Let A be a symmetric positive definite n à n matrix. (a) If k where yk Rk and βk is a scalar, show that Lk+1

Let A be a symmetric positive definite n × n matrix.
(a) If k

where yk ˆˆ Rk and βk is a scalar, show that Lk+1 is of the form

and determine xk and αk in terms of Lk, yk, and βk.
(b) The leading principal submatrix A1 has Cholesky decomposition L1hT1, where L1 = (ˆša11). Explain how part (a) can be used to compute successively the Cholesky factorizations of A2,..., An. Devise an algorithm that computes L2, L3,..., Ln in a single loop. Since A = An, the Cholesky decomposition of A will be LnLTn. (This algorithm is efficient in that it uses approximately half the amount of arithmetic that would generally be necessary to compute an LU factorization.)

Ak Yk AL+1= y B