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 $P = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$.
**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 $P = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$.