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?