I have to make an argument that the characteristic polynomial of an idempotent matrix A is precisely (maybe a constant
multiple of) $\lambda^{n-r}(1-\lambda)^{r}$, where r is the rank of A. I know that for an idempotent matrix A, the eigenvalue is either 0 or 1 and that the 1-eigenspace of A is the image of A. Furthermore, I was given the hint to use the geometric multiplicity of the eigenvalue 1 and to entertain the questions: What is the geometric multiplicity of 0 as an eigenvalue? What is the sum of these two multiplicities? Is it possible that either eigenspace is deficient?
Update: Following Steven's suggestions, I tackled the problem as follows:
-
I found that geometric multiplicities as follows:
$$
\begin{align*}
gmu(0)&=dim(ker(A-(0)I)) \\
&=dim(ker(A)) \\
&=n-rank(A-I) \\
&=n-r
\end{align*}
$$
and
$$
\begin{align*}
gmu(1)&=dim(ker(A-(1)I)) \\
&=dim(E_{1}) \\
&=dim(im(A)) \\
&=n
\end{align*}
$$ -
I then use these multiplicities to argue that the minimal polynomial must be of the form:
$$
f_{A}(\lambda)=(\lambda)^{n-r}(1-\lambda)^{r}
$$
where the roots are either 0 or 1, as suggested by the answer below. Then, matrix A is diagonalizable and similar to a diagonal matrix D with r $1$'s and (n-r) $0$'s as its diagonal entries.
Best Answer
Question: "How can one approach this argument? Any help would be appreciated."
Answer: Let $A$ be an $n \times n$ matrix with $A^2=A$. Let $v \neq 0$ be an eigenvector with eigenvalue $\lambda$. It follows $Av=\lambda v$ and
$$\lambda v=Av=A^2v=A(Av)=A(\lambda v)=\lambda^2v$$
hence $\lambda^2=\lambda$ and hence $\lambda =0,1$. Hence the eigenvalues of $A$ are $0,1$. It follows the characteristic polynomial (which has degree $n$) must look as follows:
$$\chi_A(t)=t^{n-r}(t-1)^r$$
for some integer $r$, since the eigenvalues of $A$ equal the roots ot $\chi_A(t)$.