0
votes
0answers
70 views

Eigenvalue of (0-1) matrix [on hold]

Assume I have 2 matrices, each of size nxn with only 1 and 0 as entries in both. (n>10) The first matrix (call it A) has each row summing up to 2 (ie: on each row, it has two "1" and n-2 "0"). It is ...
-3
votes
1answer
69 views

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 ...
3
votes
1answer
112 views

Is this function of a matrix convex?

Let $\mathcal{N}_{n}$ be the set of symmetric nonnegative irreducible matrices. For a matrix $A \in \mathcal{N}_{n}$ let $v^{A}$ be its Perron vector, normalized so that $||v^{A}||_{2}=1$. Define the ...
3
votes
1answer
176 views

When is this matrix singular?

Consider matrix $A$ with $(j,k)$′th entry $A_{j,k}=\sin(\omega_j t_k+\phi_j),\,\forall j,k\in\{1,2,...,n\}$, where $\omega_j,t_k,\phi_j\in \mathbf R$. 1) For $t_k=k$, what is the condition on ...
2
votes
1answer
126 views

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 ...
7
votes
1answer
588 views

Frobenius-Perron eigenvalue and eigenvector of sum of two matrices

Suppose that I have two positive matrices, $A$, and $B$, and I know their Frobenius-Perron eigenvalues ($\lambda_A$, $\lambda_B$) and eigenvectors ($v_A$, $v_B$). I'm interested in what I can say ...
4
votes
0answers
99 views

Concept of eigenvector restricted to nonnegative entries

Let $X\in \mathbb{R}^{n\times n}$ be a positive semidefinite matrix. The leading eigenvector $v\in \mathbb{R}^n$ of $X$ is the solution to the problem $\arg \max_{v:\lVert v\rVert_2=1} \lambda\quad$ ...
3
votes
1answer
153 views

Fast algorithm for maximizing smallest eigenvalue of linear combination of hermitian matrices

I have an engineering back ground. Due to work, I came across this problem \begin{align} &\max_{\lambda,y_i\in \mathbb{R}}~\lambda \\\ ...
3
votes
2answers
238 views

Eigenvectors and eigenvalues of Tridiagonal matrix with varying diagonal elements

is it possible to analytically evaluate the eigenvectors and the eigenvalues of a tridiagonal $n\times n$ matrix of the form : \begin{pmatrix} 1 & b & 0 & ... & 0 \\\ b & 2 ...
2
votes
1answer
137 views

When is there a solution to these coupled eigenvalue equations?

I am trying to find the fixed point of a dynamical system, which requires solving two coupled eigenvalue-like equations. These equations are, in general, overconstrained. I'd like to have a simple ...
14
votes
4answers
759 views

Condition for two matrices to share at least one eigenvector?

Suppose that I have two matrices $A$ and $B$, and I want them to share a common eigenvector $x$. For simplicity let's just assume that the eigenvalue associated with $x$ is $1$ for both matrices, so ...
4
votes
0answers
94 views

Preconditioner for finding the smallest eigenpairs of a large, but structured, matrix

I'm trying to find the eigenvector corresponding to the second smallest eigenvalue of a large $(4,000,000 \times 4,000,000)$ matrix $L$. $L$ is a graph Laplacian, with the following structure: $L = D ...
2
votes
2answers
149 views

Is my use of the eigendecomposition correct here?

I'm exploring different techniques to efficiently solve some matrix equations. My situation is that I have a matrix $\textbf{H} = \textbf{J}^T \textbf{J}$, where $\textbf{J}$ is a matrix with no ...
4
votes
1answer
213 views

Sum of commuting semisimple operators

Let $V$ be a finite dimensional vector space over a field $K$. An operator $T:V\to V$ is called semi-simple if every $T$-invariant subspace of $V$ has a complement(for algebraically closed fields ...
2
votes
3answers
641 views

Eigenvectors and eigenvalues of nonsymmetric Tridiagonal matrix

Hi, the question is following: We have one matrix $$\begin{pmatrix} -\beta & \Delta & 0 & 0 &\cdots & 0 & 0 & 0 \newline \beta & -(\beta+\Delta) & \Delta & ...
5
votes
1answer
329 views

Repeated Second Eigenvalue of the Adjacency Matrix of a Graph

This question is motivated by a talk I went to earlier today. Suppose we have a $d$-regular graph $G$ with $n$ vertices, with adjacency matrix $A$. Let $$\lambda_1\geq \lambda_2 \geq\dots \geq ...
5
votes
2answers
748 views

Perturbation theory for the generalized eigenvalue problem

Is there a standard reference for the perturbation theory of the generalized eigenvalue problem? More specifically, I would like to get a systematic expansion for the problem $(A_0 + \epsilon ...
1
vote
0answers
718 views

Eigenvalues of Matrix Sum

Hello, I have a linear algebra problem that I need help with. Basically, I need to get the eigenvalues and eigenvectors of several (sometimes tens of thousands) very large Hermitian matrices (6^n x ...
8
votes
4answers
2k views

Eigenvectors and eigenvalues of Tridiagonal matrix

Hi, is it possible to analytically evaluate the eigenvectors and the eigenvalues of a tridiagonal matrix of the form : $$ \mathcal{T}^{a}_n(p,q) = \begin{pmatrix} 0 & q & 0 & 0 ...
3
votes
1answer
299 views

What is the minimum of the Frobenius norm in the intersection of positive semidefinite cones?

For scalar variables $x$, we have a simple solution for the following problem. \begin{eqnarray} \min_x&&\alpha(x-a)^2+\beta(x-b)^2 \\\ \mathrm{s.t. }&&x\leq a\\\ ...
2
votes
2answers
611 views

Eigenvalues of sum of an adjacent matrix and a constant

$A$ is an adjacent matrix of a network. $la$ is the largest eigenvalue of $A$ and $Va$ is its corresponding eigenvector. I am interested in the following martix: $bA+c-dI$ ($b$, $c$, and $d$ are all ...
5
votes
1answer
998 views

Stochastic Matrix: Second largest eigenvalue and second largest absolute value of eigen value

Setup Let $A$ be a stochastic matrix. Let the eigenvalues of $A$ be $1 = \lambda_1 \geq \lambda_2 \geq \lambda_3 ... \geq -1$. Let $\lambda = \max_{x: x \perp 1} \frac{||Ax||}{|| x ||}$ Question: ...
1
vote
2answers
488 views

Relating eigenvectors of two self-adjoints operators

Suppose I have a self-adjoint operator $\mathbf{L}$ which I seperate in two parts which are themselves self-adjoint. I write this in terms of their eigenvalues/eigenvectors: $\mathbf{v} \Lambda ...
0
votes
0answers
406 views

Eigenvalues of anti-circulant matrices

Is there any theorem to find the eigenvalues of any anti-circulant matrix using the equivalent (with the same first row) circulant matrix. I found out that, for any anti-circulant matrix, the ...
4
votes
0answers
261 views

What are the eigenvectors of the Lagrange interpolation matrix?

Let $F$ be a field. Let $x_1,\ldots,x_k,y_1,\ldots,y_k\in F$ be distinct elements in the field. Consider the $k\times k$ matrix that in position $i$, $j$ has the element $\frac{\prod_{l\neq i}(y_i - ...
1
vote
1answer
206 views

Spectral analysis of sparse symmetric integer matrices

Hi all, A project I'm currently working on requires me to compute the eigenvectors / eigenvalues of sparse symmetric integer matrices. This is needed in the context of Principal Component Analysis. I ...
4
votes
1answer
853 views

Eigendecomposition after multiplying by diagonal matrix

Hello, If we possess the eigendecomposition of a positive definite matrix: $X = U \Sigma U^T$, is there an efficient way to compute the eigendecomposition of $D X D$ where $D$ is a diagonal matrix?
4
votes
3answers
2k views

Minimize trace of inverse of convex combination of matrices.

Hello! (First question--please forgive me if its unclear.) I am interested in efficient/approximate optimization techniques for minimizing a norm of a convex combination of symmetric, positive ...
1
vote
2answers
382 views

Optimizing directly on the eigenspectrum of a matrix

I have an application where I want to the eigenvalues of the graph to be involved in the objective and constraints in a flexible way (moreso than just the nuclear or frobenius norm). Whats a good ...
10
votes
5answers
794 views

Can always a family of symmetric real matrices depending smoothly on a real parameter be diagonalized by smooth similarity transformations?

This question is related to another question, but it is definitely not the same. Is it always possible to diagonalize (at least locally around each point) a family of symmetric real matrices $A(t)$ ...
3
votes
1answer
862 views

dominant eigenvector

Hi, everyone! Is there any efficient way to simplify the following tensor product $X \otimes X + X^T \otimes X^T$, where $X$ is a square $n \times n$ matrix. My goal is to efficiently compute the ...
2
votes
1answer
140 views

Statistical estimation of singular values and vectors

Hi All My question is about the well known and well studied singular value decomposition (SVD). What I am working on right now requires performing an SVD repeatedly on a slowly varying matrix. Since ...
6
votes
2answers
2k views

Conditions for smooth dependence of the eigenvalues and eigenvectors of a matrix on a set of parameters

Let $A\in\mathcal M_n$ be an $n\times n$ real [symmetric] matrix which depends smoothly on a [finite] set of parameters, $A=A(\xi_1,\ldots,\xi_k)$. We can view it as a smooth function $A:\mathbb ...