Definition of similar matrices

Definition: Two square matrices $A$ and $B$ of size $n \times n$ are called similar if there exists an invertible matrix $P$ such that $B = P^{-1} A P$.

Explanation: Similarity is a stronger condition than equivalence. While equivalence allows different change-of-basis matrices in domain and codomain, similarity requires the same change of basis in both. This corresponds to representing the same linear operator $T: V \to V$ (same domain and codomain) in different bases.

Key properties of similar matrices:

• Same eigenvalues
• Same characteristic polynomial
• Same minimal polynomial
• Same determinant, trace, and rank
• Same Jordan canonical form

Example: $A = \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 3 & 0 \\ 2 & 1 \end{pmatrix}$ are similar via $P = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$.