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 "A" − "B" is positive semi-definite. Similarly, we say that "A > B" if "A" − "B" is positive definite.
Properties.
When "A" and "B" are real scalars (i.e. "n" = 1), the Loewner order reduces to the usual ordering of R. Although some familiar properties of the usual order of R are also valid when "n" ≥ 2, several properties are no longer valid. For instance, the comparability of two matrices may no longer be valid. In fact, if
formula_1 and formula_2 then neither "A" ≥ "B" or "B" ≥ "A" holds true.
Moreover, since "A" and "B" are Hermitian matrices, their eigenvalues are all real numbers.
If "λ"1("B") is the maximum eigenvalue of "B" and "λ""n"("A") the minimum eigenvalue of "A", a sufficient criterion to have "A" ≥ "B" is that "λ""n"("A") ≥ "λ"1("B"). If "A" or "B" is a multiple of the identity matrix, then this criterion is also necessary.
The Loewner order does "not" have the least-upper-bound property, and therefore does not form a lattice.