Evaluating possible eigenvalues of a given equation

Question

Let $A$ be a $3\times3$ matrix such that $A^{2} = 4A - 4I$ Evalute the possible eigenvalues of $A$.

I have tried to multiply the equation by eigenvector $x$ and use the property of $Ax = \lambda x$ to solve it. I found that the form of my solution is close to what I am trying to find but I can't solve it. Can anyone show me the solution?

Answer 1

Since $A$ satisfies the equation $$ A^2-4A+4I=\mathbf 0\tag{1} $$ the minimal polynomial of $A$ divides $$ p(x)=x^2-4x+4=(x-2)^2 $$ This means the the minimal polynomial of $A$ is either $(x-2)$ or $(x-2)^2$. Do you know how the minimal polynomial relates to the eigenvalues?

A more direct approach is to let $v$ be an eigenvector of $A$ so $Av=\lambda v$. Then multiplying (1) on the right by $v$ gives $$ \lambda^2 v-4\lambda v+4v=\mathbf 0 $$ which implies $$ \lambda^2-4\lambda+4=0\tag{2} $$ (verify this!). But (2) may be factored as $$ (\lambda-2)^2=0 $$ so what can we conclude about $\lambda$?

Answer 2

Hint: Use the Cayley-Hamilton Theorem (which says that every square matrix satisfies its own characteristic polynomial), so, in this case, you've got: $$\lambda^2=4\lambda-4.$$

Now, solve for $\lambda.$