mathematics

Regular Hadamard matrix

Regular Hadamard 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 Regular Hadamard matrix rather than just read about it. In short: In mathematics a regular Hadamard matrix is a Hadamard matrix whose row and column sums are all equal. While the order of a Hadamard matrix must be 1, 2, or a multiple of 4, regular Hadamard matrices carry the further restriction that the order must be a square number.

Key takeaways

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

Reference excerpt

In mathematics a regular Hadamard matrix is a Hadamard matrix whose row and column sums are all equal. While the order of a Hadamard matrix must be 1, 2, or a multiple of 4, regular Hadamard matrices carry the further restriction that the order must be a square number. The excess, denoted E(H ), of a Hadamard matrix H of order n is defined to be the sum of the entries of H. The excess satisfies the bound |E(H )| ≤ n3/2. A Hadamard matrix attains this bound if and only if it is regular.

Parameters If n = 4u2 is the order of a regular Hadamard matrix, then the excess is ±8u3 and the row and column sums all equal ±2u. It follows that each row has 2u2 ± u positive entries and 2u2 ∓ u negative entries. The orthogonality of rows implies that any two distinct rows have exactly u2 ± u positive entries in common. If H is interpreted as the incidence matrix of a block design, with 1 representing incidence and −1 representing non-incidence, then H corresponds to a symmetric 2-(v,k,λ) design with parameters (4u2, 2u2 ± u, u2 ± u). A design with these parameters is called a Menon design.

Construction

A number of methods for constructing regular Hadamard matrices are known, and some exhaustive computer searches have been done for regular Hadamard matrices with specified symmetry groups, but it is not known whether every even perfect square is the order of a regular Hadamard matrix. Bush-type Hadamard matrices are regular Hadamard matrices of a special form, and are connected with finite projective planes. A regular Hadamard matrix that is also circulant corresponds to a Menon design with a regular cyclic automorphism group. The existence of such circulant examples of order greater than 4 is the subject of Ryser's conjecture on circulant Hadamard matrices, which conjectures that none exist.

History and naming Like Hadamard matrices more generally, regular Hadamard matrices are named after Jacques Hadamard. Menon designs are named after P Kesava Menon, and Bush-type Hadamard matrices are named after Kenneth A. Bush.

See also Ryser's conjecture on circulant Hadamard matrices

References C.J. Colbourn and J.H. Dinitz (Eds.), The CRC Handbook of Combinatorial Designs, 2nd ed., CRC Press, Boca Raton, Florida., 2006. W. D. Wallis, Anne Penfold Street, and Jennifer Seberry Wallis, Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices, Springer-Verlag, Berlin 1972.

Worked examples

Example 1 — a first encounter with Regular Hadamard matrix

Start with the simplest possible case. Write down what Regular Hadamard 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 Regular Hadamard 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 Regular Hadamard 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 Regular Hadamard matrix

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

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

Frequently asked questions

What is Regular Hadamard matrix in simple terms?

Why does Regular Hadamard 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 Regular Hadamard 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 Regular Hadamard matrix.