Loewner order


In mathematics, Loewner order is the partial order defined by the convex cone of positive semi-definite matrices. This order is usually employed to generalize the definitions of monotone and concave/convex scalar functions to monotone and concave/convex Hermitian valued functions. These functions arise naturally in matrix and operator theory and have applications in many areas of physics and engineering.

Definition

Let A and B be two Hermitian matrices of order n. We say that A ≥ B if AB is positive semi-definite. Similarly, we say that A > B if AB is positive definite.

Properties

When A and B are real scalars, the Loewner order reduces to the usual ordering of R. Although some familiar properties of the usual order of R are also valid when n2, several properties are no longer valid. For instance, the comparability of two matrices may no longer be valid. In fact, if
and then neither AB or BA holds true.
Moreover, since A and B are Hermitian matrices, their eigenvalues are all real numbers.
If λ1 is the maximum eigenvalue of B and λn the minimum eigenvalue of A, a sufficient criterion to have AB is that λnλ1. If A or B is a multiple of the identity matrix, then this criterion is also necessary.