The eigenvalues tag has no usage guidance.

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58 views

### Eigenvalues of a random matrix [on hold]

For test cases i generated a random real uniform distributed matrix with
entries from the intervall $[0,1]$.
Here is the MATLAB Code i used
...

**0**

votes

**0**answers

43 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 ...

**3**

votes

**1**answer

165 views

### Eigenvalues of the sum of two matrices, where one is $B=\operatorname{diag}(1, 0,\dots,0)$

I know that given two matrices $A$ and $B$, estimating the eigenvalues of $A + B$ by the eigenvalues of $A$ and $B$ is generally a non-easy problem. In particular, there are some results for matrices ...

**0**

votes

**0**answers

41 views

### An eigenvalue convergence question [closed]

The following question is from Eberhard Zeidler's book in Nonlinear Functional Analysis volume 1, question 1.5b page 40.
Let $(X,(\cdot | \cdot))$ be a Hilbert space, take the operator: ...

**2**

votes

**0**answers

185 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 ...

**0**

votes

**0**answers

31 views

### Question about “Third existence theorem for weak solutions” in Evans - Partial Differential equations [migrated]

I'm currently studying Evans excellent book "Partial Differential Equations" and I'm a bit stuck in the proof of Theorem 5 in Section 6.2.3 (P. 305, "Third Existence Theorem for weak solutions).
What ...

**1**

vote

**0**answers

79 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 ...

**3**

votes

**1**answer

102 views

### Determinant of a Certain Positive-Definite Block Matrix

Is there a lower bound for the determinant or minimum eigenvalue of the following $d$ by $d$ matrix in terms of $d$?
$$\Gamma=\left( {\begin{array}{cc}
I & B \\
B^{*} & I \\
\end{array} ...

**1**

vote

**1**answer

72 views

### Limit of largest eigenvalue [closed]

For positive definite matrix, if we increase the dimension to the infinity, is it true that the largest eigenvalue stays bounded from above?
In other words does the following limit exists:
...

**-1**

votes

**0**answers

49 views

### Spectral radius of principal submatrices for the case of hermitian matrix

A principal submatrix of a matrix $\mathbf{A}\in\mathbb{C}^{N\times N}$ is any submatrix from $\mathbf{A}$ for which the same rows and columns have been eliminated.
Assume $\mathbf{A}$ is hermitian, ...

**0**

votes

**2**answers

60 views

### Connectivity of weighted graph and zero Laplacian eigenvalues

Given an undirected graph $G$, and let $V$ denote its set of vertices and $E$ its set of edges. Suppose that there are no edges connecting the same vertex, and no more than one edge connecting any ...

**4**

votes

**1**answer

232 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 ...

**0**

votes

**0**answers

75 views

### Find motivation for calculating $\int_{2}^{X} A^2(t) A(\alpha t)dt$

I read a thesis of Kong Kar Lun (student of Tsang K.M) about the some mean value theorems for certain errors terms in analytic number theory and in which he gave the asymptotic formulas of the ...

**5**

votes

**0**answers

58 views

### Possible values of eigenvalues of Hadamard product of Hermitian matrices

One of the most important (and very well-known) result in the study of the spectrum of Hermitian matrices is Horn's conjecture (or theorem?), which provides a complete answer to the following problem:
...

**1**

vote

**0**answers

58 views

### Interpreting (Fiedler) spectral bisectioning

I would appreciate help on how to interpret the results of spectral bisectioning of a graph.
Given a $G=(V,E)$ with size $N$ represented by $Q$ its Laplacian matrix where the eigenvalues are ordered ...

**0**

votes

**0**answers

28 views

### Asymptotic behavior of the minimum eigenvalue of a certain Gram matrix with linear independence

Consider the density matrices with the following spectral decompositions:
$$\rho=\lambda_1|\nu_1\rangle+\lambda_{2}|\nu_2\rangle$$
and
$$\sigma=\gamma_1|\omega_1\rangle+\gamma_2|\omega_2\rangle$$
such ...

**0**

votes

**0**answers

22 views

### Weighted Perturbation Bound for Polar Decomposition

Setup: Let $X\in\mathbb{R}^{n\times r}$ be a matrix with orthogonal columns, with $\Sigma = X^TX$, and assume that $\Sigma$ is invertible (note, $\Sigma$ is not necessarily the identity).
Suppose we ...

**3**

votes

**5**answers

4k 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

**3**answers

136 views

### Simple Spectrum of Jacobi matrices

I want to call a matrix a Jacobi matrix (cause there may be different notions of Jacobi matrices) if it is a tridiagonal matrix with positive off-diagonal entries. Now, I read that the spectrum of ...

**7**

votes

**7**answers

581 views

### Semicircle law universality elsewhere

Wigner's semicircle distribution is:
$$f(x)=\frac{1}{2 \pi}\sqrt{4-x^2}, \ \ -2\leq x\leq 2.$$
Under reasonable conditions, the rescaled eigenvalue density of random symmetric matrices $M_n$ follows ...

**0**

votes

**0**answers

59 views

### Extracting information from a differential equation if the zero eigenvalue eigen-function is known

Given the second order linear homogeneous differential equation
$$
-\dfrac{d^2}{dx^2}\psi_m(x) + V(x)\psi_m(x)=E_m\psi_m(x)
$$
with eigen-functions $\psi_m(x)$ and eigenvalues $E_m$, what information ...

**2**

votes

**4**answers

857 views

### On the spectrum of random regular graph

For a random $d$-regular graph, where $d$ can be fixed or can grow slowly with the size of the graph $n$, what can we say about its spectrum - Do you believe it has simple spectrum?
Thank you,

**0**

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**0**answers

42 views

### Bounds on the smallest eigenvalue of a Hankel matrix

Let $H=H_n$ be a positive definite Hankel matrix of size $n$ with $\lambda_n$ is it's smallest eigenvalue.
What bounds are known on $\lambda_n$ in terms of the entries on $H$.
I can see some results ...

**4**

votes

**3**answers

221 views

### How networks with high largest eigenvalues are more robust?

In the literature, it is sometimes indicated that network with high value of largest eigenvalue (either adjacency matrix or its Laplacian counterpart) are more robust against link/node removals. ...

**0**

votes

**1**answer

40 views

### Lower bound on Spectral Gap of Rank one + Diagonal

For some $x\in\mathbb{R}^n, \|x\|_2^2=1$ and $\alpha\geq 0$, consider the positive semi-definite matrix
$$
X_\alpha := xx^T + \alpha\sum_{k=1}^nx_k^2e_ke_k^T.
$$
Suppose for simplicity that the ...

**0**

votes

**0**answers

74 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}\,\,\, ...

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vote

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69 views

### Perturbation of eigenvalues of some special matrices

In perturbation theory of linear operators, one major question is how the eigenvalues of a linear operator $A$ change under a small perturbation, $A(x) = A + xP$, with $x\in\mathbb{R}$. For instance, ...

**4**

votes

**1**answer

94 views

### An inequality for eigenvalues of the Dirichlet problem

Is either of these inequalities true?
$$\lambda(tA + (1-t)B)\geq t\lambda(A) + (1-t)\lambda(B)$$
or
$$\lambda(tA + (1-t)B)\leq t\lambda(A) + (1-t)\lambda(B),$$
where $0\leq t \leq 1$, $A,B$ are ...

**3**

votes

**1**answer

168 views

### Eigenvectors as continuous functions of matrix - diagonal perturbations

The general question has been treated here, and the response was negative. My question is about more particular perturbations. The counterexamples given in the previous question have variations not ...

**1**

vote

**0**answers

43 views

### Bounds on the spherical measure of sub-level sets of quadratic forms

I'm wondering if there are any bounds on the spherical measure of sets of the form
$$
\mu_n\left(\{y\in S^{n-1} : \frac{y_1^2}{y_2^2} < \alpha\}\right) \leq f(\alpha)
$$
where $\alpha$ is some ...

**1**

vote

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56 views

### Epsilon-net of operator norm ball around Identity

Suppose I look at the set of matrices which are invertible and satisfy
$$
\left\|A-Id\right\|_{op}<r
$$
for some $r<1$, where $Id$ is the $n\times n$ identity matrix. An $\epsilon$-net of such ...

**1**

vote

**0**answers

119 views

### Bounding the largest Singular value

D is a $n \times n$ diagonal matrix whose diagonal entries lies in $(0,1]$.
B is any $n \times n$ n.n.d. matrix.
What will be the sharpest upper bound on the largest eigenvalue of:
...

**0**

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**0**answers

17 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 $M/G$ ...

**2**

votes

**1**answer

69 views

### Is there a general way to determine the Laplacian of the eigenvalues of a real symmetric matrix? [closed]

I have a real symmetric $3\times3$ matrix $\mathbf{M}(\mathbf{r}$) which depends on $\mathbf{r} \in \mathbb{R}^3$. Each eigenvalue can be considered a scalar field $e_i(\mathbf{r})$ over ...

**4**

votes

**2**answers

1k views

### rank-one perturbation of a matrix corresponding to a specific spectrum

Let $A$ be a real symmetric matrix whose spectrum is $\lambda_1,\lambda_2,\ldots,\lambda_n$.
Let $A'$ be the matrix obtained by adding a perturbation to $A$. The requirement is that only the second ...

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vote

**0**answers

30 views

### Eigenvalue overdetermined problem

Consider the following overdetermined eigenvalue problem for $\Omega \subset \Bbb{R}^2$:
$$(1) \ \ \ \ \begin{cases} - \Delta u = \lambda u & \text{ in }\Omega \\
u = 0 ...

**2**

votes

**1**answer

116 views

### Proof of eigenvalue stability inequality via Courant-Fischer min-max theorem

Dr. Tao in his notes on eigenvalue inequalities uses Courant-Fischer min-max theorem to prove the eigenvalue stability inequality. Specifically, I am looking for proof of Eq. (13) where Dr. Tao ...

**0**

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**0**answers

56 views

### On the least singular value of a random matrix and its minors

Let $A$ be a $n \times n$ random matrix with entries i.i.d from the continuous uniform distribution U([-1,1]). Is it true that the least singular value of $A$ and all its minors is greater than ...

**1**

vote

**1**answer

76 views

### On the least singular values

Let $A$ be a square matrix of size $n \times n$ ($n>2$) and let $B$ be $A$ if we delete the last row and column (size $(n-1) \times (n-1)$). Let $\sigma (A)$ be the least singular value of $A$ and ...

**6**

votes

**5**answers

406 views

### The maximal eigenvalue of a symmetric Toeplitz matrix

Let $0\le x\le 1$ be a real number. Denote by $A_n(x)=(a_{ij})$ the $n$ by $n$ matrix such that $a_{ij}=x^{|i-j|}$ and let $\lambda_n(x)$ be the maximal eigenvalue of $A_n(x)$.
Is there any ...

**2**

votes

**1**answer

76 views

### Reducing eigenvalues of symmetric PSD matrix towards 0: effect on ratios of original matrix elements?

Let $\boldsymbol{S}$ be $k \times k$ positive semi-definite real symmetric matrix with eigen decomposition $\boldsymbol{S} = \boldsymbol{X} \boldsymbol{\Lambda} \boldsymbol{X}'$ ...

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votes

**0**answers

94 views

### Bound on the ratio of top 2 eigenvalues

Let $P$ be a $(n+1) \times (n+1)$ stochastic matrix such that $P_{ij}=\tau$ if $i \neq j$ and $P_{ii} = (1 - n\tau)$ where $0<\tau < \frac{1}{n+1}$. It is clear that the largest eigenvalue of ...

**2**

votes

**1**answer

110 views

### the impossibility of exactly computing eigenvalues [closed]

I is well known that there is no explicit formula for the eigenvalues of a general matrix (see e.g. Wikipedia). This result is a consequence of (1) Abel's theorem, stating that there is no explicit ...

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votes

**2**answers

237 views

### No Exceptional Eigenvalues of Weight 1/2 Maass Forms on $\Gamma_0(4)$?

Some colleagues and I were wondering if there is a citation out there which shows there are no exceptional eigenvalues, $\lambda$, of classical weight 1/2 Maass forms on $\Gamma_0(4)$, which is to say ...

**3**

votes

**1**answer

558 views

### Why are 1 and -1 eigenvalues of this matrix?

This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix $\mathbf{A}$.
First, let's define two matrices:
...

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votes

**1**answer

46 views

### Characterisation of a matrix ordering property

Let $n$ be a positive integer; we consider all matrices mentioned henceforth to be $n$-by-$n$ matrices. Let $A$ and $B$ be matrices wherein all entries are nonnegative (such matrices will be called ...

**2**

votes

**2**answers

166 views

### Matrices with real spectrum

Assume you have a non-symmetric real square matrix of all whose eigenvalues are real. Can anything be said about it? Is it unitarily equivalent to a symmetric matrix?
EDIT: Is it at least similar to ...

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118 views

### Eigenvalue of product of self adjoint compact operators

Suppose A is a self adjoint $m \times m$ real matrix with eigenpairs $\{e_j, \lambda_j\}$ such that $\lambda_j > \lambda_{j + 1}$. Let $B$ be another self adjoint real $m \times m$ matrix such that ...

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39 views

### Multiplicity of Ritz eigenvalues

Consider a Krylov subspace $K_m=\mathrm{span}\{v,Pv,...,P^{m-1}v\}$, for $P$ a square matrix and a nonzero vector $v$. Let $H_m$ represent the projection of $P$ (seen as an application) restricted to ...

**1**

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79 views

### An exact fraction of a matrix

Let $A$ be a $n \times m$ real matrix with $n<<m$ and of rank $r<n$. It is known that $A$ has exactly two distinct non-zero singular values: $\sigma_{\max}$ and $\sigma_{2}$, and also that ...