User a. lerario - MathOverflow

Map with prescribed Jacobian

2012-07-17T14:55:46Z

Recently I came up with the following problem.

Suppose $U$ is an open subset of $\mathbb{R}^n$ and we are given a continuous map $M:U\to GL(n;\mathbb{R})$. Does anybody know if there are conditions for the existence of a $C^1$ map $f:U\to \mathbb{R}^n$ such that $$Jf(x)=M(x)\quad \forall x \in U$$ (here $Jf(x)$ is the Jacobian matrix of $f$ at $x$)?

I know this question is somehow a little bit general; I will really appreciate even a reference for something related to this problem. :)

Least singular value gaussian orthogonal ensemble.

2012-02-29T19:20:25Z

Hello everybody, here is my question:

Assume A is a random symmetric $nxn$ matrix whose entries are independent, normally distributed with mean zero and variance 2 on the diagonal and 1 off diagonal (to my knowledge such a random matrix is said to belong to the 'Gaussian orthogonal ensemble'). The joint pdf for the eigenvalues of A is well known, but i was wondering:

does there exist a precise formula for the probability that all the eigenvalues of A have norm greater then epsilon?

Equivalently which is the probability that the least singular value of A is smaller than epsilon?

I am computing the intrinsic volume of singular symmetric matrices of (Frobenius or trace-square) norm one and I need this precise formula to perform the epsilon limit using tubes. I am sorry if this is a well known result, but I was not able to find it in the literature. In case I would really appreciate a reference for this.

Thanks everyone!

Answer by A. Lerario for How much can a diagonal matrix change the eigenvalues of a symmetric matrix?

2011-09-23T09:59:37Z

In general there is no relation: for example consider the simplest case $S$ itself is diagonal and invertible. Letting $D_1=S^{-1}$ then $A$ can be any diagonal matrix $D_2$. The only considerations you can do are related to the presence of the zero eigenvalues using Binet formula for determinants. Notice also that in general $A$ itself can be nonsymmetric, and its eigenvalues can be complex. However small perturbations, i.e. small $D_1$ and $D_2$, result in a small perturbation of the eigenvalues of $S$ in the complex plane.

Answer by A. Lerario for Small neighborhoods of singularities on varieties

2011-07-23T11:25:53Z

Indeed the following theorem to me seems exactly you were looking for (see J. Bochnak, M. Coste, M-F. Roy, "Real algebraic geometry", Theorem 9.3.6 [Local conic structure]):

Let $E$ be a semialgebraic susbet of $\mathbb{R}^n$ and $x$ be a nonisolated point of $E.$ Let also $D_\epsilon$ be the closed $\epsilon$-ball around $x$ and $S_\epsilon$ its boundary. Set $K=S_\epsilon \cap E$. Then there for $\epsilon>0$ small enough the pair $(D_\epsilon,E∩D_\epsilon)$ is semialgebraically homeomorphic to the pair $(CS_\epsilon,CK)$, where $C$ denotes taking the cone. Moreover the semialgebraic homeomorphism can be chosen as to preserve the distance from $x.$

Two words of remarks on the previous statement:

Every real or complex algebraic set in $\mathbb{R}^n$ or in $\mathbb{C}^n\simeq \mathbb{R}^{2n}$ is a semialgebraic set.
The point $x$ is any nonisolated point of $E$ (no matter singular - in whatever meaning this word has for a general semialgebraic set - or regular).

Answer by A. Lerario for Canonical form for a pair of quadratic forms

2011-07-16T00:39:17Z

Hi, I just read your question. I hope it's not too late to answer, but I think the following paper contains exactly what you are looking for:

R. C. Thompson: Pencils of complex and real symmetric and skew matrices, Linear Algebra and its Applications, Volume 147, March 1991, 323-371.

The author classifies - up to congruence - any pairs of real symmetric matrices, which amounts to give the classification for any pair of real quadratic forms.

You'll probably find of some interest also this paper of mine: http://arxiv.org/abs/1106.4678

Comment by A. Lerario

2012-07-17T20:01:17Z

I was writing tautological for the case $M$ is differentiable. For $M$ continuous - I am not an expert - do I understand correctly that closedness of $\omega_i$ in the distributional sense is a sufficient condition?

Comment by A. Lerario

2012-07-17T15:52:39Z

This is a nice answer, though somehow tautological. Also notice that we are not allowed to take derivatives of the matrix $M$ since it is assumed to depend only continuously on $x.$

Comment by A. Lerario

2012-03-01T01:44:19Z

I don't see an immediate way to relate this to the GOE case... :)