mathematics

Krylov subspace

Krylov subspace 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 Krylov subspace rather than just read about it. In short: In linear algebra, the order-r Krylov subspace generated by an n-by-n matrix A and a vector b of dimension n is the linear subspace spanned by the images of b under the first r powers of A (starting from A 0 = I {\displaystyle A^{0}=I} ), that is, K r ( A , b ) = span { b , A b , A 2 b , … , A r − 1 b } . {\displaystyle {\mathcal {K}}_{r}(A,b)=\operatorname {span} \,\{b,Ab,A^{2}b,\ldots ,A^{r-1}b\}.} Background The…

Key takeaways

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

Reference excerpt

In linear algebra, the order-r Krylov subspace generated by an n-by-n matrix A and a vector b of dimension n is the linear subspace spanned by the images of b under the first r powers of A (starting from A 0 = I {\displaystyle A^{0}=I} ), that is,

K r ( A , b ) = span { b , A b , A 2 b , … , A r − 1 b } . {\displaystyle {\mathcal {K}}_{r}(A,b)=\operatorname {span} \,\{b,Ab,A^{2}b,\ldots ,A^{r-1}b\}.}

Background The concept is named after Russian applied mathematician and naval engineer Alexei Krylov, who published a paper about the concept in 1931.

Properties

K r ( A , b ) , A K r ( A , b ) ⊂ K r + 1 ( A , b ) {\displaystyle {\mathcal {K}}_{r}(A,b),A\,{\mathcal {K}}_{r}(A,b)\subset {\mathcal {K}}_{r+1}(A,b)} . Let r 0 = dim ⁡ span { b , A b , A 2 b , … } {\displaystyle r_{0}=\operatorname {dim} \operatorname {span} \,\{b,Ab,A^{2}b,\ldots \}} . Then { b , A b , A 2 b , … , A r − 1 b } {\displaystyle \{b,Ab,A^{2}b,\ldots ,A^{r-1}b\}} are linearly independent unless r > r 0 {\displaystyle r>r_{0}} , K r ( A , b ) ⊂ K r 0 ( A , b ) {\displaystyle {\mathcal {K}}_{r}(A,b)\subset {\mathcal {K}}_{r_{0}}(A,b)} for all r {\displaystyle r} , and dim ⁡ K r 0 ( A , b ) = r 0 {\displaystyle \operatorname {dim} {\mathcal {K}}_{r_{0}}(A,b)=r_{0}} . So r 0 {\displaystyle r_{0}} is the maximal dimension of the Krylov subspaces K r ( A , b ) {\displaystyle {\mathcal {K}}_{r}(A,b)} . The maximal dimension satisfies r 0 ≤ 1 + rank ⁡ A {\displaystyle r_{0}\leq 1+\operatorname {rank} A} and r 0 ≤ n {\displaystyle r_{0}\leq n} . Consider dim ⁡ span { I , A , A 2 , … } = deg p ( A ) {\displaystyle \dim \operatorname {span} \,\{I,A,A^{2},\ldots \}=\deg \,p(A)} , where p ( A ) {\displaystyle p(A)} is the minimal polynomial of A {\displaystyle A} . We have r 0 ≤ deg p ( A ) {\displaystyle r_{0}\leq \deg \,p(A)} . Moreover, for any A {\displaystyle A} , there exists a b {\displaystyle b} for which this bound is tight, i.e. r 0 = deg p ( A ) {\displaystyle r_{0}=\deg \,p(A)} .

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Krylov subspace

Start with the simplest possible case. Write down what Krylov subspace 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 Krylov subspace 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 Krylov subspace 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 Krylov subspace

In research: Krylov subspace 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 Krylov subspace 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: Krylov subspace is common in secondary-school and first-year university syllabi. It links to neighbouring topics Invariant subspaces, Numerical linear algebra, Operator theory, so understanding it makes those chapters shorter.
In everyday life: Look for Krylov subspace 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 Krylov subspace in 20 minutes

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

Frequently asked questions

What is Krylov subspace in simple terms?

Why does Krylov subspace 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 Krylov subspace?

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 Krylov subspace.