Let A be an n X n symmetric and idempotent matrix of constan

Let A be an n X n symmetric and idempotent matrix of constants. Let r = rank(A). Find constants C1, C2, and c3 and an n x 1 vector of constants a such that c1 + a\' y+ c2y?Ay has a central X^2 distribution with degrees of freedom c3.

Solution

As A i idempotent we have A²=A

r = rank of A (i.e. no of non zero rows in matrix A

c1,c2 and c3 are constants of A

c1+A\'y+c2y\'Ay has a sum of c3 independent variables.

Hence it follows chi square distribution and df = c3