Why are eigenvalues of nilpotent matrices equal to zero?

Question

If $A$ is a $ \displaystyle 10 \times 10 $ matrix such that $A^{3} = 0$ but $A^{2} \neq 0$ (so A is nilpotent) then I know that $A$ is not invertible, but why does at least one eigenvalue of $A$ have to be equal to zero? How would one show that all eigenvalues of $A$ are equal to zero?

Answer 1

If $v$ is a non-zero eigen vector corresponding to an eigenvalues $\lambda$ we have, by definition, $Av=\lambda v$. Then $A^2v= A( Av)= A(\lambda v)= \lambda^2v$. It easily follows that $\lambda^n$ is an eigenvalue for $A^n$ but the latter is the zero matrix, for which all eigenvalues are zero, hence $\lambda=0$.

Answer 2

Suppose $\lambda$ is an eigenvalue of the nilpotent matrix $A,$ and $u$ its associated eigenvector. Then $$Au = \lambda u, u \neq 0$$ multiplying by $A$ on the right shows $$A^2u = \lambda Au = \lambda^2 u$$ and by induction $$A^n u = \lambda^n u$$

If $A$ is nilpotent, then $A^k = 0$ for some $k>0.$ that implies $\lambda^k = 0\to \lambda = 0.$

Answer 3

Suppose $n$ is the smallest integer for which $A^n=0$. Since $A$ is non-zero $\implies$ $n \gt 1$

let the characteristic polynomial for $A$ be

$$ A^m+c_1A^{m-1}+\cdots + c_m = 0 $$ here $m \le n$. multiplying the equation by $A^{n-1}$ gives $$ c_mA^{n-1} =0 $$ hence $c_m=0$. this procedure can be repeated to show that all coefficients except the first are zero, hence the equation is simply: $$ A^m=0 $$ and we must have $m=n$

since the eigenvalues are roots of the equation: $$ x^m = 0 $$ it follows that they must all be zero

Answer 4

Alternatively, do the contrapositive. If $A$ has a non-zero eigenvalue, $\lambda,$ then $A^{k} \neq 0 $ for all $k.$

Proof: there exists $v \neq 0,$ such that $$ Av = \lambda v, $$ so $$ A^{k} v = \lambda^{k} v \neq 0. $$ Done.