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A Relationship Between Eigenvectors of Commuting Matrices

If square matrices  
\[A, \: B\]
  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  
\[B\]
.

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