In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
a regular Hadamard matrix is a
Hadamard matrix
In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows in ...
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 be a
square number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
. 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''), ≤ ''n''
3/2. A Hadamard matrix attains this bound if and only if it is regular.
Parameters
If ''n'' = 4''u''
2 is the order of a regular Hadamard matrix, then the excess is ±8''u''
3 and the row and column sums all equal ±2''u''. It follows that each row has 2''u''
2 ± ''u'' positive entries and 2''u''
2 ∓ ''u'' negative entries. The orthogonality of rows implies that any two distinct rows have exactly ''u''
2 ± ''u'' positive entries in common. If ''H'' is interpreted as the
incidence matrix
In mathematics, an incidence matrix is a logical matrix that shows the relationship between two classes of objects, usually called an incidence relation. If the first class is ''X'' and the second is ''Y'', the matrix has one row for each element ...
of a
block design
In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as ''blocks'', chosen such that frequency of the elements satisfies certain conditions making the collection of bl ...
, with 1 representing incidence and −1 representing non-incidence, then ''H'' corresponds to a symmetric 2-(''v'',''k'',''λ'') design with parameters (4''u''
2, 2''u''
2 ± ''u'', ''u''
2 ± ''u''). A design with these parameters is called a Menon design.
Construction
{{unsolved, mathematics, Which square numbers can be the order of a regular Hadamard matrix?
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
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
.
History and naming
Like Hadamard matrices more generally, regular Hadamard matrices are named after
Jacques Hadamard
Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations.
Biography
The son of a teac ...
. Menon designs are named after
P Kesava Menon
Puliyakot Keshava Menon (1917 – 22 October 1979) was an Indian mathematician best known as Director of the Joint Cipher Bureau.
His sudden demise on 22 October 1979, ended active research in the areas of number theory, combinatorics, alg ...
, and Bush-type Hadamard matrices are named after Kenneth A. Bush.
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
Anne Penfold Street (1932–2016) was one of Australia's leading mathematicians, specialising in combinatorics. She was the third woman to become a mathematics professor in Australia, following Hanna Neumann and Cheryl Praeger. She was the auth ...
, and
Jennifer Seberry Wallis, Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices, Springer-Verlag, Berlin 1972.
Matrices