Begin a discussion of the structure of the hessian.

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@@ -360,6 +360,96 @@ occur order-one times, one could reasonably hope to perform such a procedure.
 If they occur exponentially often in the system size, there is little hope of
 keeping track of the resulting weights, and analytic continuation is intractable.
 
+\subsection{The structure of stationary points}
+
+\begin{eqnarray}
+  \operatorname{Hess}\operatorname{Re}\beta\mathcal S
+  &=\left[\matrix{
+    \partial_x\partial_x\operatorname{Re}\beta\mathcal S &
+    \partial_y\partial_x\operatorname{Re}\beta\mathcal S \cr
+    \partial_x\partial_y\operatorname{Re}\beta\mathcal S &
+    \partial_y\partial_y\operatorname{Re}\beta\mathcal S
+  }\right] \\
+  &=\left[\matrix{
+    \hphantom{-}\operatorname{Re}\beta\partial\partial\mathcal S &
+    -\operatorname{Im}\beta\partial\partial\mathcal S \cr
+    -\operatorname{Im}\beta\partial\partial\mathcal S &
+    -\operatorname{Re}\beta\partial\partial\mathcal S
+  }\right]
+\end{eqnarray}
+
+The eigenvalues and eigenvectors of the Hessian are important for evaluating
+thimble integrals, because those associated with upward directions provide a
+local basis for the surface of the thimble. Suppose that $v_x,v_y\in\mathbb
+R^N$ are such that
+\begin{equation}
+  (\operatorname{Hess}\operatorname{Re}\beta\mathcal S)\left[\matrix{v_x \cr v_y}\right]
+  =\lambda\left[\matrix{v_x \cr v_y}\right]
+\end{equation}
+where the eigenvalue $\lambda$ must be real because the hessian is real symmetric. The problem can be put into a more obviously complex form by a change of basis. Writing $v=v_x+iv_y$, we find
+\begin{eqnarray}
+  &\left[\matrix{0&-i(\beta\partial\partial\mathcal S)^*\cr i\beta\partial\partial\mathcal S&0}\right]
+  \left[\matrix{v \cr iv^*}\right]\\
+  &\qquad=\left[\matrix{1&i\cr i&1}\right]
+  (\operatorname{Hess}\operatorname{Re}\beta\mathcal S)
+  \left[\matrix{1&i\cr i&1}\right]^{-1}
+  \left[\matrix{1&i\cr i&1}\right]
+  \left[\matrix{v_x \cr v_y}\right] \\
+  &\qquad=\lambda\left[\matrix{1&i\cr i&1}\right]\left[\matrix{v_x \cr v_y}\right]
+  =\lambda\left[\matrix{v \cr iv^*}\right]
+\end{eqnarray}
+It therefore follows that the eigenvalues and vectors of the real hessian satisfy the equation
+\begin{equation} \label{eq:generalized.eigenproblem}
+  \beta\partial\partial\mathcal S v=\lambda v^*
+\end{equation}
+a sort of generalized
+eigenvalue problem. If we did not know the eigenvalues were real, we could
+still see it from the second implied equation, $(\beta\partial\partial\mathcal
+S)^*v^*=\lambda v$, which is the conjugate of the first if $\lambda^*=\lambda$.
+
+Something somewhat hidden in the structure of the real hessian but more clear
+in its complex form is that each eigenvalue comes in a pair, since
+\begin{equation}
+  \beta\partial\partial\mathcal S(iv)=i\lambda v^*=-\lambda(iv)
+\end{equation}
+Therefore, if $\lambda$ is an eigenvalue of the hessian with eigenvector $v$,
+than so is $-\lambda$, with associated eigenvector $iv$, rotated in the complex
+plane. It follows that each stationary point has an equal number of descending
+and ascending directions, e.g. the index of each stationary point is $N$. For a
+stationary point in a real problem this might seem strange, because there are
+clear differences between minima, maxima, and saddles of different index.
+However, we can quickly see here that for a such a stationary point, its $N$
+real eigenvectors which determine its index in the real problem are accompanied
+by $N$ purely imaginary eigenvectors, pointing into the complex plane and each
+with the negative eigenvalue of its partner.
+
+These eigenvalues and vectors can be further related to properties of the
+complex symmetric matrix $\beta\partial\partial\mathcal S$. Suppose that
+$u\in\mathbb R^N$ satisfies the eigenvalue equation
+\begin{equation}
+  (\beta\partial\partial S)^\dagger(\beta\partial\partial S)u
+  =\sigma u
+\end{equation}
+for some positive real $\sigma$ (real because $(\beta\partial\partial
+S)^\dagger(\beta\partial\partial S)$ is self-adjoint). The square root of these
+numbers, $\sqrt{\sigma}$, are the definition of the \emph{singular values} of
+$\beta\partial\partial\mathcal S$. A direct relationship between these singular
+values and the eigenvalues of the hessian immediately follows by taking an
+eigenvector $v\in\mathbb C$ that satisfies \eref{eq:generalized.eigenproblem},
+and writing
+\begin{eqnarray}
+  \sigma v^\dagger u
+  &=v^\dagger(\beta\partial\partial S)^\dagger(\beta\partial\partial S)u
+  =(\beta\partial\partial Sv)^\dagger(\beta\partial\partial S)u\\
+  &=(\lambda v^*)^\dagger(\beta\partial\partial S)u
+  =\lambda v^T(\beta\partial\partial S)u
+  =\lambda^2 v^\dagger u
+\end{eqnarray}
+Thus if $v^\dagger u\neq0$, $\lambda^2=\sigma$. It follows that the eigenvalues
+of the real hessian are the singular values of the complex matrix
+$\beta\partial\partial\mathcal S$, and their eigenvectors coincide up to a
+constant complex factor.
+
 \subsection{Gradient flow and the structure of thimbles}
 
 The `dynamics' describing thimbles is defined by gradient descent on the real