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4
votes
3answers
192 views

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 ...
0
votes
0answers
19 views

Eigenvalue bounds [migrated]

I have a symmetric(pxp), positive semi definite matrix Big_Omega(pxp). If somebody says find the eigenvalue bounds of the matrix such that w1*Identity(pxp) <= Big_omega(pxp) <= ...
3
votes
1answer
76 views

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 ...
0
votes
1answer
161 views

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 ...
1
vote
0answers
81 views

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 ...
6
votes
2answers
239 views

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 ...
1
vote
0answers
109 views

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 ...
0
votes
1answer
96 views

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 ...
1
vote
0answers
62 views

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 ...
0
votes
1answer
78 views

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, ...
1
vote
0answers
56 views

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 ...
0
votes
0answers
49 views

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 ...
-3
votes
1answer
87 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
188 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 ...
0
votes
1answer
117 views

Which graph topology has the greatest eigenvalue?

I am looking at comparing multiple graph topologies based on their spectra. From the set of all $N\times N$ adjacency matrices, is there any result which points to the adjacency matrix with the ...
4
votes
1answer
122 views

Epidemic threshold

Need some help / ideas to proceed. Stuck for a while on this. In the literature of epidemic theory, it is found that the epidemic threshold is $1/\lambda_{max}(A)$ where $\lambda_{max}(A)$ is the ...
1
vote
0answers
92 views

A fractional calculus eigenvalue problem

One set of eigenfunction for the following fractional integral operator is $f(z)=e^{-bz}$ for any constant Re$b>0$, with eigenvalue $\lambda=\frac{\Gamma(\alpha)}{b^\alpha}$, $$\int_z^\infty ...
2
votes
1answer
231 views

Gaussian kernel eigenfunctions

I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple and provide me some references. What is the eigenfunction of a multivariate Gaussian kernel: ...
2
votes
0answers
112 views

Distributions of eigenvalues for matrix normal distribution: related references

I am interested in the distribution of the eigenvalues of matrices that are sampled from the matrix normal distribution. I am sampling from $p(X \mid M,U,V)$ and let's assume that I know the ...
1
vote
1answer
115 views

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 ...
2
votes
1answer
93 views

Dimension independent computational complexity of singular value decomposition

Suppose $X$ is a $m \times n$ real matrix, which has only $k$ number of nonzero elements ($k \ll mn$). Given a vector $y$, the sparsity of $X$ allows $X y$ to be computed in $O(k)$ time which is ...
1
vote
2answers
187 views

Eigenvalue computation using inverse iteration

I have a positive definite matrix $A$. I need to compute the max eigen value of $A$ using inverse iteration. The problem is that there are duplicate maximum eigen values and so inverse iteration ...
2
votes
1answer
165 views

quadratic programming on hypercube

I want to maximize a quadratic form $\mathbf x^T\mathbf Q\mathbf x$ and also want to find out which vector $\mathbf x$ maximizes the quadratic form when $\mathbf Q$ is an $n\times n$ positive ...
2
votes
3answers
275 views

Analytic perturbation of the eigenvalues/eigenvectors of non-Hermitian matrix

Consider a matrix function $A(x)$, analytically depending on single parameter $x$. Consider the eigenvalue/eigenvector pair of $A(0)$, namely $\lambda(0)$ and $w(0)$. The question is whether we can ...
3
votes
2answers
264 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
3answers
247 views

eigenvalue of Laplacian matrix

If we have a Laplacian matrix $\boldsymbol{A}$ such that \begin{align} &A_{ii} >0 \\ &A_{ii}=-\sum_{j\neq i}A_{ij} \end{align} with known eigenvalues $\lambda_i$. Define the matrix ...
0
votes
1answer
71 views

The influence of eigendecomposition on the periodicity of a (rank 2) Hermitian matrix (of functions)

Let $\boldsymbol{R}(u,v);~u,v\in\mathbb{R}$ be a Hermitian matrix (of Hermitian functions) with entries \begin{equation} r_{ij}(u,v) = 1 + Ae^{-2\pi i \phi_{ij}(ul_0 + vm_0)}; ...
3
votes
2answers
323 views

What is known about the spectrum of a Cauchy matrix?

Math people: A Cauchy matrix is an $m$-by-$n$ matrix $A$ whose elements have the form $a_{i,j} = \frac{1}{x_i-y_j}$, with $x_i \neq y_j$ for all $(i, j)$, and the $x_i$'s and $y_i$'s belong to a ...
-2
votes
3answers
237 views

relationship between eigenvalues/eigenvectors of A, B and AB [closed]

What is the eigenvalue/eigenvector relationship between matrix A,B and AB?
1
vote
0answers
63 views

Effect of removing a Hamiltonian cycle on the Laplacian spectrum

Notation: $\lambda_{\max}(G)$ is the largest eigenvalue of the Laplacian matrix of the graph $G$ (aka the Laplacian index of $G$). Now suppose $G$ is a Hamiltonian graph with Hamiltonian cycle $C$. ...
1
vote
0answers
92 views

positiveness of the inverse solution to Sylvester equation

I need to construct a non-negative matrix with desired eigenvalues. To that end, I came up with a block matrix of the following form: $$ \mathbf{M} = \begin{vmatrix} \mathbf{A} & \mathbf{b} \\\ ...
8
votes
0answers
266 views

Eigenvalues of permutations of a real matrix: how complex can they be?

This is sort of complementary to this thread. I’ll repeat the definitions here: For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and ...
20
votes
4answers
1k views

Eigenvalues of permutations of a real matrix: can they all be real?

For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and define the total spectrum $TS(M)$ as the union of all their spectra (counting ...
1
vote
3answers
2k views

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 ...
0
votes
2answers
220 views

Eigenvalues of an amplification matrix

Let $A$ and $B$ square real matrices. I know that the matrix $A+B$ has 1 as eigenvalue of multiplicity 1 and the others eigenvalues have their modulus <1. Can we say something about the eigenvalues ...
0
votes
1answer
153 views

Can an accumulation point be an eigenvalue?

For an discrete (separable) infinite-dimensional Hilbert Space with a compact operator, 0 is always an accumulation point (https://www.math.ucdavis.edu/~hunter/book/ch9.pdf). Does this mean its part ...
1
vote
0answers
59 views

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 ...
1
vote
0answers
70 views

On generalization of Wigner semi circle

I want to analyse noise model for a matrix M whose entries are not real numbers. The matrix is a collection of N permutation matrices of size nxn i.e, M is NnxNn. Because its a collection of ...
1
vote
0answers
110 views

nodal lines in the dirichlet problem

In the Dirichlet problem if nodal lines do not touch $\partial\Omega$ (unit disk), what happens to the eigenvalues? Thanks for help.
0
votes
1answer
164 views

eigenvalues of two nonnegative matrices

Let $A$ and $B$ be symmetric non-negative matrices. If $A\geq B$ (i.e., $A-B$ is a nonnegative matrix), can we say that $\lambda_i(A) \geq \lambda_i(B)$ for all $i$, where $\lambda_i$ denotes the ...
2
votes
1answer
572 views

Non symmetric matrices with real eigenvalues

Consider the following block matrix $A=\pmatrix{A_1 & A_2\cr kA_2^\top & A_3}$ where $A_1$ is a symmetric matrix, $A_3$ is diagonal matrix and all entries of $A$ are real and non-negative. ...
1
vote
0answers
74 views

Characterizing the singular values of a matrix with structure

Suppose we have a function from $\mathbb{R}^2\to\mathbb{C}$, $$f(x,y) = e^{\imath\pi x g(y)}$$ where $g(y)$ is periodic in $y\in[-T, T),\ T<\infty$ (e.g., a sinusoid) and $0\leq x < \infty$ ...
2
votes
1answer
187 views

Multiple eigenvalues over imperfect fields

Let $K$ be a field. For a matrix $A\in GL_n(K)$ we can find the Jordan normal form $A'$ in $GL_n(\overline{K})$, where $\overline{K}$ is the algebraic closure of $K$. We write $j_\alpha(A)$ for the ...
0
votes
0answers
153 views

Summation of eigenvalues of tri-diagonal matrix smaller than specific value

Is there any analytic expression for summation of eigen-values of a tri-diagonal matrix which are smaller than a constant value? Or even a rough approximation for it. How about case of a general ...
1
vote
0answers
63 views

Eigenvectors of contraction times projection

Suppose $A$ is a real $n\times n$ matrix with real eigenvalues: $$ 1=\lambda_1>|\lambda_2|\ge \ldots\ge |\lambda_n|>0. $$ Suppose $B$ is an involution, for simplicity let us assume that $B$ is ...
0
votes
1answer
106 views

Spectrum of a Laplacianized matrix

Suppose that $A$ is a positive matrix and that we let $R$ be the diagonal matrix of $A$'s row-sums. What can be said about the spectrum of $R-A$? I am particularly interested in the largest eigenvalue ...
0
votes
0answers
66 views

Laplacian using SDP

Is there any suggestion about how could one construct a model that uses semidefinite programming that minimizes sum of k smallest eigenvalues of Laplacian matrix? I found two papers that have done ...
0
votes
0answers
90 views

Ratio of Eigen values and Mutual Independence

Given a matrix $X$. Calculating the Eigen values of $XX^T$ and using the ratio of maximum and minimum eigen values normally gives the condition number of the matrix. If $X$ contains $M$ observations ...
0
votes
3answers
504 views

Relating the angle between two vectors to max and min eigenvalues

Hi. I am busy working through a paper i came accross online on portfolio optimization. The paper may be accessed on the following link: http://ssrn.com/abstract=1483412 I am struggling, in ...
1
vote
1answer
134 views

The smallest eigenvalue from an equitable partitions

Suppose that $G$ is a connected graph with equitable partition $\pi$. Then the eigenvalues of the divisor multigraph $G / \pi$ are all eigenvalues of $G$. (Perhaps excluding some pathological cases) ...