TOEPLITZ DETERMINANTS AND POSITIVE SEMIDEFINITENESS

We explore the role that the determinants of real, symmetric, Toeplitz matrices play in testing for their positive semidefiniteness. We show that the "leading principal minor" test used to test for positive definiteness is not sufficient in general to test for positive semidefiniteness of Toeplitz matrices, except in certain cases. We derive several properties and show in which cases the leading principal minor test is indeed sufficient. We then present a simple method for testing the positive semidefiniteness of all symmetric Toeplitz matrices.