Canonical Forms and Similarity Classes of Square Matrices over a Field

Abstract

The rows of xI−A constitute an F[x]-basis of kerθ _A , the invariant factor decomposition of M(A), the invariance theorem, the uniqueness of the rational canonical form. The minimum polynomial μ _A (x), the Cayley–Hamilton theorem.

The primary decomposition of M(A) given the irreducible factorisation of χ _A (x), the primary canonical form. The Jordan normal form. Separable polynomials and formal derivatives. The separable Jordan form, the real Jordan form.

Endomorphisms and automorphisms of M(A). Irreducible μ _A (x), the centraliser Z(A). M(A) cyclic and the Euler Φ _q -function. The orbit-stabiliser theorem. Frobenius’ theorem. Decomposition of End M(A) and Aut M(A), Shoda’s method, size of similarity classes over a finite field.