Matrix/Equivalent/Eigenvalue/Fact

${\displaystyle {}n}$

${\displaystyle {}{\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}}$

eigenvector of ${\displaystyle {}M}$ for the eigenvalue ${\displaystyle {}a}$ if and only if

${\displaystyle {}{\begin{pmatrix}x'_{1}\\\vdots \\x'_{n}\end{pmatrix}}=B{\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}\,}$

is an eigenvector of the matrix ${\displaystyle {}BMB^{-1}}$ for the eigenvalue ${\displaystyle {}a}$. In particular, ${\displaystyle {}M}$ and ${\displaystyle {}BMB^{-1}}$