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

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### 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 ...

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vote

**1**answer

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 $...

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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...

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**1**answer

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

**1**answer

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 ...

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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^...

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votes

**1**answer

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 $R_s = U_s\Lambda_sU_s^*$. ...

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votes

**1**answer

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 $(\...

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**1**answer

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}}...

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**1**answer

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 ...

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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 ...

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**2**answers

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 ...

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**2**answers

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,\...

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**0**answers

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 ...

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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 $...

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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&...

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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) ...

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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}'=\...

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**1**answer

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 ...

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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 ...

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**1**answer

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 \\
&...

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**1**answer

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}^{...

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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 ...

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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 ...

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**1**answer

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-...

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**1**answer

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-...

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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 ...

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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 ...

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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) ...

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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}
...

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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 ...

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**1**answer

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 ...

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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 ...

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**1**answer

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 $\...

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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&...

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### 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
$Y=U\Lambda U^T$...

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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 ...

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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 $...

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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 &...

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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 ${\bf X}$ ...

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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 ...

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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 ...

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**2**answers

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 ...

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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)

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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 = ...

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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} & \...

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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 ...

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### 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 ...

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**1**answer

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?

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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 ...