The purpose of this question is to ask about the Fourier transform of the map which associate to an $n$ by $n$ matrix its $n$ eigenvalues, or some function of the $n$ eigenvalues. The main motivation comes from random matrices. (But additional motivation comes from discrete harmonic analysis.)

(There are various possible variations and I am not sure which one is more fruitful and accessible so I indicate several possibilities.)

A. The functions

1) Let $F$ be the function which associate to an $n$ by $n$ symmetric matrix its $n$ eigenvalues $\lambda_1 \le \lambda_2<\cdots\le\lambda_n$.

2) $G$ - a variation on $F$: Here we consider the set of eigenvalues rather than the ordered sequence of eigenvalues.

3) Let $f=\lambda_n$ be the real function which assigns to a matrix its maximum eigenvalue.

4) Other functions of the eigenvalues, like the determinant.

B. The domain of matrices

Case 1: Suppose that the entries of the matrices to start with are i.i.d. Gaussian and $F$ is considered as a function from a $n^2$-dimensional (or ${{n+1} \choose {2}}$-dimensional) product space to $R^n$.

Case 2: Suppose that the entries are $\pm 1$ Bernoulli variables (namely, each entry is +1 with probability 1/2 and -1 with probability 1/2) and $F$ is considered as a function from the $n^2$-dimensional (or ${n+1} \choose {2}$-dimensional) discrete cube to $R^n$.

Case 3: More variations- other classes of random matrices.

C. The question

Question: What is known about the Fourier expansion/Fourier-Walsh expansion of these functions $F$, $G$, $f$? What can be expected?

D. Remarks

1) The case I am most comfortable in asking the question is the Bernoulli case. So $f=\sum \hat f(S)W_S$ where $S$ is a subset of the locations in the matrix and $W_S= \prod x_{ij}:${$i,j$}$ \in S$. For the function $F$, $\hat F(S)$ is an $n$-dimensional vector rather than a real number.

2) The case which seems easiest to handle is the case of Gaussian variables. I am not sure what is the best expansion we want to use. Perhaps the expansion into spherical harmonics is a good choice.

3) The function $G$ looks natural but since the targets are sets I am not sure at all how to transform.

4) The function $f$ of maximum eigenvalue is natural and much studied and we can also talk about other functions of the eigenvalues like the determinant.

E. Determinants (added April 2013)

What can be said about the Fourier transform of the determinant?

  • $\begingroup$ would you be interested in Fourier expansion of n-point correlation functions? these are known. $\endgroup$ – Carlo Beenakker Dec 18 '12 at 14:05
  • $\begingroup$ Yes I will be happy to learn about that as well. $\endgroup$ – Gil Kalai Dec 18 '12 at 15:03
  • 1
    $\begingroup$ Dear Carlo, But also note that my question is not about the Fourier transform of the distribution of all eigenvalues or the maximum one but of the function that send the entries of the matrix to these things. $\endgroup$ – Gil Kalai Dec 18 '12 at 15:19
  • $\begingroup$ How about this: dlmf.nist.gov/35.2 $\endgroup$ – Suvrit Dec 18 '12 at 17:18
  • $\begingroup$ As Yuval Filmus pointed out the formula for the determinant for Bernoulli matrices is its Fourier expansion ! :) (and I suppose that this is so in the Gaussian case as well). $\endgroup$ – Gil Kalai Jul 23 '13 at 4:43

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