such that Mox where i is a column of ones' s vector, x=[ 1, 2,,n]and i=i2iai Proof As definition Mox=(I--iix=x--iix Using the idempotent matrix Mo to calculate 2=G-j)2, where j=200 2=IG) from gauss 2. 7 Trace of matrix The trace of a square k x k matrix is the sums of its diagonal elements tr(A Some useful results are tr(ca)=ctr(a)) (A)=tr(A'), tr(A+B)=tr(B)+tr(A), tr(ABCD)= tr(BCDa)=tr(CDaB)=tr(DAbC)such that M0x =         x1 − x¯ x2 − x¯ . . . xn − x¯         , where i is a column of ones’s vector, x = [x1, x2, ..., xn] ′ and ¯x = 1 n Pn i=1 xi . Proof. As definition, M0x = (I − 1 n ii′ )x = x − 1 n ii′x = x − ix. ¯ Exercise: Using the idempotent matrix M0 to calculate P200 j=1(j−¯j) 2 , where ¯j = 1 200 P200 j=1(j) from Gauss. 2.7 Trace of Matrix The trace of a square k × k matrix is the sums of its diagonal elements: tr(A)=Pk i=1 aii. Some useful results are: 1. tr(cA) = c(tr(A)), 2. tr(A) = tr(A′ ), 3. tr(A + B) = tr(B) + tr(A), 4. tr(ABCD) = tr(BCDA) = tr(CDAB) = tr(DABC). 5