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of R. Although some familiar properties of the usual order of R are also valid when n ≥ 2, several properties are no longer valid.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 $A=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$ and $B=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$ 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 $S$ 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 $\max (S)$ such that any upper bound U on $S$ obeys $\max (S)$ ≤ 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).

See also

• Trace inequalities

References

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