In some old course notes I'm reading to touch up on statistical forecasting methods, the book often makes reference to "non-negative definite" matrices. I know what a semi-positive definite, positive definite, and indefinite matrix are, but I've never heard this terminology before. Further, online resources don't really seem to mention it, so I'm unsure what exactly it is equivalent to as the "non-negative definite" property isn't directly applied anywhere. Any ideas?
asked Dec 14, 2015 at 23:27 Eric Hansen Eric Hansen 919 1 1 gold badge 6 6 silver badges 17 17 bronze badges$\begingroup$ My guess would be a positive semidefinite matrix, i.e. a matrix whose eigenvalues are non-negative. But we would have to look at the context to be sure. $\endgroup$
Commented Dec 14, 2015 at 23:33$\begingroup$ @Rahul The context is mainly centered around trying to invert a really really big square matrix, and noting that "being nonnegative definite" is a great property. $\endgroup$
Commented Dec 14, 2015 at 23:37$\begingroup$ I have seen this used in a book by Rao and Mitra, where they use a different definition of positive semi-definite. There non-negative definite means $x^*Ax\geq 0$, while positive semi-definite means $x^*Ax\geq 0$ and $x^*Ax = 0$ for some $x\ne 0$. This is unlike the definition typically used for positive semi-definite, in the sense that it enforces that $A$ is singular. Also in some other languages strictly positive definite is used to refer to $x^*Ax>0$, while positive definite refers to $x^*Ax\geq 0$. This mismatch occurs also in literature on positive definite functions and kernels. $\endgroup$