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-3
votes
0answers
26 views

finding eigenvector [on hold]

I have where λ1 = λ2 = 6 and λ2 = λ3 = 0. I wish to find the eigenvectors for these eigenvalues above. I've tried to turn it into equations and trying to solve them (this is for λ1 & λ2): ...
0
votes
0answers
50 views

Bounding the off-diagonal entries of a hermitian matrix [on hold]

I asked this question on stackexchange and didnt receive any answers, am therefore posting it here. The Pauli matrices are $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, $X = ...
1
vote
1answer
43 views

How to retrieve eigenvectors from shifted QR algorithm?

I understand that the key to retrieve eigenvectors in the non-shifted QR algorithm is to accumulate the transformations at each steps in the following way: $Q = \Pi_i Q_i$ Can we accumulate the ...
6
votes
4answers
457 views

Minimum negative eigenvalue of zero-one matrices

The following question must have been answered decades ago. For $n$ fixed, what is the most negative eigenvalue among all trace zero zero-one matrices (that is, all entries are either zero or one, ...
3
votes
0answers
51 views

Eigenspaces of the Laplace operator on a unit ball

I am interested in structures of the eigenspaces of the Laplace operator on the $n$-dimensional unit ball with Neumann or Dirichlet boundary conditions as representations of the special orthogonal ...
-2
votes
1answer
94 views

Eigenvalues of cyclic tridiagonal matrix [closed]

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 ...
1
vote
1answer
111 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
1answer
180 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
1answer
62 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 ...
2
votes
0answers
116 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
2answers
93 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. ...
1
vote
0answers
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
1answer
55 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
1answer
91 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 ...
1
vote
1answer
80 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 ...
2
votes
0answers
35 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
1answer
127 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 ...
1
vote
0answers
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
1answer
79 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
1answer
131 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 & ...
0
votes
1answer
60 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 ...
8
votes
2answers
459 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
1answer
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
1answer
79 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 & \\ ...
3
votes
2answers
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
2answers
66 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 ...
3
votes
0answers
70 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
2answers
162 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 $A+cI$, ...
2
votes
1answer
76 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 ...
5
votes
1answer
167 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
2answers
106 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
1answer
88 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
1answer
155 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
1answer
247 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
1answer
152 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
3answers
142 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 & ...
4
votes
2answers
159 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 ...
6
votes
1answer
193 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
0answers
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
1answer
185 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
0answers
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 ...
4
votes
1answer
146 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 ...
5
votes
0answers
198 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
0answers
76 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
0answers
101 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 ...
-3
votes
1answer
163 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 $A < 2 (n - 1).$
1
vote
1answer
239 views

Upper bound for sum of absolute values of eigenvalues of Hermitian matrix

Given a hermitian, but not necessarily positive, sparse matrix $C = (c_{ij}) \in \mathbb{C}^{n \times n}$ and $n \ggg 1$ ($n \approx 2^{100}$) with eigenvalues $\lambda_1 \le \lambda_2 \le \dots \le ...
0
votes
0answers
95 views

Eigenvalue bounds from eigenvalues of Schur complement

Is it possible to lower bound the minimal eigenvalue of a symmetric PSD matrix $M= \begin{pmatrix}A & C\\ C^*&B\end{pmatrix}$ from the knowledge of the eigenvalues of $M$'s Schur complement ...
1
vote
0answers
162 views

3-regular (cubic) graph with adjacency eigenvalue 1

Suppose $A\in\{0,1\}^{n\times n}$ is the adjacency matrix of a 3-regular (cubic) graph $G=(V,E)$; that is, all $n$ vertices $v\in V$ in the graph have three neighbors. Is there a nice necessary ...
2
votes
0answers
337 views

Largest eigenvalues distribution of tridiagonal symmetric random matrix

I would like to find the largest eigenvalue distribution of the following tridiagonal symmetric random matrix in an analytic way. All the ${\lambda}_i$ are distributed the same way with chi-square ...