Similarity of matrices in an equivalence relation.

Theorem: Similarity of matrices is an equivalence relation on the set of $n \times n$ matrices.

Proof:

1. Reflexive: $A = I^{-1}AI$, so $A \sim A$.
2. Symmetric: If $B = P^{-1}AP$, then $A = PBP^{-1} = (P^{-1})^{-1}B(P^{-1})$, so $A \sim B$.
3. Transitive: If $B = P^{-1}AP$ and $C = Q^{-1}BQ$, then $C = Q^{-1}P^{-1}APQ = (PQ)^{-1}A(PQ)$, so $A \sim C$.

Since similarity is an equivalence relation, it partitions the set of $n \times n$ matrices into similarity classes. All matrices in the same class share the same:

• Eigenvalues
• Determinant and trace
• Characteristic and minimal polynomials
• Jordan canonical form

The Jordan form serves as a canonical representative for each similarity class (over $\mathbb{C}$).