derivatives and random matrices

In connection with our IGL project, I remembered a result that might be useful, and is not widely enough known.

Let \(z_1\leq \ldots\leq z_N\) be a sequence of real numbers forming the spectrum of a Hermitian operator \(A\) acting in \(N\)-dimensional Hermitian space \(U\):
\[P_A(z):=\det(zE-A)=\prod_k (z-z_k).\]

Proposition: Let \(V\) be a codimension \(1\) hyperplane in \(U\), and \(A(V)\) the restriction of \(A\) to it. Then
\[P_{A(V)}(z)= P_A(z)\sum_k\frac{|v_k|^2}{z-z_k},\]
where \((v_1,\ldots,v_k)\) are the coordinates of the vector of norm \(1\) defining the hyperplane \(V\) in the eigenbasis of \(A\).

This follows more or less immediately from the standard formulae for the determinant of the block matrices, applied to \[ \left( \begin{array}{cc}0&v^\dagger\\v& zE-A \end{array}\right).\]

Corollary: The average of the polynomials \(P_{A(V)}\) over \(V\) uniformly drawn from the corresponding unit sphere is the derivative of \(P_A\): \[\mathbb{E}_V P_{A(V)}(z)=\frac{1}{N}P_A'(z).\]

Indeed, for \((v_1,\ldots,v_k)\) uniformly distributed in the unit sphere, the vector of their squared norms \((|v_1|^2,\ldots,|v_k|^2)\) is uniformly distributed in the unit simplex, and its center of gravity is \((1/N,\ldots,1/N)\).

Iterating we obtain that the expected characteristic polynomial of \(A\) restricted to a uniformly distributed \(N-k\)-dimensional flat in \(U\) is the \(k\)-th derivative of \(P_A\) rescaled so that its leading coefficient is \(1\).

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