Notes and counterexamples on positive (semi) definite properties of some matrix products

Abstract

In the present paper, we give some notes and counterexamples to show that the positive (semi) definite property of the Khatri-Rao and Tracy-Singh products of partitioned matrices are in general incorrect and show also that the matrix triangle inequality for the usual and Kronecker sums need not hold. Moreover, we show that the Fiedler’s inequality as revisited by Johnson’s works [10] cannot be strict. Keywords: Kronecker product (sum), Hadamard product, Tracy-singh product, Khatri-rao product, Positive (semi) definite matrix, 2010 MSC: 15A45, 15A69, 15B48