**Similarity** is a relation between square matrices that captures when two matrices represent the same linear operator under different choices of basis.

**Definition:** Two $n \times n$ matrices $A$ and $B$ are **similar** if there exists an invertible matrix $P$ such that:

$$B = P^{-1}AP$$

**Key difference from equivalence:** Similarity uses the **same** change-of-basis matrix $P$ on both sides (with $P^{-1}$ on the left), whereas equivalence uses independent matrices $P$ and $Q$.

**Similarity invariants** (properties preserved under similarity):
- Determinant
- Trace
- Rank
- Characteristic polynomial
- Eigenvalues
- Minimal polynomial
- Jordan canonical form

Similar matrices represent the same linear operator $T: V \to V$ with respect to different bases.