The eigenvalues tag has no usage guidance.

**-1**

votes

**0**answers

25 views

### Eigenvalues of signed networks [on hold]

A signed graph is a graph where edges can be positive or negative. A cycle in a signed network is called balanced cycle when it has an even number of negative edges otherwise it is called unbalanced ...

**-3**

votes

**1**answer

89 views

### Eigenvalues of cyclic tridiagonal matrix [on hold]

The following matrix is the result of a special kind of balanced signed graph of order $n$. In the Matrix $n_1,n_2,..,n_k$ are positive integers, which satisfy $\sum
n_i=n.$ Prove that this matrix ...

**-4**

votes

**0**answers

63 views

### Eigenvalues of tridiagonal matrix [on hold]

The following matrix $T$ is result of my research on special type of balanced signed graphs of order(No. of nodes) $n$. In the matrix $T$ $n_1,n_2,...,n_k$ are positive integers such that $\sum ...

**-1**

votes

**0**answers

48 views

### What will draw a shape for $L = \left\{ {\lambda \in \mathbb{C}:{s_4}(\lambda ) = {s_3}(\lambda )} \right\}$ [on hold]

Let $P(\lambda ) = \left( {\begin{array}{*{20}{c}}
{{\lambda ^2} - 1} & 0 \\
0 & {{\lambda ^2} - 2\lambda } \\
\end{array}} \right)$ and $\lambda \in \mathbb{C}$( $λ$ is a complex ...

**1**

vote

**1**answer

106 views

### Modified interlacing of eigenvalues

Let $A$ be a real symmetric matrix of order $n$ and $B=\begin{bmatrix}v &v &v &v\end{bmatrix}$ where $v$ is a non zero real column vector of dimension $n$. Consider $$C=\begin{bmatrix}A ...

**4**

votes

**1**answer

168 views

### exact definition of Fiedler vector

For a given N-vertex similarity graph $ G=(V,A) $ the eigenvalues of the unrenormalized (graph) Laplacian may be denoted as
$$ 0= \mu_0 \leq \mu_1 \leq ... \leq \mu_N $$
where the corresponding ...

**1**

vote

**2**answers

87 views

### Spectral radius of a non-negative matrix after moving and replicating an element

Let $A$ be a non-negative square matrix and its spectral radius (i.e., it's largest eigenvalue) be $\rho(A)$. I need to do the following operation to $A$ and compare the resulting spectral radii.
...

**3**

votes

**0**answers

111 views

### Conditions for continuity of non-simple eigenvectors

Here, http://math.stackexchange.com/a/1146455, it is noted that eigenprojections are continuous, but eigenvectors are not. Are there any conditions where the eigenvalues are not simple, but the ...

**1**

vote

**1**answer

59 views

### Spectral radius's relation with row sum

Let $A$ be a non-negative $N \times N$ square matrix with $a_{i,i}=0, 1 \leq i \leq N$. Also, let $r_i$ be the $i$-th row sum of $A$.
I know that $\rho(A)$, the spectral radius of $A$, is bounded as ...

**1**

vote

**0**answers

32 views

### Characterization of eigenvector

Let's say we have the following optimization problem. (All the $\Sigma_{ii}$'s are positive definite.)
$\max u^\top \Sigma_{12} v\quad$
$\text{subject to}\quad u^\top \Sigma_{11} u = 1\quad and\quad ...

**-6**

votes

**1**answer

54 views

### Does $\|(I-A)^{-1}\|_{2} \leq 1/(1-\|A\|_{2})$ holds for matrices with spectral radius smaller then 1?

Given a symmetric positive semidefinite matrix matrix $A$, if its spectral radius $0<\rho(A)<1$, does the inequality $\|(I-A)^{-1}\|_{2} \leq 1/(1-\|A\|_{2})$ hold true?
$\|A\|_{2}$ denotes ...

**1**

vote

**1**answer

79 views

### Largest element in inverse of a positive definite symmetric matrix [closed]

If I have an $n \times n$ positive definite symmetric matrix $A$, with eigenvalues $\lambda_{1}>\lambda_{2}\cdots>\lambda_{n}$, can I claim that the highest value which matrix $A^{-1}$ can have ...

**1**

vote

**1**answer

88 views

### Find a square, stochastic matrix (w/ non-neg entries) of odd size, not a permutation matrix, with an eigenvalue other than 1 on the unit circle

...or prove that none exists.
Note that such a matrix M couldn't be primitive, so there would be at least one entry equal to zero in every power M^k (Perron-Frobenius theory).
Preferably the ...

**8**

votes

**1**answer

128 views

### Exact eigenvalues of a specific tridiagonal matrix

I'm studying the following tri-diagonal matrix
$$
X = \begin{pmatrix}
0 & x_0 & 0 & 0 &\cdots & 0 & 0 & 0 \\\
x_0 & 0 & x_1 & 0 &\cdots & 0 & ...

**2**

votes

**0**answers

34 views

### Nonnegative Inverse Eigenvalue Problem (NIEP),

Does the NIEP, currently open for $n\ge 5$, have any good, practical applications?
For the easy case, $n=2$, I am able to prove some of the results that agree with the current literature.
In some of ...

**1**

vote

**1**answer

126 views

### Can we claim that all the terms in a matrix are less than equal to 1 if spectral radius is less than 1?

I have a a full column rank matrix A, and using this I want to construct a matrix with spectral radius less than 1. I do that using,
H = $I-\alpha A^{T} A$ ($I$ is identity matrix), where the term ...

**2**

votes

**1**answer

74 views

### Spectral radius of perturbed bipartite graphs

I am looking into how perturbation(s) on a bipartite graph affect its spectrum (specifically its spectral radius or largest eigenvalue). Actually, I'm not exactly looking into bipartite but my ...

**0**

votes

**1**answer

78 views

### Large Tridiagonal Matrix - Eigenvalues

Consider large tridiagonal matrix (where $a$ and $b$ are real numbers):
$M =
\begin{pmatrix}
a^2 & b & 0 & 0 & \cdots \\
b & (a+1)^2 & b & 0 & \cdots & \\
...

**1**

vote

**0**answers

51 views

### inverse of asymptotic Toeplitz matrix with band limited associated function

I am reviewing a controversial paper, and the main result, a revolution within my field, comes down to whether or not the following is true. I strongly believe it is not, but would need confirmation.
...

**2**

votes

**1**answer

76 views

### How to find the eigenvalues equation of this PDE problem

Given the problem:
$$(\kappa(x)X^{'})^{'}+\lambda\rho(x)X=0$$
for $0<x<l$ with $X(0)=X(l)=0$
where $\kappa(x)=\kappa_{1}^{2}$ for $x<a$, $\kappa(x)=\kappa_{2}^{2}$ for $\kappa>a$. ...

**8**

votes

**2**answers

450 views

### A log inequality for positive definite trace-one matrices

Let $\{v_i\}_{i=1}^N$ be a set of $n$-dimensional real vectors and let $X=X^\top\in\mathbb{R}^{n\times n}$ be a positive definite trace-one matrix. I would like to prove (or disprove) the following ...

**-1**

votes

**1**answer

67 views

### Closed formula for a homogeneous second order linear ODE [duplicate]

Let $A, B, C\geq 0$ be constants. Is there an explicit formula to a nontrivial solution to the homogeneous linear ODE
$$y''(t) -(A+B\,\sin t)\,y'(t) -C\, y(t)=0$$
for $t\in(0,2\pi)$ with periodic ...

**0**

votes

**1**answer

59 views

### Eigenvalue-related statements [closed]

(I understand this question might not be appropriate for this website, but it has been asked on MathStackexchange and did not receive any replies even with a bounty)
How can I prove that the ...

**3**

votes

**2**answers

103 views

### Eigenspace of convex combination of two idempotent matrices

Let $H_1,H_2\in\mathbb{Q}^{n\times n}$ be idempotent and symmetric matrices. For any $0<\mu<\frac{1}{2}$, consider the matrix
$$H_\mu:=\mu H_1+(1-\mu)H_2.$$
I'm looking for a description of ...

**1**

vote

**2**answers

65 views

### Non-asymptotic bound on the variance of largest singular value of gaussian matrix

Let $A$ be a gaussian matrix of size $d \times n$ where all the coefficients are drawn i.i.d. from $ \mathcal{N}(0, 1)$ and denote by $s_{\text{max}}$ its largest singular value.
Theorem 2.6 of ...

**7**

votes

**0**answers

102 views

### Classes for which the Spectrum determines a Convex Shape

Given a planar domain $\Omega \subset \Bbb{R}^2$ bounded and open we can associate to it the spectrum of the Laplace operator with Dirichlet boundary condition. It is known that there are planar ...

**6**

votes

**0**answers

396 views

### Spaces of matrices with same eigenvalue/Great circles in O(n)-orbits

Let $Sym^2(V)$ be the set of symmetric matrices of a real $n$-dimensional vector space $V$. Given an element $\underline{\lambda}=[\lambda_1,\ldots \lambda_n]\in \mathbb{RP}^n$, where ...

**3**

votes

**0**answers

68 views

### Spectra of certain totally positive matrices

Let $S$ be the set of $3 \times 3$ matrices $A$ satisfying the following conditions:
All minors are $>0$ (i.e., $A$ is a strictly totally positive matrix);
all principal minors are $>1$, ...

**4**

votes

**2**answers

153 views

### Relation between eigenvalues of $A$ and $A^TA$?

For an $n\times n$ diagonizable matrix $A$, is there a relation between the eigenvalues of $A$ and the eigenvalues of $A^TA$?
I ask this because I am looking into the relation between $A$ and $A+cI$, ...

**4**

votes

**2**answers

157 views

### Non-asympototic version of Gelfand's formula

Let $A$ be a $n\times n$ matrix. Let $\|A\|$ be the spectral norm of $A$, and $\rho(A)$ be the spectral radius. I am wondering whether the following statement is true.
There exists universal ...

**7**

votes

**1**answer

282 views

### Can I find the gap between the two least eigenvalues of this special matrix A(t)?

I am interested in finding the gap between the two least eigenvalues of $A(t)$, a Hermitian $N\times N$ sparse matrix whose diagonal elements are $a_it+b_i\,(1\leq i\leq N)$, and all off-diagonal ...

**5**

votes

**1**answer

163 views

### Eigenvalues of $X$ in the metric of $Y$

What does this statement describe? $X$ and $Y$ are matrices.
The eigenvalues of $X$ in the metric of $Y$.
I've not seen this language used before in this fashion and I don't really know what ...

**2**

votes

**2**answers

104 views

### Quick tests to differentiate eigenvalues

Given two real square symmetric matrices $A$ and $B$ are there any quick tests to make sure at least one of their eigenvalues differ without computing the eigenvalues and likely more robust or looking ...

**2**

votes

**1**answer

87 views

### Eigenvalues of product of symplectic matrices

I have 2 symplectic matrices $X_{1},X_{2} \in \mathbb{R}^{2n\times2n}$. The matrix $X=X_{1} \cdot X_{2}$ is also symplectic.
Question: Are there any theorems which allow me to express eigenvalues of ...

**6**

votes

**1**answer

152 views

### Extension of Wigner's semicircle law?

It is well-known that the semicircle law holds for a wide class of matrices with independent and identically distributed (mean zero) entries.
My question is: is there any study about the more general ...

**6**

votes

**1**answer

245 views

### Why are some solutions of these diophantine equations off the usual patterns?

This is inspired by a recent question about complete multipartite integral graphs. I am wondering if more can be said about tripartite integral graphs with block sizes $a<b<c$. It is easy to see ...

**5**

votes

**1**answer

147 views

### Sum of the absolute eigenvalues of A>=B

Kindly help me to prove/disprove the following statement.
Let $A$ be a symmetric matrix of order $n \times n$ with all the diagonal entry equal to $0$, and other non-diagonal entry equal to $k$ ...

**3**

votes

**3**answers

138 views

### Can a block matrix with at least 3 zero blocks of different size on the diagonal and 1's everywhere else have only integer eigenvalues?

Let $M=\begin{pmatrix}
\begin{array}{cccccccc}
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
1 & 1 & 0 & 0 & ...

**5**

votes

**2**answers

736 views

### 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\\
&& ...

**6**

votes

**1**answer

190 views

### Eigenvalue inequality for regular graphs

I recently proved an inequality relating some of the eigenvalues of a regular graph with each other, and I was wondering if it is already known. I was unable to find it online, and a quick skim ...

**0**

votes

**0**answers

46 views

### Eigenvalues of sum of two fuchsian matrices

Dear mathoverflow users,
I am trying to solve a problem concerning eigenvalues and sum of matrices.
In particular: consider the expression
$$
A=\frac{E}{x-x_1}+\frac{F}{x-x_2},
$$
and suppose to know ...

**-1**

votes

**1**answer

184 views

### How bad could $\|A^k\|$ be when $\rho(A) < 1-\delta$ [closed]

(Sorry, I do hate editing this many many times but let me try the last time)
Gelfand's formula says that
$$\lim_{k\rightarrow \infty} \|A^k\|^{1/k} = \rho(A)$$
I am wondering whether there is any ...

**0**

votes

**0**answers

33 views

### On the computation of generalized eigenvalues of a low-rank approximation using SVD

I have trouble deriving an expression of the generalized eigenvalues of a matrix pair, found in http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4618700 .
The setup is the following and can ...

**1**

vote

**0**answers

96 views

### Limit for eigenvalues of the Dirichlet problem

If $\Omega$ is a bounded domain in $\mathbb{R}^n$, let $\lambda(\Omega)$ be an eigenvalue of the problem
$$-\Delta\,u=\lambda\,u\,\,\mbox{in}\,\,\, \Omega, \, u=0\,\,\,\mbox{on}\,\,\, ...

**4**

votes

**1**answer

141 views

### Is the sum of spectral projections a projection?

Let $T$ be a closed operator on a Hilbert space with discrete spectrum. Then for $\{\lambda_1,...\lambda_n\}\in\sigma(T)$ one can define the spectral projections
...

**3**

votes

**0**answers

100 views

### Relating Numerical Range and Perron-frobenius theorem for positive matrices?

Let $A$ be any matrix with all entries positive (which means Perron-Frobenius theorem can be applied). Then its numerical range is defined as the set of complex numbers
$$W(A)=\{x^HAx\lvert ...

**0**

votes

**0**answers

31 views

### Neumann eigenvalues of quotients of manifolds with boundaries

I was wondering what is generally known about Neumann eigenvalues of Laplacian of $M/G$ where, $M$ is a manifold with boundary and $G$ is acting freely discontinuously on $M$? I understand that $M/G$ ...

**5**

votes

**0**answers

195 views

### Eigenvalues of a certain product of matrices with special structure

Let $d$ and $c$ be positive integers and $q = dc$. Let $G$ be a $q$-by-$q$ positive semi-definite real matrix with eigenvalues all $\le 1$, and define the $q$-by-$2q$ matrix $A = ...

**1**

vote

**0**answers

75 views

### Eigenvalues of a partitioned self-adjoint matrix

This is a repost of the same question on MSE (with no reply/comment):
http://math.stackexchange.com/questions/1454314/eigenvalues-of-a-partitioned-self-adjoint-matrix
I would be grateful just for a ...

**-3**

votes

**1**answer

160 views

### Eigenvalues of real symmetric matrix [closed]

Suppose $A$ is a $n \times n$ real symmetric matrix with entries $a_{ij}\geq 1 $ and $a_{ii} = 0 $. Is it possible to have sum of the absolute eigenvalues of
$A < 2 (n - 1).$