 views

optimization with eigenvector composition [on hold]

as the figure shows, I would like to minimize v, and also to solv the corresponding x vector. The solution has shown in the figure. I cannot solve x vector on my own. Help, with thanks...
-2
votes
1answer
87 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
84 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 ...
3
votes
0answers
148 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
114 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
103 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
121 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
751 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
107 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
635 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
121 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
58 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
60 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
43 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
18 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
119 views

Finding a combined matrix that can minimise the mean square error.

Suppose we have a rank deficient of $k$ different covariance matrices $R_{k} \in C^{N \times N}$.Different $R_{k}$ are computed by utilizing the Toeplitz structure and they are all positive ...
0
votes
0answers
48 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
413 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
122 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
88 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
64 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
105 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
98 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
53 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
67 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
31 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
173 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
151 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
157 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
193 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
198 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
105 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
64 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
312 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 $...
2
votes
0answers
109 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
217 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
161 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
119 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
114 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
673 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 = ...
1
vote
0answers
61 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} & \...
3
votes
1answer
250 views

Upper bounds on the condition number of the eigenvector matrix

Let $A$ be an $n\times n$ real matrix with entries in a fixed interval $[a_\min,a_\max]$, with $a_\min$, $a_\max>0$. Question: Are there any upper bounds on the condition number of the ...
4
votes
2answers
1k views

SOLVED: How to retrieve Eigenvectors from QR algorithm that applies shifts and deflation

After having googled for several days without locating a definitive answer, I will try my luck here! I have implemented a version of the QR algorithm to calculate Eigenvalues and hopefully ...
-1
votes
1answer
46 views

Finding a matrix with shared eigen vectors with a given matrix [closed]

If I have a known matrix A, is there a method to find a matrix B that share all the eigen vectors of Matrix A?
4
votes
1answer
195 views

complexity of computing the singular vector corresponding to the smallest singular value

It is known that the singular value decomposition of an $m \times n$ matrix $A$ is in general of complexity of the order $m n^2$, assuming that $m \ge n$. But what if we only want to compute say the ...