---
tag: 9G9S92T1VJ
kind: concept
name: Loewner order
msc: [15B57, 47A63]
specializes: [0QV2JS9RJ3]
sources:
  - url: https://en.wikipedia.org/wiki/Loewner_order
same_as:
  wikidata: Q25303887
  wikipedia: Loewner order
---

## Summary

The Loewner order is the [[partial order|0QV2JS9RJ3]] on [[Hermitian matrices|0P17NGQQMQ]] in which $A \succeq B$ means that $A - B$ is [[positive semidefinite|01MMVW8CKC]].
It extends the usual order of real numbers to matrices, but it is not a total order.
It is the order used in semidefinite programming, matrix inequalities and the theory of operator monotone functions.

## Explanations

### L1 human

For numbers, $a \ge b$ means that $a - b$ is not negative.
For Hermitian matrices, $A \succeq B$ means that $A - B$ is not negative in the matrix sense: $x^* (A - B) x \ge 0$ for every vector $x$.
So $A$ is bigger than $B$ in every direction.
Two matrices can be incomparable: $\operatorname{diag}(1, 0)$ is bigger in the first direction and $\operatorname{diag}(0, 1)$ in the second.