The eigenvalue tag has no wiki summary.

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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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### 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 ...

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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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### 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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### When is this matrix singular?

Consider matrix $A$ with $(j,k)$â€˛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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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:
...

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### 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 ...

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107 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 ...

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### 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 ...

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175 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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228 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 ...

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### 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)}; ...

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### 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 ...

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

What is the eigenvalue/eigenvector relationship between matrix A,B and AB?

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### 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 $C$.
...

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### 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} \\\
...

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

This is sort of complementary to this thread. Iâ€™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 ...

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### 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 ...

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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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### 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 ...

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### 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 ...

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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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### 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 ...

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### 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.

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### 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 ...

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### 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.
...

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### 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 $0\leq x < \infty$
...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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) ...

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### Condition for block symmetric real matrix eigenvalues to be real

I have a (2nx2n) block symmetric matrix that in the simplest case (n=2) looks like:
$$
M_2 = \begin{bmatrix}
a_1 & 0 & b_{1,2} & -b_{1,2}\\\
0 & -a_1 & b_{1,2} & -b_{1,2}\\\
...

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### Singular Value Decomposition of Noisy Matrices

I am an engineer who makes measurements of a variable over a grid
of, say, $m\times n$. Since these are actual measurements, the true
values are always corrupted by noise, and what I measure is a ...

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### Eigenvalues of a Symmetric Positive Semi-Definite (PSD) matrix after rank one update

I have a Symmetric Positive Semi-Definite matrix $A$ which i know its eigenvalue and eigenvectors. let $v$ and $u$ be a random column vector. i want to know if it is possible to have eigenvalues of ...

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### Checking whether this would be bounded

It may be better to post this question here. Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric
positive-semi-definite matrix with exactly $m/2$ positive eigenvalues
and every entry of ...

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### can eigenvector be found without computing the eigenvalue [closed]

Is there any ways to compute the eigen vector without computing explicitly the associated eigenvalue?
Actually, I'd like to compute the largest eigenvalue of a positive matrix from its eigen vector, ...

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### Prove log of eigenvalues are dense in R?

Suppose you have the set of all possible $n$ x $n$ square adjacency matrices where $n$={1,2,3,4...}. For each matrix, compute the logarithm of the largest eigenvalue. Is it true that the set of ...

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### Upper bounds on eigenvalues of PSD matrix?

Suppose A is a symmetric positive semidefinite matrix. Is there a way to upper bound the largest eigenvalue using properties of its row sums or column sums?
For instance, the Perronâ€“Frobenius ...

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### Eigenvalues vs.matrix sparsity

For an n X n matrix whose entries are constrained to be in some [x,y], is the maximum absolute eigenvalue of the matrix a function of its sparsity?
Is there a closed-form expression that states this ...

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### Destroying the structure of a linear system while preserving its maximum eigenvalue

I have an asymmetric square matrix with non-negative real entries in the range [0,10], representing the edge-weights of a directed network. Assume that the network is a linear system. My general ...