What is the rank of $\alpha I + \beta P$ with $P^2=P$ and $\alpha,\beta>0$?

Question

I have been working with projection matrices lately and I have one additional question:

Is it true a statement that
linear combination of identity matrix $I$ of rank $n$ and any projection matrix $P$ of rank less than $n$ where coefficients are positive i.e. matrix $A = {\alpha}I+{\beta}P$, with $ {\alpha}>0, {\beta}>0 ,$
has always rank $n$ ?

If so how to prove it?

If so the same is true of course for powers $A^m$.

Answer 1

Saying that $\alpha I+\beta P$ has rank $n$ is the same as saying that $P-\lambda I$ has rank $n$, where $\lambda=-\alpha/\beta$, that is, that $\lambda$ is not an eigenvalue of $P$. What are the eigenvalues of a projection matrix? Hint: $P^2=P$.

Answer 2

Pick a basis of the image of the projection and of its kernel. Then they combine to form a basis of the whole space and in this basis the projection is

$$\begin{pmatrix}1\\ &\ddots\\ &&1\\ &&&0\\ &&&&\ddots\\ &&&&&0 \end{pmatrix}$$

so $\alpha I+\beta P$ is

$$\begin{pmatrix}\alpha+\beta\\ &\ddots\\ &&\alpha+\beta\\ &&&\alpha\\ &&&&\ddots\\ &&&&&\alpha \end{pmatrix}$$