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.

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.

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.

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of R. Although some familiar properties of the usual order of R are also valid when n ≥ 2, several properties are no longer valid.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 ${\displaystyle A={\begin{bmatrix}1&0\\0&0\end{bmatrix}}\ }$ and ${\displaystyle B={\begin{bmatrix}0&0\\0&1\end{bmatrix}}\ }$ then neither A ≥ B or B ≥ A holds true. In other words, the Loewner order is a partial order, but not a total order.

Moreover, since A and B are Hermitian matrices, their eigenvalues are all real numbers. If λ₁(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) ≥ λ₁(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. It is bounded: for any finite set ${\displaystyle S}$ of matrices, one can find an "upper-bound" matrix A that is greater than all of S. However, there will be multiple upper bounds. In a lattice, there would exist a unique maximum ${\displaystyle \max(S)}$ such that any upper bound U on ${\displaystyle S}$ obeys ${\displaystyle \max(S)}$ ≤ U. 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).