Operator monotone function

Key takeaways

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

Reference excerpt

In linear algebra, operator monotone functions are an important type of real-valued function, fully classified by Charles Löwner in 1934. They are closely related to operator concave and operator convex functions, and are encountered in operator theory and in matrix theory, and led to the Löwner–Heinz inequality. Operator monotone functions are called in other contexts complete Bernstein function, Nevanlinna function, Pick function or class (S) function.

Definition

A function f : I → R {\displaystyle f:I\to \mathbb {R} } defined on an interval I ⊆ R {\displaystyle I\subseteq \mathbb {R} } is said to be operator monotone if whenever A {\displaystyle A} and B {\displaystyle B} are Hermitian matrices (of any size/dimensions) whose eigenvalues all belong to the domain of f {\displaystyle f} and whose difference A − B {\displaystyle A-B} is a positive semi-definite matrix, then necessarily f ( A ) − f ( B ) ≥ 0 {\displaystyle f(A)-f(B)\geq 0} where f ( A ) {\displaystyle f(A)} and f ( B ) {\displaystyle f(B)} are the values of the matrix function induced by f {\displaystyle f} (which are matrices of the same size as A {\displaystyle A} and B {\displaystyle B} ). The function f {\displaystyle f} is said to be n-matrix monotone (or just n-monotone) if the above holds for any matrices A {\displaystyle A} and B {\displaystyle B} of size n {\displaystyle n} (but not necessarily of other sizes). Notation This definition is frequently expressed with the notation that is now defined. Write A ≥ 0 {\displaystyle A\geq 0} to indicate that a matrix A {\displaystyle A} is positive semi-definite and write A ≥ B {\displaystyle A\geq B} to indicate that the difference A − B {\displaystyle A-B} of two matrices A {\displaystyle A} and B {\displaystyle B} satisfies A − B ≥ 0 {\displaystyle A-B\geq 0} (that is, A − B {\displaystyle A-B} is positive semi-definite). With f : I → R {\displaystyle f:I\to \mathbb {R} } and A {\displaystyle A} as in the theorem's statement, the value of the matrix function f ( A ) {\displaystyle f(A)} is the matrix (of the same size as A {\displaystyle A} ) defined in terms of its A {\displaystyle A} 's spectral decomposition A = ∑ j λ j P j {\displaystyle A=\sum _{j}\lambda _{j}P_{j}} by

f ( A ) = ∑ j f ( λ j ) P j , {\displaystyle f(A)=\sum _{j}f(\lambda _{j})P_{j}~,} where the λ j {\displaystyle \lambda _{j}} are the eigenvalues of A {\displaystyle A} with corresponding projectors P j . {\displaystyle P_{j}.} The definition of an operator monotone function may now be restated as: A function f : I → R {\displaystyle f:I\to \mathbb {R} } defined on an interval I ⊆ R {\displaystyle I\subseteq \mathbb {R} } said to be operator monotone if (and only if) for all positive integers n , {\displaystyle n,} and all n × n {\displaystyle n\times n} Hermitian matrices A {\displaystyle A} and B {\displaystyle B} with eigenvalues in I , {\displaystyle I,} if A ≥ B {\displaystyle A\geq B} then f ( A ) ≥ f ( B ) . {\displaystyle f(A)\geq f(B).}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Operator monotone function

Start with the simplest possible case. Write down what Operator monotone function 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 Operator monotone function 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 Operator monotone function 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 Operator monotone function

In research: Operator monotone function 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 Operator monotone function 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: Operator monotone function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Linear algebra stubs, Matrix theory, Operator theory, so understanding it makes those chapters shorter.
In everyday life: Look for Operator monotone function 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 Operator monotone function in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Operator monotone function 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 Operator monotone function out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Operator monotone function in simple terms?

In linear algebra, operator monotone functions are an important type of real-valued function, fully classified by Charles Löwner in 1934. They are closely related to operator concave and operator convex functions, and are encountered in operator theory and in matrix theory, and led to the Löwner–He…

Why does Operator monotone function 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 Operator monotone function?

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 Operator monotone function.