Eigenvalues

This chapter looks for a basis in which a matrix representation becomes diagonal, so that the map simply scales each basis direction. Such a basis turns matrix powers, coupled differential equations and Markov chains into independent scalar modes. The chapter assumes linear maps, kernels and matrices and reads growth and decay using stability of systems.

Lessons

1. Eigenvalues and eigenvectors: directions that a matrix scales, found as roots of the characteristic polynomial.
2. Diagonalisation: when the eigenvectors form an eigenbasis, giving $A=PDP^{-1}$.
3. Spectral applications: multiplicities, matrix powers and the long-term behaviour of each mode.
4. Linear ODE systems: solving $\mathbf{x}'=A\mathbf{x}$ with exponential modes in place of powers.
5. Markov systems: transition matrices, stationary distributions and long-run limits.
6. Seen questions: exam problems on stability, diagonalisation, ODE systems and Markov models.

Where it leads

This is the last Algebra chapter. Its exponential modes match the characteristic roots of second-order constant-coefficient equations, and the eigenvalues of a Hessian give the curvature at stationary points in Calculus.

Vocabulary checkpoints

• An eigenvalue, eigenvector and eigenspace are a scalar, a non-zero vector and a subspace respectively.
• Algebraic and geometric multiplicity count different things and comparing them shows whether diagonalisation is possible.
• In Markov systems, a transition matrix, a probability vector and a stationary distribution are three different objects.