Description
Problem 1:

Show that if
_{a}
_{w}^{T}
is symmetric and positive definite,
A= ^{11}
w
K
K and K − ww^{T} / a
are symmetric and positive definite.
11
(Hint: Use the definition of positive definite matrix. Assume x =
scalar and y is an n1 dimensional vector.)
then a_{11} 0 and both
where is a y
Problem 2:
Use necessary conditions to test positive definiteness of the symmetric matrix A. If conditions are satisfied, compute the Cholesky factorization:

3 −1
−1
1
0
− 3
a)
A= −1
3 −1
b)
A =
0
2
1
−1
−1
3
− 3 1
3
Problem 3:
Solve the following system of equations by Cholesky factorization (if the coefficient matrix is positive definite) or by Gaussian elimination (otherwise)

4 x _{1}
+ 2 x _{2}
− 2 x _{4}
= 6
2 x _{1} +10 x _{2}
− 6 x _{3} + 2 x _{4}
= 36
− 6 x _{2} + 8 x _{3}
= −30
− 2 x _{1}
+ 2 x _{2}
+ 4 x _{4}
= 6