Similarity of matrices

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.