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.
Let A and B be two Hermitian matrices of order n. We say that A ≥ B if A − B is positive semi-definite.
Although it is commonly discussed on matrices (as a finite-dimensional case), the Loewner order is also well-defined on operators (an infinite-dimensional case) in the analogous way.
For instance, the comparability of two matrices may no longer be valid.
Moreover, since A and B are Hermitian matrices, their eigenvalues are all real numbers.
The Loewner order does not have the least-upper-bound property, and therefore does not form a lattice.
of matrices, one can find an "upper-bound" matrix A that is greater than all of S. However, there will be multiple upper bounds.
But in the Loewner order, one can have two upper bounds A and B that are both minimal (there is no element C < A that is also an upper bound) but that are incomparable (A - B is neither positive semidefinite nor negative semidefinite).