Let \(\mathbf{1}_{n}\) be the vector in \(\mathbb{R}^{n}\) whose components are all one. Show that \(J_{n}=\) \(\mathbf{1}_{n} \mathbf{1}_{n}^{\prime} / n\) is a projection matrix, i.e., that \(J_{n}^{2}=J_{n}\), and that it has rank one: \(\operatorname{tr}\left(J_{n}ight)=1\). Show also that \(I_{n}-J_{n}\) is the complementary projection of rank \(n-1\).

Each of the quadratic forms in Exercise 1.15 can be expressed in the form \(Y^{\prime} M_{r} Y\), where each \(M_{r}\) is a projection matrix of order \(m n \times m n\). Show that each matrix is a Kronecker product

\[
J_{n} \otimes J_{m} ; \quad\left(I_{n}-J_{n}ight) \otimes J_{m} ; \quad J_{n} \otimes\left(I_{m}-J_{m}ight) ; \quad \text { and } \quad\left(I_{n}-J_{n}ight) \otimes\left(I_{m}-J_{m}ight) .
\]

Find the rank of each matrix.