Find the canonical form – choosing among the Jordan, rational, Smith, and spectral forms, and computing them from a matrix, from its minimal and characteristic polynomials, or from a polynomial identity it satisfies.
Determinants and eigenvalues – determinant formulas, trace and determinant as functions of the eigenvalues, the classical matrix groups.
Minimal and characteristic polynomials – Cayley–Hamilton and the relation between the two polynomials.
Jordan canonical form – elementary divisors and generalized eigenspaces.
Rational canonical form – invariant factors and companion matrices, over any field.
Smith normal form – matrices over a PID up to equivalence.
The spectral theorem – unitary and orthogonal diagonalization, and simultaneous diagonalization.