This problem considers whether a Cholesky type factorization can be used on a matrix which is not positive definite. The assumption is that given an invertible symmetric matrix
A, then
A
=
C
T
C
where
C
is upper triangular with possibly complex-valued entries. In this problem this will be referred to as a generalized Cholesky factorization. What will be shown is that a generalized Cholesky factorization is possible as long as the leading principal minors of
A
are nonzero. Note that a method that avoids the use of complex-valued factors is considered in Exercise 3.14.
(a) The following matrix is invertible and symmetric but not positive definite. Find a matrix
C
satisfying the stated assumption.
(b) Using the factorization found in part (a), solve
Do you obtain the same answer you would get if you did not use the factorization?
(c) What conditions must be imposed on the entries of the following symmetric matrix so it is invertible and has a generalized Cholesky factorization.
(d) Answer the question posed in part (c) for a symmetric 3 × 3 matrix.
Exercise 3.14
Assuming pivoting is not necessary, then a symmetric matrix
A
can be factored as
A
=
LDL
T
, where
L
is a lower triangular matrix with ones on its diagonal and
D
is a diagonal matrix.
(a) Find the
LDL
T
factorization of the matrix
(b) In Section 3.1, it was shown how an LU factorization results in solving two matrix equations (for
y
and
x). Explain how an
LDL
T
results in solving three matrix equations. Use this to solve
(c) Explain why the flop count for solving
Ax
=
b, where
A
is a symmetric
×
matrix, using an
LDL
T
factorization is approximately half of the flop count when using an
LU factorization.