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of R. Although some familiar properties of the usual order of R are also valid when n ≥ 2, several properties are no longer valid.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 A = [ 1 0 0 0 ]   {\displaystyle A={\begin{bmatrix}1&0\\0&0\end{bmatrix}}\ } and B = [ 0 0 0 1 ]   {\displaystyle B={\begin{bmatrix}0&0\\0&1\end{bmatrix}}\ } then neither A ≥ B or B ≥ A holds true.

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.

See also

  • Trace inequalities

References

  • Pukelsheim, Friedrich (2006). Optimal design of experiments. Society for Industrial and Applied Mathematics. pp. 11–12. ISBN 9780898716047.
  • Bhatia, Rajendra (1997). Matrix Analysis. New York, NY: Springer. ISBN 9781461206538.
  • Zhan, Xingzhi (2002). Matrix inequalities. Berlin: Springer. pp. 1–15. ISBN 9783540437987.