**2**

votes

**2**answers

353 views

### Can one estimate the distribution of eigenvalues of a matrix by its Cauchy/Stieltje transform?

Given a real symmetric $n$ dimensional matrix $A$, with eigenvalues $\lambda_i$ I am defining its Cauchy transform as the function, $f_A(z) = \sum_i \frac{1}{z-\lambda_i}\,$
Is there any information ...

**23**

votes

**3**answers

3k views

### Trapped rays bouncing between two convex bodies

At some point during my research I was confronted with this problem, but I did not dedicate serious time to it. Anyway it stayed in the back of my mind and I'm still interested in hints for it. ...

**11**

votes

**5**answers

1k views

### Can always a family of symmetric real matrices depending smoothly on a real parameter be diagonalized by smooth similarity transformations?

This question is related to another question, but it is definitely not the same.
Is it always possible to diagonalize (at least locally around each point) a family of symmetric real matrices $A(t)$ ...

**10**

votes

**2**answers

3k views

### Conditions for smooth dependence of the eigenvalues and eigenvectors of a matrix on a set of parameters

Let $A\in\mathcal M_n$ be an $n\times n$ real [symmetric] matrix which depends smoothly on a [finite] set of parameters, $A=A(\xi_1,\ldots,\xi_k)$. We can view it as a smooth function $A:\mathbb ...

**1**

vote

**5**answers

307 views

### About adding a negative definite rank-1 matrix to a symmetric matrix

If $B$ is a symmetric matrix then how do its eigenvalues compare to the eigenvalues of $B - vv^T$? ( where $v$ is a vector of the same dimension as $B$)
I guess that the eigenvalues of $B - vv^T$ ...

**1**

vote

**0**answers

376 views

### Possibility of Disconnected Subgraphs of a $k$ Connected $r$ regular Graph under a given condition

Context: Given a adjacency matrix A of a $r$-regular graph $G$ (not complete graph $K_{r+1}$) . $G$ is $k$ connected.
The matrix A can be divided into 4 sub-matrices based on adjacency of vertex $x ...

**33**

votes

**10**answers

8k views

### real symmetric matrix has real eigenvalues - elementary proof

Every real symmetric matrix has at least one real eigenvalue. Does anyone know how to prove this elementary, that is without the notion of complex numbers?

**8**

votes

**3**answers

707 views

### Functions of Pseudodifferential Operators

Suppose I have a self-adjoint pseudo-differential operator $A$ on $\mathbb{R}^n$ and a continuous function $f$ (possibly bounded, or Schwartz, or compactly supported) on its spectrum. Then I can ...

**16**

votes

**4**answers

1k views

### High multiplicity eigenvalue implies symmetry?

It is well known that on any compact Riemannian symmetric space $X$, the eigenvalues of the Laplacian have very high multiplicity (comparable with the Weyl bound), and the resulting actions ...

**14**

votes

**2**answers

645 views

### Eigenvalues of an “oblique diagonal” matrix

I am looking for guidance about the behavior of powers of a particular matrix (call it $A_n$ for $n\ge2$), which has come up in a counting problem about quantum knot mosaics (a good reference for ...

**6**

votes

**3**answers

850 views

### Some explanation about Dynin's formalism

I have seen this claim on the Wikipedia page for the Yang-Mills Millenium problem by Alexander Dynin. He is a mathematician working at the Department of Mathematics of Ohio State University and so, I ...

**4**

votes

**2**answers

616 views

### Exotic spectrum of Laplace operator

Given a closed Riemannian manifold and a generalized Laplace $\Delta$ operator,
it is well known that $\Delta$ has discrete spectrum $(\lambda_n)_n$ (arranged in a increasing way, not counting ...

**3**

votes

**3**answers

3k views

### eigenvalues of sum of a non-symmetric matrix and its transpose (A+A^T)

Suppose we have a matrix $M$ such that $M$ is non-symmetric real and has positive eigenvalues. Do we have a relation between eigenvalues/eigenvectors of $(M+M^T)$ and those of $M$?
What if $M$ and ...

**9**

votes

**3**answers

1k views

### The first eigenvalue of the laplacian for complex projective space

What is the exact value of the first eigenvalue of the laplacian for complex projective space viewed as $SU(n+1)/S(U(1)\times U(n))$?

**5**

votes

**1**answer

1k views

### The Guinand-Weil explicit formula without entire function theory

I'll admit from the outset that this question is slightly vague. The actual question appears at the end of the post.
The explicit formula of Guinand and Weil can be written in the following way:
For ...

**2**

votes

**1**answer

590 views

### Global Lichnerowicz Formula Proof (in the Kahler case)?

For a Kahler manifold $M$, let us denote its Dirac operator $\overline{\partial} + \overline{\partial}^\ast$, with respect to a metric $g$, by $D$. Moreover, let us dentoe the Levi-Civita connection ...

**7**

votes

**1**answer

733 views

### 0 eigenvalue for a symmetric tridiagonal matrix

Let $T\in \mathbb{R}^{n\times n}$ be a symmetric tridiagonal matrix having the off--diagonal entries equal to -1. The diagonal entries are all positive, $a_i>0$, $i=\overline{1,n}$, and there ...

**5**

votes

**1**answer

265 views

### Spectrum of this ODE

I noticed something interesting studying this Sturm-Liouville Problem:
$$ \frac{d}{dx}\left(\sqrt{(1-x^{2})}\frac{df}{dx} \right)+\frac{\left(n \alpha x+\alpha^2 x^{2} + ...

**4**

votes

**3**answers

386 views

### Estimates for the diameter of a (nice) surface?

The Question
Let $M$ be a compact, connected, orientable surface without boundary of bounded genus smoothly embedded in $\mathbb{R}^3$; define the diameter $d_M$ of $M$ as the maximum minimal ...

**3**

votes

**0**answers

255 views

### The Bilu-Linial conjecture and Ramanujan graphs

The Bilu-Linial conjecture claims that every $d-$regular graph has a $2-$lift such that for the signing matrix has its eigenvalues between $[-2\sqrt{d-1},2\sqrt{d-1}]$ (the ``signing matrix" is the ...

**3**

votes

**1**answer

357 views

### The first eigenvalue of the Schrödinger operator is simple.

Hello,
let $(M,g)$ be a compact and connected Riemannian manifold (possibly with $\partial M\neq \emptyset$). We consider the Friedrichs extension of $L=-\Delta +V: C^{\infty}(M,\mathbb{R})\subset ...

**3**

votes

**3**answers

667 views

### Boundness of Laplacian eigenfunctions

Let $A$ be a bounded domain in $\mathbb R^d$, $d>1$, and $\{u_k\}$ is the set of all $L^2$-normalized Laplacian eigenfunctions on $A$ with Dirichlet boundary condition (i.e., $\|u_k\|_2 = 1$).
Is ...

**9**

votes

**1**answer

223 views

### $C^k$ one-parameter family of metrics

Consider a smooth Riemannian manifold $M$ and a $C^k$ one-parameter family of Riemannian metrics $g_t$ on $M$. Here $k$ could be any integer, $k$ could be infinity, when the one-parameter family $g_t$ ...

**0**

votes

**1**answer

159 views

### Decomposition of a regular graph and connected subgraphs

I have asked almost same question earlier. I have been told that my question was poorly written, so I am trying to write it more clearly in this post. Also, this time I would be a little different in ...

**0**

votes

**1**answer

217 views

### Spectrum of the operator PAP, with A self-adjoint and P strictly positive

Let $A$ be an unbounded self-adjoint operator with spectrum $\sigma(A)=\mathbb R$ in a Hilbert space $\mathcal H$. Let $P$ be a bounded operator in $\mathcal H$ satisfying $P\ge1$ and
$$
{\rm ...