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.
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.
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of R. Although some familiar properties of the usual order of R are also valid when n ≥ 2, several properties are no longer valid.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
A =
\begin{bmatrix}
1 & 0 \\
0 & 0
\end{bmatrix}\
and
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 λn(A) the minimum eigenvalue of A, a sufficient criterion to have A ≥ B is that λn(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 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 such that any upper bound U on obeys ≤ 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).
See also
References
- {{cite book|last1=Pukelsheim|first1=Friedrich|title=Optimal design of experiments|date=2006|publisher=Society for Industrial and Applied Mathematics|isbn=9780898716047|pages=11–12}}
- {{cite book|last1=Bhatia|first1=Rajendra|title=Matrix Analysis|date=1997|publisher=Springer|location=New York, NY|isbn=9781461206538}}
- {{cite book|last1=Zhan|first1=Xingzhi|title=Matrix inequalities|date=2002|publisher=Springer|location=Berlin|isbn=9783540437987|pages=1–15}}