I'm studying my midterm exam and solving the problem set. Unfortunately, there is no solution manual for this set. I will show what I did and I will ask my specific question. Firstly, the question is the following:
Let $v$ $∈$ $\Bbb{R}^{n}$ be a unit vector (i.e., $⟨v,v⟩$ $=$ $v^T$ $v$ $= 1$). Define the $n × n$ matrix $P$ := $v$ $v^T$
The question is
a) what is the Range Space of $P$,
b) what is the Null Space of $P$,
c) what is the characteristic polynomial of $P$.
I observed that $P$ matrix is a projection matrix. Namely, $P^2$=$P$ and $P^T$=$P$.
Then, I found that the minimal polynomial is $m(s) = s(s-1)$.
Therefore the eigenvalues are $0$ and $1$. I do not know the numbers of the $0$ and $1$ eigenvalues.
What should I do for the next step so that I can find the Range Space, Null Space and characteristic polynomial.
Thanks.
Since $P^T=P$ and $P^2=P$, then you know that $P$ is an orthogonal projection, not merely a a projection. So the range and the nullspace will be orthogonal to each other.
Projection matrices always have minimal polynomial dividing $s(s-1)$; they have minimal polynomial equal to $s-1$ if and only if they are the identity, and minimal polynomial equal to $s$ if and only if it is the zero matrix.
This matrix is clearly not the zero matrix, since $Pv = vv^Tv = v\neq\mathbf{0}$.
Construct an orthonormal basis that has $v$ as one of its vectors, $\beta=[v=v_1,\ldots,v_n]$. Then we have $v_i^Tv_i = \langle v_i,v_i\rangle = 1$ if $i=j$, and $v_i^Tv_j = \langle v_i,v_j\rangle = 0$ if $i\neq j$. Therefore, $$Pv_j = (vv^T)v_j = v_1(v_1^Tv_j) = \langle v_1,v_j\rangle v_1 = \delta_{1j}v,$$ where $\delta_{ij}$ is Kronecker's Delta.
Thus, the range is $\mathrm{span}(v)$, the nullspace is $(\mathrm{span}(v))^{\perp}$ the orthogonal complement of $v$. The characteristic polynoial is therefore $s^{n-1}(s-1)$.