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# Definition of similar matrices

Put content here**Definition:** Two square matrices $A$ and $B$ of size $n \times n$ are called **similar** if there exists an invertible matrix $P$ such that $B = P^{-1} A P$.
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**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.
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**Key properties of similar matrices:**
- Same eigenvalues
- Same characteristic polynomial
- Same minimal polynomial
- Same determinant, trace, and rank
- Same Jordan canonical form
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**Example:** $A = \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 3 & 0 \\ 2 & 1 \end{pmatrix}$ are similar via $P = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$.

# Parents

* Similarity of matrices