Let 1 n 1 n be the vector in R n R n whose components are all one Show that J n J n 1 n 1 n n 1 n 1 n n is a projection matrix, i e , that J 2 n J n J n 2 J n , and that it has rank one tr(J n ) 1 Extra left or missing ight Show also that I n J n I n J n is the complementary projection of rank n 1 ...

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Let 1 n 1 n be the vector in R n R n whose components are all

Question:

Let $1_{n}$ be the vector in $R^{n}$ whose components are all one. Show that $J_{n} =$ $1_{n} 1_{n}^{'} / n$ is a projection matrix, i.e., that $J_{n}^{2} = J_{n}$ , and that it has rank one: tr(J_n) = 1 $Extra \left or missing \right$ . 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 $Y^{'} M_{r} Y$ , where each $M_{r}$ is a projection matrix of order $m n \times m n$ . Show that each matrix is a Kronecker product