Question

6) In econometrics we frequently encounter matrices that are both symmetric and idempotent. Such a matrix A has the properties A",4 and A#AA. Use these properties to show that OS a“ 1, where a" is the ith diagonal element of A. [7 points]

Answer 1

Given A2-A-AT Consider a 2x 2 matrix a121 - A a22 412 a a12 A12 a221 1a12 a2 a12. a11a12 + a12a2 a22 a11a12 + a12a2 Thus aîi + aî2 -ai ...(1) C11" 12722 a12 ...(2) af2 + a22-022 (3) If a1,-0 then (1) gives. a, = a11 and thus al 1-0 or 1, similarly, a 2 = a22 and thus a22-0 or 1, 0. from equation (2), we get If a12

a11 + a22-1 Equation (1) gives a11(a11 -1)--a2 .. (4) As the right side is negative, so the left side must be negative, so (i) if a11 s 0 then (4) is not satisfied (ii) If a 1,2 1, then left side is either positive or 0. But as per our consideration a 12 0, so (4) is not satisfied (iii) If 0 < a11< 1, then (4) is satisfied. As a11 + a22-1 and 0 a11 < 1 thus 0 < a22 < 1. Thus it is proved that 0 3 at S 1 For a3 x 3 matrix 11 a12 a 13 13 a 23 a SAVING AUTORECOVERY FILE LINEAR AL:

a11 a12 a131 ra1 a, A2 a12 a22 a2 3 1 012 a22 a23 13 13 a 23 a 33J La 13 a 23 a a21 a22a3 a11a12 + a12a22 + a13a23 2a22 a13023 a11a13 a12a23 1112 13 C33 a12a13 22 23 23"423 a12a13 + a22a23 + a23a3 Q13a33 23 т а 11 a12 a 13 13 а 23 a So comparing these data we get 13) da + a 2 + a2,-a22 (6) a23 + a33-a33 13 Thus a1 -a11a11(a11 1)(ai a) JA

As above we have shown that the right side is s 0 thus if 0 1 then only the equality will be satisfied. a11 Similarly, it can be shown that0 1. a22, a33 This can be generalised to any matrix of size n.