.unnumbered}
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Consider the vector space of functions $\zoon\to\Re,$ equipped with the inner product $$\begin{aligned}
\langle f,g\rangle = 2^{-n} \sum_{x\in\zoon}f(x)g(x).\end{aligned}$$ For $S\subseteq\oneton,$ define $\chi_S\colon \zoon\to\moo$ by $\chi_S(x) =(-1)^{\sum_{i\in S} x_i}.$ Then $\{\chi_S\}_{S\subseteq\oneton}$ is an orthonormal basis for the inner product space in question. As a result, every function $f\colon \zoon\to\Re$ has a unique representation of the form $$\begin{aligned}
f=\sum_{S\subseteq\oneton} \hat f(S)\chi_S,\end{aligned}$$ where $\hat
f(S)=\langle f,\chi_S\rangle$. The reals $\hat f(S)$ are called the *Fourier coefficients of $f.$* The orthonormality of $\{\chi_S\}$ immediately yields *Parseval’s identity*: $$\begin{aligned}
  \sum_{S\subseteq\oneton} \hat f(S)^2 
    = \langle f,f\rangle = \Exp_{x\in\zoon}[f(x)^2]. \label{eqn:parsevals}\end{aligned}$$

Matrices. {#matrices. .unnumbered}
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The symbol $\Re^{m\times n}$ refers to the family of all $m\times n$ matrices with real entries. A matrix $A\in\Re^{n\times n}$ is called *strictly diagonally dominant* if $$\begin{aligned}
\abs{A_{ii}} > 
\sum_{\substack{j=1\\j\ne i}}^n \abs{A_{ij}}, &&i=1,2,\dots,n.\end{aligned}$$ A well-known result in matrix perturbation theory, due to Gershgorin [@gershgorin31disks], states that the eigenvalues of a matrix lie in the union of certain disks in the complex plane centered around the diagonal entries of the matrix. We will need the following very special case, which corresponds to showing that the eigenvalues are all nonzero.

\[thm:gershgorin\] Let $A\in\Re^{n\times n}$ be strictly diagonally dominant. Then $A$ is nonsingular.

Fix a 