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# Similarity of matrices in an equivalence relation.

Put content here**Theorem:** Similarity of matrices is an **equivalence relation** on the set of $n \times n$ matrices.
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**Proof:**
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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 $A \sim C$.
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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
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The Jordan form serves as a canonical representative for each similarity class (over $\mathbb{C}$).

# Parents

* Similarity of matrices