a. The Cholesky algorithm yields the matrix
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
where x is indeterminate. We set x equal to 0 and proceed. We obtain the matrix
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. [Return]
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