Suppose that A is a positive definite.
a. Show that we can write A = D − L − Lt, where D is diagonal with dii > 0 for each 1 ≤ i ≤ n and L is lower triangular. Further, show that D − L is nonsingular.
b. Let Tg = (D − L) −1Lt and P = A − Ttg ATg. Show that P is symmetric.
c. Show that Tg can also be written as Tg = I − (D − L) −1A.
d. Let Q = (D − L) −1A. Show that Tg = I − Q and P = Qt [AQ−1 − A + (Qt)−1A]Q.
e. Show that P = QtDQ and P is positive definite.
f. Let λ be an eigenvalue of Tg with eigenvector x ≠ 0. Use part (b) to show that xtPx > 0 implies that |λ| < 1.
g. Show that Tg is convergent and prove that the Gauss-Seidel method converges.