Value-at-Risk

Chapter 2, Page 77
Exercise 10

Identify all factorizations of the following matrices that are obtainable with our Cholesky algorithm. Take only positive square roots when selecting nonzero diagonal elements g_i,i. In each case, is the original matrix positive definite, singular positive semidefinite, or neither of these?

[2.86]

[2.87]

[2.88]

Solution

a. The Cholesky algorithm yields the matrix

[s1]

Because the algorithm completes successfully with no 0 diagonal elements, the original matrix is positive definite.

b. At the fifth step of the Cholesky algorithm, we obtain

[s2]

where x is indeterminate. We set x equal to 0 and proceed. We obtain the matrix

[s3]

Because this has a 0 diagonal element, we conclude that the original matrix is singular positive semidefinite.

c. The Cholesky algorithm fails. The matrix is neither positive definite nor singular positive semidefinite.