Homework #7 Solution

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Problem 1:

Show that if

a

wT

is symmetric and positive definite,

A= 11

w

K

K and K wwT / a

are symmetric and positive definite.

11

(Hint: Use the definition of positive definite matrix. Assume x =

scalar and y is an n-1 dimensional vector.)

then a11 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


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