Exercise Solution 7.3

Matrix [7.22] has eigenvalues 2.6001, 0.9988, 0.4214, and –0.0004. Having both positive and negative eigenvalues, the matrix is indefinite. Setting ε = 0.0005, we apply [7.21]. The resulting matrix is

[s1]

This has eigenvalues 2.6006, 0.9993, 0.4219, and 0.0001. All are positive, so the matrix is positive definite.