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.
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- Definition of similar matricesDefinition: Two square matrices $A$ and $B$ of size $n \times n$ are called similar if there exists an invertible...
- Similarity of matrices is an equivalence relationTheorem: Similarity of matrices is an equivalence relation on the set of $n \times n$ matrices. Proof: 1. Reflexive:...
- Definition of similarity transformDefinition: A similarity transform (or similarity transformation ) is the operation of conjugating a matrix $A$ by an...
- Similar matrices have the same eigenvalues and the same characteristic polynomials.Theorem: If $A$ and $B$ are similar ($B = P^{-1}AP$), then they have the same eigenvalues and the same characteristic...
- Every square matrix is similar to one in Jordan form.Theorem: Every square matrix over $\mathbb{C}$ is similar to a matrix in Jordan canonical form . That is, for any $A...
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