Prove that if $v$ is an eigenvector for the matrix $A$, then $A^2v=c^2v$

Pretty much all I have is:

$Av=cv$ where $v$ is a nonzero vector


2 Answers 2




c is a scalar you can factor it out



But you know that $Av=cv$ So



Apply $A$ both sides of $Av=cv$ (i.e. $A^2v=Acv$), and use that $A$ commutes with multiplication by constants...


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