Similar Matrices

Last updated Mar 20, 2022 Edit Source

• Math linear

Two matrices $A$ and $B$ are said to be similar if there exists an invertable matrix $P$ such that:

$$ A = PBP^{-1} $$

When you have such similar matrices and you know that $v$ is an eigenvector of $B$, $Pv$ must be an eigenvector of $A$ with the same eigenvalue.

$$ A(Pv) = PBP^{-1}Pv = PBv = P\lambda v = \lambda Pv $$

Building from the definition of the eigenvalues, it is clear that $Pv$ is an eigenvector of $A$ with eigenvalue $\lambda$.