Handbook of Linear Algebra, 2nd Edition

Chapter 6

Canonical Forms for Similarity

Leslie Hogben

Iowa State University and American Institute of Mathematics

Intuitively, a canonical form is a representative of an equivalence class that has a particularly simple form (or a form well suited to a specific purpose). More precisely, let S be a given set (e.g., of matrices), and let ~ be an equivalence relation on S (e.g., similarity). For any a ∈ S, let Sa = {b ∈ S : b ~ a} denote the equivalence class that contains a. Then ∪a∈SSa = S and for each a, b ∈ S, either Sa = Sb (if a ~ b) or Sa ∩ Sb = Ø (if a ≁ b). Let the set C ⊂ S be a distinguished set of elements of S, one from each equivalence class, i.e., ∪a∈CSa = S and Sa ∩ Sb = Ø whenever a, b ∈ C and a ≠ b. For a given a ∈ S, c ∈ C is ...