\[A, \: B\]
commute, \[AB=BA\]
.If
\[\mathbf{v}\]
is an eigenvector of \[A\]
then \[A \mathbf{v} = \lambda_1 \mathbf{v}\]
for some scalar \[\lambda\]
.Then
\[BA \mathbf{v}= B \lambda_1 \mathbf{v}\]
.But
\[AB=BA\]
hence \[AB \mathbf{v}= B \lambda_1 \mathbf{v}\]
.Hence
\[B \mathbf{v}\]
is an igenvector of \[A\]
Similarly if
\[\mathbf{w}\]
is an eigenvector of \[B\]
, \[A \mathbf{w}\]
is an eigenvector of \[B\]
.