In an exercise I was given an 3d inner product space and a basis for a subspace and had to orthogonalize it and complete it into an orthogonal basis for the whole space. Then, I was told to find the minimal and characteristic polynomials for the orthogonal projection on the span of the first two basis elements.
I thought to look for the representing matrix w.r.t the orthogonal basis itself, which just gives me $$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1& 0 \\ 0 & 0 & 0 \end{pmatrix},$$ from which I conclude $\rho (t)=t(t-1)^2$ and $m(t)=t(t-1)$, in particular orthogonal projections are always diagonalizable.
Is my reasoning correct?