Loewner order

Key takeaways

Loewner order belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Loewner order to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Loewner order from memory before moving on to harder problems.

Reference excerpt

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of R. Although some familiar properties of the usual order of R are also valid when n ≥ 2, several properties are no longer valid.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 = [ 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. 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 S {\displaystyle 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 ) {\displaystyle \max(S)} such that any upper bound U on S {\displaystyle S} obeys max ( S ) {\displaystyle \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).

Worked examples

Example 1 — a first encounter with Loewner order

Start with the simplest possible case. Write down what Loewner order claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Loewner order before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Loewner order ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Loewner order

In research: Loewner order appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Loewner order in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Loewner order is common in secondary-school and first-year university syllabi. It links to neighbouring topics Linear algebra, Matrix theory, so understanding it makes those chapters shorter.
In everyday life: Look for Loewner order outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Loewner order in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Loewner order means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Loewner order out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Loewner order in simple terms?

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.

Why does Loewner order matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Loewner order?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Loewner order.