i MakesThe classical eigensolvers of Part 3 stretch. Insert a factor of \(i\) into the exponent and stretching becomes rotation: the eigenvalues move onto the unit circle, lengths are preserved, and a quantum computer can run the evolution natively. That single letter is the entire difference.
Every eigensolver is a way of applying a function of the operator and letting the spectrum sort itself out. This post surveys the classical half: the methods that stretch.
Part 1 built the spectral theorem. Before we survey the tools that compute eigenvalues, here is why you would want to: three industries, three matrices, three ways the same decomposition does real work.
e to a Matrix (and Why You'd Want To)A first-principles account of why \(\lvert\psi(t)\rangle = e^{-iHt}\lvert\psi_0\rangle\) solves the Schrödinger equation, and what that formula means geometrically. The whole story is carried by one running example, the \(2\times2\) Hermitian matrix \(H=\begin{pmatrix}2&1\\1&2\end{pmatrix}\).