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# Similarity of matrices

Put content here**Similarity** is a relation between square matrices that captures when two matrices represent the same linear operator under different choices of basis.
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**Definition:** Two $n \times n$ matrices $A$ and $B$ are **similar** if there exists an invertible matrix $P$ such that:
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$$B = P^{-1}AP$$
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**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 $Q$.
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**Similarity invariants** (properties preserved under similarity):
- Determinant
- Trace
- Rank
- Characteristic polynomial
- Eigenvalues
- Minimal polynomial
- Jordan canonical form
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Similar matrices represent the same linear operator $T: V \to V$ with respect to different bases.

# Parents

* Matrices