Math 1080
Homework #7
Problem 1:
Show that if
=
w K
a w
A
T
11
is symmetric and positive definite, then
a11 0
and both
K and
11
T K − ww / a
are symmetric and positive definite.
=
y
x
(Hint: Use the definition of positive definite matrix. Assume where
is a
scalar and y is an n-1 dimensional vector.)
Problem 2:
Use necessary conditions to test positive definiteness of the symmetric matrix A. If
conditions are satisfied, compute the Cholesky factorization:
a)
− −
− −
− −
=
1 1 3
1 3 1
3 1 1
A b)
−
−
=
3 1 3
0 2 1
1 0 3
A
Problem 3:
Solve the following system of equations by Cholesky factorization (if the coefficient
matrix is positive definite) or by Gaussian elimination (otherwise)
2 2 4 6
6 8 30
2 10 6 2 36
4 2 2 6
1 2 4
2 3
1 2 3 4
1 2 4
− + + =
− + = −
+ − + =
+ − =
x x x
x x
x x x x
x x x
