Exercise Solution 2.10

1. The Cholesky algorithm yields the matrix

   [s1]

   

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

2. 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.

3. The Cholesky algorithm fails in the final step, calling for a square root of –1. The matrix is neither positive definite nor singular positive semidefinite.