Matrix Similarity

Definition

$\exists P\text{ s.t. }\mathbf{A}=\mathbf{PB}{\mathbf{P}}^{-1}\iff A\sim B$

Two matrices $\mathbf{A}$ and $\mathbf{B}$ are similar if there exists $\mathbf{P}$

Similar matrices represent the same linear map under two different bases, with $\mathbf{P}$ being the change of basis matrix.

Facts

Similarity invariant

Similar matrices share all properties of their shared underlying operator

• Rank of Matrix
• Determinant
• Invertibility
• Nullity
• Characteristic Polynomial
• Eigenvalue
• Algebraic Multiplicity
• Geometric Multiplicity
• Trace