mathematics

Hilbert matrix

Hilbert matrix 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 Hilbert matrix rather than just read about it. In short: In linear algebra, a Hilbert matrix, introduced by Hilbert (1894), is a square matrix with entries being the unit fractions H i j = 1 i + j − 1 . {\displaystyle H_{ij}={\frac {1}{i+j-1}}.} For example, this is the 5 × 5 Hilbert matrix: H = [ 1 1 2 1 3 1 4 1 5 1 2 1 3 1 4 1 5 1 6 1 3 1 4 1 5 1 6 1 7 1 4 1 5 1 6 1 7 1 8 1 5 1 6 1 7 1 8 1 9 ] . {\displaystyle H={\begin{bmatrix}1&{\frac {1}{2}}&{\frac {1}{3}}&{\frac {1}…

Key takeaways

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

Reference excerpt

In linear algebra, a Hilbert matrix, introduced by Hilbert (1894), is a square matrix with entries being the unit fractions

H i j = 1 i + j − 1 . {\displaystyle H_{ij}={\frac {1}{i+j-1}}.}

For example, this is the 5 × 5 Hilbert matrix:

H = [ 1 1 2 1 3 1 4 1 5 1 2 1 3 1 4 1 5 1 6 1 3 1 4 1 5 1 6 1 7 1 4 1 5 1 6 1 7 1 8 1 5 1 6 1 7 1 8 1 9 ] . {\displaystyle H={\begin{bmatrix}1&{\frac {1}{2}}&{\frac {1}{3}}&{\frac {1}{4}}&{\frac {1}{5}}\\{\frac {1}{2}}&{\frac {1}{3}}&{\frac {1}{4}}&{\frac {1}{5}}&{\frac {1}{6}}\\{\frac {1}{3}}&{\frac {1}{4}}&{\frac {1}{5}}&{\frac {1}{6}}&{\frac {1}{7}}\\{\frac {1}{4}}&{\frac {1}{5}}&{\frac {1}{6}}&{\frac {1}{7}}&{\frac {1}{8}}\\{\frac {1}{5}}&{\frac {1}{6}}&{\frac {1}{7}}&{\frac {1}{8}}&{\frac {1}{9}}\end{bmatrix}}.}

The entries can also be defined by the integral

H i j = ∫ 0 1 x i + j − 2 d x , {\displaystyle H_{ij}=\int _{0}^{1}x^{i+j-2}\,dx,}

that is, as a Gramian matrix for powers of x. It arises in the least squares approximation of arbitrary functions by polynomials. The Hilbert matrices are canonical examples of ill-conditioned matrices, being notoriously difficult to use in numerical computation. For example, the 2-norm condition number of the matrix above is about 4.8×105.

Historical note Hilbert (1894) introduced the Hilbert matrix to study the following question in approximation theory: "Assume that I = [a, b], is a real interval. Is it then possible to find a non-zero polynomial P with integer coefficients, such that the integral

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Hilbert matrix

Start with the simplest possible case. Write down what Hilbert matrix 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 Hilbert matrix 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 Hilbert matrix 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 Hilbert matrix

In research: Hilbert matrix 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 Hilbert matrix 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: Hilbert matrix is common in secondary-school and first-year university syllabi. It links to neighbouring topics Approximation theory, Determinants, Matrices (mathematics), so understanding it makes those chapters shorter.
In everyday life: Look for Hilbert matrix 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 Hilbert matrix in 20 minutes

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

Frequently asked questions

What is Hilbert matrix in simple terms?

In linear algebra, a Hilbert matrix, introduced by Hilbert (1894), is a square matrix with entries being the unit fractions H i j = 1 i + j − 1 . {\displaystyle H_{ij}={\frac {1}{i+j-1}}.} For example, this is the 5 × 5 Hilbert matrix: H = [ 1 1 2 1 3 1 4 1 5 1 2 1 3 1 4 1 5 1 6 1 3 1 4 1 5 1 6 1 7…

Why does Hilbert matrix 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 Hilbert matrix?

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 Hilbert matrix.