# 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 $$1$$.