Eigenvalues and eigenvectors
A matrix turns almost every arrow it acts on. Are there directions it only stretches, and what do they reveal about the matrix?
Where you are 5 inside
What this area is about
For a square matrix A, an eigenvector is a nonzero vector \mathbf{v} with A\mathbf{v} = \lambda\mathbf{v}: along it, A does nothing but multiply by the number \lambda, the eigenvalue. Those directions and numbers describe what A does far better than its entries, and they make repeated application of A easy to predict.
How these fit together
Procedures. The characteristic equation finds the eigenvalues first, as the roots of \det(A - \lambda I) = 0, the numbers that make A - \lambda I fail to be invertible. Finding an eigenvector puts each eigenvalue back and solves (A - \lambda I)\mathbf{x} = \mathbf{0} for its eigenspace.
Results. Diagonalization shows that an n \times n matrix with n linearly independent eigenvectors is PDP^{-1} with D diagonal. Complex eigenvalues covers a real matrix whose characteristic equation has non-real roots: for a 2 \times 2 one, in suitable axes it rotates and scales.
Application. Matrix powers and dynamical systems reads the long run of \mathbf{x}_{k+1} = A\mathbf{x}_k off the eigenvalues, including the steady state of a Markov matrix.
A path through it
Start with the characteristic equation, then finding eigenvectors, then diagonalization, then dynamical systems. Complex eigenvalues branches off after the characteristic equation and also needs the polar form of a complex number; dynamical systems sends its spiral cases there.
This area is the base for linear systems of differential equations and Markov chains.