Fallacies in computational testing of matrix positive definiteness/semidefiniteness

The status of computational tests for establishing matrix positive semidefiniteness and positive definiteness is reviewed. Two pervasive real-time tests that have been used for many years in varied applications to ensure that computed covariances encountered in Kalman filter applications are positive definite and discussed. Structural representations of covariance matrices are reviewed as a prelude to constructing a counterexample and demonstrating that it refutes these real-time tests. It is maintained that the latter are bogus approaches despite the fact that they are pervasive. It is suggested that such bogus tests arose as an attempt to fill the need for a quick check (over the entire mission time) of the massive number of matrices computationally encountered in real-time applications