Proof: If $P^2=P$, then $P$ is diagonalizable

Question

$\textbf{Definition:}$ Let $P$ be a matrix. We say is diagonalizable matrix iff

$\exists$ an invertible matrix $A$: $\exists$ a diagonal matrix $D$: $A^{-1}PA=D$.

I asked a question earlier as to how to prove the following proposition. I am on a mission to get to the end of the full proof as I have listed everything below to help see where I am going. I put the part where I am stuck currently.

$\textbf{Proposition:}$ Let $P$ be an $n\times n$ matrix.

If $P^2=P$, then $P$ is diagonalizable. Link to help for future reference.

$\textbf{Proof:}$ Let $P$ be an $n\times n$ matrix. Assume $P^2=P$. [First, show $Im(I-P)=ker(P)$.]

$\subseteq$ Let $y\in Im(I-P)$ [Show $y\in ker(P)$. It suffices to show $P(y)=0$.] Then, $y=(I-P)(z)$ for some $z\in \mathbb{R}^n$. Thus, the following holds true:

$$\begin{equation}\begin{aligned} P(y)&=P((I-P)(z)) \text{ as } y=(I-P)(z)\\ &=P(z-P(z)) \\ &=P(z)-\underbrace{P^2}_{=P}(z)) \\ &=P(z)-P(z) \\ &=0. \end{aligned}\end{equation}$$

$\supseteq$ Let $y\in ker(P)$. [Show $y\in Im(I-P)$.] Then,

$$\begin{equation}\begin{aligned} (I-P)(y)&=I(y)-P(y)\\ &=I(y)-0\\ &=y. \end{aligned}\end{equation}$$ As $y=(I-P)(y)$, $y\in Im(I-P)$.

Thus, it has been shown that $ker(P)=Im(I-P)$.

$\textbf{Question/Part Stuck On:}$ In the case $P\neq I$, since $Im(I-P)=ker(P)$, there must be an eigenspace that corresponds to an eigenvalue zero. Why must there be an eigenspace that corresponds to an eigenvalue of zero and also an eigenspace that corresponds to an eigenvalue of one here? It does not make sense to me as to why $X^2-X=X(X-1)=0$ gives us our eigenvalues of $1$ and $0$. I have also seen others go from knowing $v=(v-P(v))+P(v)$ to all of the sudden knowing the direct sum of these Eigenvalues yields the whole space which is also quite a jump for me. I am lost as to where to go next in this proof, and any help would be greatly appreciated.

Answer 1

You are wrong when you assert that, if $P\neq\operatorname{Id}$, then $P$ has an eigenspace that corresponds to the eigenvalue $1$. Supose that $P$ is the null matrix.

Since $P^2=P$, then $P^2-P=0$. So, if $Q(x)=x^2-x$, $Q(P)=0$. So, the minimal polynomial $m_P(x)$ of $P$, which must divide $Q(x)$ is one of these polynomials: $Q(x)$, $x$, or $x-1$. In each case, it has no roots other than $0$ and $1$. Since the eigenvalues of $P$ are the roots of $m_P$, $P$ cannot have other eigenvalues.

Answer 2

Note that the eigenvalues of $P$ are $0$ or $1$

View $P$ as a linear operator on $\Bbb{R}^n$. Use $\text{ker}\; P=\text{Im}\;(I-P)$ to prove $\Bbb{R}^n=\text{Im} P \oplus \text{ker} P$ and then prove the matrix of $P$ under some basis has the form $$\text{diag}\;(1,1,...,1,0,0,...,0)$$ where the number of $1's$ is the rank of $P$.

The conclusion follows from the above fact!

Answer 3

We have $I = P + (I-P)$ and so any $x$ can be written as $x=x_1+x_2$ where $x_1 \in {\cal R}P$ and $x_2 \in {\cal R}(I-P)$. Furthermore, if $Py_1 = (I-P)y_2$, then multiplying by $P$ shows that $Py_1 = 0$ and $(I-P) y_2 = 0$ from which we get ${\cal R}P \cap {\cal R}(I-P) = \{0\}$. Hence $\mathbb{R}^n = {\cal R}P \oplus {\cal R}(I-P)$.

Note that if $x \in {\cal R}P $ we have $Px = x$ and if $x \in {\cal R}(I-P)$ we have $Px = 0$. In particular, if $b_1,...,b_{k_1}$ form a basis for ${\cal R}P$ and $c_1,...,c_{k_2}$ form a basis for ${\cal R}(I-P)$ then $b_1,...,b_{k_1},c_1,...,c_{k_2}$ forms a basis for $\mathbb{R}^n$ and $P$ is diagonal in this basis.