mathematics

Weinstein–Aronszajn identity

Weinstein–Aronszajn identity is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Weinstein–Aronszajn identity rather than just read about it. In short: In mathematics, the Weinstein–Aronszajn identity states that if A {\displaystyle A} and B {\displaystyle B} are matrices of size m × n and n × m respectively (either or both of which may be infinite) then, provided A B {\displaystyle AB} (and hence, also B A {\displaystyle BA} ) is of trace class, det ( I m + A B ) = det ( I n + B A ) , {\displaystyle \det(I_{m}+AB)=\det(I_{n}+BA),} where I k {\displaystyle I_{k}} i…

Key takeaways

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

Reference excerpt

In mathematics, the Weinstein–Aronszajn identity states that if A {\displaystyle A} and B {\displaystyle B} are matrices of size m × n and n × m respectively (either or both of which may be infinite) then, provided A B {\displaystyle AB} (and hence, also B A {\displaystyle BA} ) is of trace class,

det ( I m + A B ) = det ( I n + B A ) , {\displaystyle \det(I_{m}+AB)=\det(I_{n}+BA),}

where I k {\displaystyle I_{k}} is the k × k identity matrix. It is closely related to the matrix determinant lemma and its generalization. It is the determinant analogue of the Woodbury matrix identity for matrix inverses.

Proof The identity may be proved as follows. Let M {\displaystyle M} be a matrix consisting of the four blocks I m {\displaystyle I_{m}} , A {\displaystyle A} , B {\displaystyle B} and I n {\displaystyle I_{n}} :

M = ( I m A B I n ) . {\displaystyle M={\begin{pmatrix}I_{m}&A\\B&I_{n}\end{pmatrix}}.}

Because Im is invertible, the formula for the determinant of a block matrix gives

det ( I m A B I n ) = det ( I m ) det ( I n − B I m − 1 A ) = det ( I n − B A ) . {\displaystyle \det \!{\begin{pmatrix}I_{m}&A\\B&I_{n}\end{pmatrix}}=\det(I_{m})\det(I_{n}-BI_{m}^{-1}A)=\det(I_{n}-BA).}

Because In is invertible, the formula for the determinant of a block matrix gives

det ( I m A B I n ) = det ( I n ) det ( I m − A I n − 1 B ) = det ( I m − A B ) . {\displaystyle \det \!{\begin{pmatrix}I_{m}&A\\B&I_{n}\end{pmatrix}}=\det(I_{n})\det(I_{m}-AI_{n}^{-1}B)=\det(I_{m}-AB).}

Thus

det ( I n − B A ) = det ( I m − A B ) . {\displaystyle \det(I_{n}-BA)=\det(I_{m}-AB).}

Substituting − A {\displaystyle -A} for A {\displaystyle A} then gives the Weinstein–Aronszajn identity.

Applications Let λ ∈ R ∖ { 0 } {\displaystyle \lambda \in \mathbb {R} \setminus \{0\}} . The identity can be used to show the somewhat more general statement that

det ( A B − λ I m ) = ( − λ ) m − n det ( B A − λ I n ) . {\displaystyle \det(AB-\lambda I_{m})=(-\lambda )^{m-n}\det(BA-\lambda I_{n}).}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Weinstein–Aronszajn identity

Start with the simplest possible case. Write down what Weinstein–Aronszajn identity 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 Weinstein–Aronszajn identity 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 Weinstein–Aronszajn identity 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 Weinstein–Aronszajn identity

In research: Weinstein–Aronszajn identity 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 Weinstein–Aronszajn identity 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: Weinstein–Aronszajn identity is common in secondary-school and first-year university syllabi. It links to neighbouring topics Determinants, Linear algebra stubs, Matrix theory, so understanding it makes those chapters shorter.
In everyday life: Look for Weinstein–Aronszajn identity 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.

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How to study Weinstein–Aronszajn identity in 20 minutes

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

Frequently asked questions

What is Weinstein–Aronszajn identity in simple terms?

Why does Weinstein–Aronszajn identity 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 Weinstein–Aronszajn identity?

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 Weinstein–Aronszajn identity.