The eigenvalues tag has no wiki summary.

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### when can I say that $UV^T$ is a permutation matrix? [closed]

suppose we have two p.s.d matrices A and B: so we can diagonalize them like this:
A= $UΛU^T$ and $B=VΣV^T$
1: on what condition for $A$ and $B$ I can say that $UV^T$ is a permutation matrix?
2: how ...

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### what is the best algorithm complexity known for finding a p.s.d symmetric matrix eigenvalues? and eigenvectors? [closed]

I dont know if I can ask this question here or not, if I should delete it tell me:(
what is the best algorithm complexity known for finding a p.s.d symmetric matrix eigenvalues? and eigenvectors?

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### Independence of Eigenvalues of Wishart

This question regards a previous post, but it is not immediately obvious the two are related, so I ask it anyways: are the eigenvalues of a Wishart matrix $\mathbf{S}$ $=$ ...

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### Dipole Transition Integrals - Acceleration Form, What's Wrong?

I should have posted this question in a physics forum, but I think by posting in MathOverflow I may get more responses.
The following question may sound stupid, since I'm sure I was wrong somewhere, ...

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### Eigenvalue of a linear map over finite field

Let $ F_q $ be a finite field with $ q $ elements.
Let $ g $ be a multiplicative generator of $ F_{q^2}^* $.
It implies that
$ <g^{q+1}> = F_q^* $.
Let $ l $ be a prime greater than $ q^2-1 ...

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### Eigenvalues of Symmetric Tridiagonal Matrices

Suppose I have the symmetric tridiagonal matrix:
$ \begin{pmatrix}
a & b_{1} & 0 & ... & 0 \\\
b_{1} & a & b_{2} & & ... \\\
0 & b_{2} & a & ... & 0 ...

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### Eigenvalue problem

I am studying torsional Alfven waves in spicules.
In this concern I have encountered the following equation:
$
\left(1-m^2 e^{-αz}\right)y''(z)+\left(4π i m ...

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### inverse of partial differential operator

I have a bounded degree hermitian partial differential operator over $\mathbb{R}^3$:
$D=\sum_{i,j,k,l,m,n\in{\{0,1,..5\}}} a_{i,j,k,l,m,n} x^iy^jz^k \frac{\partial^l}{\partial x^l} ...

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### Anti-bidiagonal matrix with main anti-diagonal {1,2,3,…} and first sub-anti-diagonal {-1,-2,-3,…} has eigenvalues lambda={1,-2,3,-4,…}

Consider the anti-bidiagonal matrix $B_6\in\mathbb{R}^{6\times 6}$, defined along its anti-diagonals as follows
$$
B_6=\begin{bmatrix} & & & & & 6\\
& & & ...

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### the eigenvalues of a generalized circulant matrix

A $2k\times 2k$ circulant matrix $\ C$ takes the form
\begin{align}
C= \begin{bmatrix} c_0 & c_{2k-1} & \dots & c_{2} & c_{1} \\
c_{1} & c_0 & c_{2k-1} & & c_{2} ...

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### What is the significance of the median eigenvalue?

When I look at the spectral density plots of my (usual) laplacian graphs, they spike at the median eigenvalue. But what significance for the graph/matrix (which originates from a network) does the ...

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### Linearly constrained eigenvalue problem

Suppose I'd like to:
\begin{align}
\mathop{\text{min}}_\mathbf{x} && \mathbf{x}^T\mathbf{A}\mathbf{x} \\
\text{subject to:} && \mathbf{x}^T \mathbf{M} \mathbf{x} = 1\\
&& ...

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### Zeroes of Sturm-Liouville solutions as a function of the (complex) eigenvalue

Given the Sturm-Liouville type (time independent Schroedinger) equation
\begin{equation}
\frac{d^2 y}{d x^2} - \left(\mu + V(x)\right) y = \lambda \, y,\quad x \in \mathbb{R}
\end{equation}
where ...

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### Estimating singular values of integral operators

I would like to estimate the singular values of certain trace class integral operators. For the sake of concreteness, consider on $L^2({\mathbb R},dx)$ the integral operator $$(Tf)(x)=\int_{\mathbb ...

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### MInors related problem [closed]

A matrix $A$ has $m$ rows and $n$ colums, such that $m \leq n$. We know that each row of $A$ has the norm $1$ (the norm of an element $x=(x_1,x_2,...,x_n) \in \mathbb{R}^n$ is ...

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### Asymptotic eigenvalue analysis for a sparse random matrix

We have an asymptotic analysis problem for the eigenvalue performance of the following random matrix:
$H=\{h_{ij}\}_{N_r\times N_t}$,
where each entry $h_{ij}$ is with a probability $p$ to obey the ...

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### Eigenvalues of this matrix

I have a linear map that is defined by $$T:\text{lin}(1,...,x^m) \rightarrow \text{lin}(1,...,x^m) \text{ with}$$ $$x^k \mapsto 2w(k-m)x^{k+1}+(k^2-k-w^2)x^k-2kwx^{k-1}+(k-k^2)x^{k-2}$$
Let me give a ...

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### Eigenvalues of product of diagonal positive matrix and symmetric matrix [closed]

Assume that we have two real symmetric matrices A and B, where A is a positive diagonal matrix, and B is a symmetric matrix with one eigenvalue λ = 0. Assume that H= AB;
is it possible to proof that ...

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### Distribution of Wishart Sample Eigenvalues for Multiple Roots

I am interested in finding an asymptotic approximation to the latent roots $l_1>\dots>l_p$ of a white noise Wishart matrix $nS\sim W_p(n,I)$ as $n\rightarrow\infty$ (where $p$ is fixed). In ...

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### Trace inequality for matrices with determinant 1

Let $A$ and $B$ be two matrices with $\det(A)=\det(B)=1$. Does it follow that
$\sqrt{\mathrm{tr}(A^TB^TBA-I)}\le\sqrt{\mathrm{tr}(A^TA-I)}+\sqrt{\mathrm{tr}(B^TB-I)}$
I suspect that this can be ...

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### Toda Flow Embeddings

What are strategies for generating the following types of pictures:
Here's what's going on here. Take a toda flow in 3 variables. The equations of motion are:
...

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### Proving that the eigenvalues of a certain matrix product are positive

Let $A$ be an $m \times n$ matrix, and define:
\begin{align*}
U &= {\rm diag} \{ \frac{1}{\beta_j} \}, \beta_j = \sum_{k=1}^m |a_{kj}|, j = 1 \dots n \\
V &= {\rm diag} \{ \frac{1}{\alpha_i} ...

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### How do eigenvalues change if we duplicate a row and column of a symmetric matrix

Let ${\bf A}$ be a size $n \times n$ symmetric positive semidefinite matrix with the first column being ${\bf a}_1$. If we define a new matrix,
\begin{align}
{\bf B} = \left[\begin{array}{cc} a_{11} ...

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### Is there an easy way to tell if all eigenvalues of a unitary or self-adjoint matrix only have eigenvalues of multiplicity two?

I am interested in a class of $2n\times 2n$ unitary matrices with complex entries (if you prefer, we can replace "unitary" with "self-adjoint").
I know that all the eigenvalues of matrices in this ...

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### The relation between eigenvalue and singular value of non-symmetric square matrix

The problem bothers me for a long time.
Suppose, we have two matrix $A$ and $B$, where $A$ is a $m$ by $n$ complex matrix while $B$ is a $n$ by $m$ complex matrix.
Apparently, $AB$ and $BA$ have the ...

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### Existence of a real eigenvalue

I have a matrix $M \in \mathbb{R}^{(n+1) \times (n+1)}$ that is tridiagonal.
In numerical computations I found out that I always find a real eigenvalue. My question is: Is there a theorem that ...

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### Alike looking matrices imply convergence of eigenvalues?

This is a question about convergence of eigenvalues which essentially came up in studying the spectrum of St.-Liouville operators.
We want to look at matrices that agree in most of their entries and ...

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### Equivalence of Positive Matrix in Infinite Dimensional Vector Space

What is the corresponding linear operator on an infinite dimensional vector space, say a Banach space or Hilbert space, to the nonnegative matrix on a finite dimensional vector space? What is the ...

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### Is this function well studied?

Let $A_1,\dots,A_L$ be $N\times N$ hermitian matrices. Define the simplex
\begin{align}
\mathcal{S}=\left\{[x_1,\dots,x_L]\mid x_i\geq 0,~\sum_{i=1}^{L}x_i=1 \right\}
\end{align}
and consider the ...

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### Comparison of the smallest eigenvalues of two tridiagonal matrices

Let $n\geq2$ be an integer and $E_{ii}$ for an integer $2\leq i\leq n$ be the $n\times n$-matrix with its $ii$-entry equal to 1 and remaining entries equal zero. Furthermore, let ...

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### Eigenfunctions to 2nd-order Differential Operators: Relation between Frobenius Series Solution and Eigenfunction Normalised to the Delta Function

Consider the 2nd-order linear ODE $x f^{''}(x) + x (\beta - 2 \alpha x) \kappa / \sigma f^{'}(x) - 1 / \sigma \left[ 2 \alpha \kappa - \lambda^2 (\beta - 2 \alpha x)^2 \right] f(x) = 0$, where ...

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### Is Rellich's function valued theorem valid for a rank defficient function valued matrix?

Theorem (Rellich). Let $\boldsymbol{A}(t) : \mathbb{R}\rightarrow\mathbb{C}^{n \times n}$ be a Hermitian matrix function that depends
on $t$ analytically.
(i) The $n$ roots of the characteristic ...

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### How could I prove this equality for eigenvalues of Laplacian matrix?

I would be glad if you have some comments that how I could prove following statement.
Suppose that graph $G =(N, E)$ be given. The the following program computes the $k$-smallest eigenvalues of the ...

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### Maximising a Rayleigh quotient over a subspace

Let $M\in\mathbb{R}^{n\times n}$ be symmetric positive definite and consider a matrix $Q\in\mathbb{R}^{n\times m}$ ($m<n$) with orthonormal columns ($Q^TQ=I$). I'm interested in finding an exact ...

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### Linear dynamical systems: interpretation of Frobenius eigenvector

Consider a positive linear dynamical system. $\frac{dx}{dt}=Ax$, where $A$ is a quasipositive/Metzler/essentially nonnegative matrix. By its properties, the vector $x$ will remain positive for all ...

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### Proving that a specific kernel is positive definite

Most theoretical papers concerning kernels assume that they are given a positive definite kernel. In this question, we want to show that a specific kernel is positive definite.
We are interested in ...

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### Eigenvalues of matrix products [closed]

Hi my problem is with row stochastic matrices. Its known that if we keep multiplying this row stochastic matrices, we will get a rank one row stochastic matrix. Rank one means that all the rows has ...

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### Does small Perron-Frobenius eigenvalue imply small entries for integral matrices?

Suppose that $M$ is an $n \times n$ matrix where each entry is a positive integer. Then $M$ is Perron-Frobenius and so has unique largest real eigenvalue $\lambda_{\textrm{PF}}$.
Does an upper ...

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### Relationship betwen eigenvectors

Suppose that we have two matrices A and B. Matrix B is taken from A with one row and one column deleted. On the other hand A is n*n matrix and B is (n-1)*(n-1) matrix and is created by deleting last ...

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### Parametrization of real diagonalizable matrices with given eigenvalues

Complex diagonalizable matrices with given eigenvalues can be conveniently parametrized as $A=T^{-1} \Lambda T$, where $T$ is any invertible matrix, and $\Lambda=diag(\lambda_1,...,\lambda_N)$ with ...

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### Using Marchenko - Pastur type Theorems on Regression Analysis

Sometimes when doing regression analysis, we estimate our function $g(x) = E(Y |X =x )$ using an orthonormal series, and in particular we use an approximate series $g_{p_n}(x) = \sum_{k=1}^{p_n} ...

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### Dominant eigenvalue of sum of tridiagonal and diagonal matrices

Suppose I have a tridiagonal square matrix $A$ of some nice form, for which I know the eigenvalues $\lambda_1<\dots<\lambda_n$. $A$ is also essentially nonnegative (nonnegative everywhere except ...

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### Any generic way to move a psd matrix to its neighbors?

Given a two positive matrices $A,B$. For simplicity, let's assume that $Tr A=Tr B=1$. Assume that $\|A-B\|_1\leq\varepsilon$, for some small $\varepsilon>0$, where $\|\cdot\|_1$ is the $l_1$-norm, ...

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### Lanczos algorithm with thick restart on a dynamic matrix

currently, I'm working on a way to compute the 2 biggest eigenvalues of a real, symmetric, huge and sparse matrix that changes a few entries from time to time. The problem should be solved using an ...

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### derivative of the dominant eigenvalue

In my reaseach, recently, I came across of this problem where I have to compute, analytically, the derivative of the dominant eigenvalue of the following matrix.
Let $D$ be a diagonal real $n ...

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### Probabilistic Interpretation of First Dirichlet Eigenvalue?

The first Dirichlet eigenvalue of a compact domain $\Omega\subset\mathbb{R}^n$ with smooth boundary is the smallest positive number for which there exists a non-trivial solution to
$$
-\Delta\psi = ...

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### Eigenvalues of a “Half-Kronecker ” Product

The Problem:
Given a 2 by 2 matrix $C$(the matrix elements of C are given), and two other
2 by 2 matrices $A$ and $B$(the matrix elements of A and B are given).
Now we can construct a new matrix ...

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### Which matrix/operator in a cone has the largest negative spectral part?

Background:
Let $\mathcal{K}$ be set (convex cone, if you like) of symmetric matrices of order $n$. Each matrix $A \in \mathcal{K}$ can be decomposed in a unique way as $A=A_{+}-A_{-}$, where ...

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### Relationship of eigenvalue/eigenvector of hermitian matrix R and QRQ (Q is diagonal)

For a hermitian matrix R and a diagonal one Q, is there any relationship between eigenvalues/eigenvectors of R and QRQ?
To be specific, assuming the eigenvalue decomposition of R is R=VDV*, then can ...

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### Algorithms to compute largest gap between smallest nonzero eigenvalues of sparse symmetric matrix

I am looking mainly for implementations but also for theoretical algorithms to compute gaps between smallest positive eigenvalues of symmetric, singular matrix or real numbers.
To be precise, I want ...