Woodbury matrix identity

Key takeaways

Woodbury matrix 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 Woodbury matrix identity to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Woodbury matrix identity from memory before moving on to harder problems.

Reference excerpt

In mathematics, specifically linear algebra, the Woodbury matrix identity – named after Max A. Woodbury – says that the inverse of a rank-k correction of some matrix can be computed by doing a rank-k correction to the inverse of the original matrix. Alternative names for this formula are the matrix inversion lemma, Sherman–Morrison–Woodbury formula or just Woodbury formula. However, the identity appeared in several papers before the Woodbury report. The Woodbury matrix identity is

( A + U C V ) − 1 = A − 1 − A − 1 U ( C − 1 + V A − 1 U ) − 1 V A − 1 , {\displaystyle \left(A+UCV\right)^{-1}=A^{-1}-A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1},}

where A, U, C and V are conformable matrices: A is n×n, C is k×k, U is n×k, and V is k×n. This can be derived using blockwise matrix inversion. While the identity is primarily used on matrices, it holds in a general ring or in an Ab-category. The Woodbury matrix identity allows cheap computation of inverses and solutions to linear equations. However, little is known about the numerical stability of the formula. There are no published results concerning its error bounds. Anecdotal evidence suggests that it may diverge even for seemingly benign examples (when both the original and modified matrices are well-conditioned).

Discussion To prove this result, we will start by proving a simpler one. Replacing A and C with the identity matrix I, we obtain another identity which is a bit simpler:

( I + U V ) − 1 = I − U ( I + V U ) − 1 V . {\displaystyle \left(I+UV\right)^{-1}=I-U\left(I+VU\right)^{-1}V.}

To recover the original equation from this reduced identity, replace U {\displaystyle U} by A − 1 U {\displaystyle A^{-1}U} and V {\displaystyle V} by C V {\displaystyle CV} . This identity itself can be viewed as the combination of two simpler identities. We obtain the first identity from

I = ( I + P ) − 1 ( I + P ) = ( I + P ) − 1 + ( I + P ) − 1 P , {\displaystyle I=(I+P)^{-1}(I+P)=(I+P)^{-1}+(I+P)^{-1}P,}

thus,

( I + P ) − 1 = I − ( I + P ) − 1 P , {\displaystyle (I+P)^{-1}=I-(I+P)^{-1}P,}

and similarly

( I + P ) − 1 = I − P ( I + P ) − 1 . {\displaystyle (I+P)^{-1}=I-P(I+P)^{-1}.}

The second identity is the so-called push-through identity

( I + U V ) − 1 U = U ( I + V U ) − 1 {\displaystyle (I+UV)^{-1}U=U(I+VU)^{-1}}

that we obtain from

U ( I + V U ) = ( I + U V ) U {\displaystyle U(I+VU)=(I+UV)U}

after multiplying by ( I + V U ) − 1 {\displaystyle (I+VU)^{-1}} on the right and by ( I + U V ) − 1 {\displaystyle (I+UV)^{-1}} on the left. Putting all together,

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Woodbury matrix identity

Start with the simplest possible case. Write down what Woodbury matrix 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 Woodbury matrix 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 Woodbury matrix 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 Woodbury matrix identity

In research: Woodbury matrix 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 Woodbury matrix 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: Woodbury matrix identity is common in secondary-school and first-year university syllabi. It links to neighbouring topics Lemmas in linear algebra, Matrices (mathematics), Matrix theory, so understanding it makes those chapters shorter.
In everyday life: Look for Woodbury matrix 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.

How to study Woodbury matrix identity in 20 minutes

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

Frequently asked questions

What is Woodbury matrix identity in simple terms?

In mathematics, specifically linear algebra, the Woodbury matrix identity – named after Max A. Woodbury – says that the inverse of a rank-k correction of some matrix can be computed by doing a rank-k correction to the inverse of the original matrix.

Why does Woodbury matrix 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 Woodbury matrix 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 Woodbury matrix identity.