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
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Sign up to join this communityProve 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
$$Av=cv$$
$$AAv=Acv$$
c is a scalar you can factor it out
$$AAv=c(Av)$$
$$A^2v=c(Av)$$
But you know that $Av=cv$ So
$$A^2v=c(cv)=c^2v$$
Apply $A$ both sides of $Av=cv$ (i.e. $A^2v=Acv$), and use that $A$ commutes with multiplication by constants...