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of R. Although some familiar properties of the usual order of R are also valid when n ≥ 2, several properties are no longer valid.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 [math]\displaystyle{ A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\ }[/math] and [math]\displaystyle{ B = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}\ }[/math] 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 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.

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.