Consider a real differentiable function $f:\mathbb{R}\to\mathbb{R}^N$ and define a matrix $A_{ij}=\mathbb{E}[\frac{\partial}{\partial x}[f_i(x) f_j(x)]]$ where the expectation is with respect to some measure on $f$'s domain. As it is symmetric all its eigenvalues are real, and I would like to derive conditions on $f$ so that if $\lambda$ is an eigenvalue of $A$ then $-\lambda$ is also an eigenvalue. That is, I ask:

  • Q1: When do the eigenvalues of $A$ can be paired as $\pm\lambda$?
  • Q2: If there a nessesary and sufficient condition on $f$ so that this will be the case?


EDITED: changed $A$'s definition to include expectation over $x$; otherwise the second question is meaningless.

  • $\begingroup$ So, apparently, $\;f(x)=\left(f_1(x),f_2(x),\ldots,f_n(x)\right)\;$ and all the $\;f_i$'s are differentiable as real valued functions, right? $\endgroup$ – DonAntonio Nov 1 '13 at 23:33
  • $\begingroup$ The eigenvalues will be functions of $x$; you want this condition to hold for all $x$? $\endgroup$ – Gerry Myerson Nov 1 '13 at 23:34
  • 2
    $\begingroup$ Well, a first necessary and rather trivial condition is that $\;n\;$ is even...unless you accept, for example, thirds of eigenvalues of the form $\;\lambda, -\lambda,-\lambda\;$ ... $\endgroup$ – DonAntonio Nov 1 '13 at 23:36
  • 1
    $\begingroup$ @DonAntonio: yes, assume all the $f_i$ are differentiable and $n$ is even. $\endgroup$ – Uri Cohen Nov 1 '13 at 23:40
  • $\begingroup$ @GerryMyerson: yes, indeed as I put it $A=A(x)$; I'm actually interested in a broader question of $A=\mathbb{E}[A(x)]$ with respect to some measure on x, but I wanted to keep the question simple. $\endgroup$ – Uri Cohen Nov 1 '13 at 23:43

Assme that your measure is in the form $d\mu=\phi(x)dx$, with $d\mu(\mathbb{R})=1$. Then $A_{i,j}=\int(f_if_j)'\phi(x)dx=-\int f_if_j\phi '(x)dx$ if $\phi$ tends quickly to $0$ at infinity and if $\phi '$ exists. A necessary condition is $trace(A)=0$, that is $\int (\sum_if_i^2)\phi '(x)dx=0$. This condition is sufficient when $n=2$. If, for instance, $\phi(x)=e^{-|x|}$, then the condition is $\int_0^{\infty}\sum_i(f_i^2(x)-f_i^2(-x))e^{-x}dx=0$. Yet, the general case seems to me very difficult especially if $d\mu$ is not given.


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