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 a Hadamard matrix represents two perpendicular vectors, while in combinatorial terms, it means that each pair of rows has matching entries in exactly half of their columns and mismatched entries in the remaining columns. It is a consequence of this definition that the corresponding properties hold for columns as well as rows. The n-dimensional parallelotope spanned by the rows of an n×n Hadamard matrix has the maximum possible n-dimensional volume among parallelotopes spanned by vectors whose entries are bounded in absolute value by 1. Equivalently, a Hadamard matrix has maximal determinant among matrices with entries of absolute value less than or equal to 1 and so is an extremal solution of Hadamard's maximal determinant problem.

Certain Hadamard matrices can almost directly be used as an error-correcting code using a Hadamard code (generalized in Reed–Muller codes), and are also used in balanced repeated replication (BRR), used by statisticians to estimate the variance of a parameter estimator.

Let H be a Hadamard matrix of order n. The transpose of H is closely related to its inverse. In fact:

Suppose that M is a complex matrix of order n, whose entries are bounded by |Mij| ≤ 1, for each i, j between 1 and n. Then Hadamard's determinant bound states that

Equality in this bound is attained for a real matrix M if and only if M is a Hadamard matrix.

Examples of Hadamard matrices were actually first constructed by James Joseph Sylvester in 1867. Let H be a Hadamard matrix of order n. Then the partitioned matrix

is a Hadamard matrix of order 2n. This observation can be applied repeatedly and leads to the following sequence of matrices, also called Walsh matrices.

In this manner, Sylvester constructed Hadamard matrices of order 2k for every non-negative integer k.[2]

Sylvester's matrices have a number of special properties. They are symmetric and, when k ≥ 1 (2k  > 1), have trace zero. The elements in the first column and the first row are all positive. The elements in all the other rows and columns are evenly divided between positive and negative. Sylvester matrices are closely connected with Walsh functions.

It can be shown by induction that the image of the Hadamard matrix under the above homomorphism is given by

The most important open question in the theory of Hadamard matrices is that of existence. The Hadamard conjecture proposes that a Hadamard matrix of order 4k exists for every positive integer k. The Hadamard conjecture has also been attributed to Paley, although it was considered implicitly by others prior to Paley's work.[3]

Sylvester's 1867 construction yields Hadamard matrices of order 1, 2, 4, 8, 16, 32, etc. Hadamard matrices of orders 12 and 20 were subsequently constructed by Hadamard (in 1893).[4] In 1933, Raymond Paley discovered the Paley construction, which produces a Hadamard matrix of order q + 1 when q is any prime power that is congruent to 3 modulo 4 and that produces a Hadamard matrix of order 2(q + 1) when q is a prime power that is congruent to 1 modulo 4.[5] His method uses finite fields.

The smallest order that cannot be constructed by a combination of Sylvester's and Paley's methods is 92. A Hadamard matrix of this order was found using a computer by Baumert, Golomb, and Hall in 1962 at JPL.[6] They used a construction, due to Williamson,[7] that has yielded many additional orders. Many other methods for constructing Hadamard matrices are now known.

In 2005, Hadi Kharaghani and Behruz Tayfeh-Rezaie published their construction of a Hadamard matrix of order 428.[8] As a result, the smallest order for which no Hadamard matrix is presently known is 668.

As of 2008, there are 13 multiples of 4 less than or equal to 2000 for which no Hadamard matrix of that order is known.[9] They are: 668, 716, 892, 1004, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, and 1964.

Two Hadamard matrices are considered equivalent if one can be obtained from the other by negating rows or columns, or by interchanging rows or columns. Up to equivalence, there is a unique Hadamard matrix of orders 1, 2, 4, 8, and 12. There are 5 inequivalent matrices of order 16, 3 of order 20, 60 of order 24, and 487 of order 28. Millions of inequivalent matrices are known for orders 32, 36, and 40. Using a coarser notion of equivalence that also allows transposition, there are 4 inequivalent matrices of order 16, 3 of order 20, 36 of order 24, and 294 of order 28.[10]

Many special cases of Hadamard matrices have been investigated in the mathematical literature.

Reid and Brown in 1972 showed that there exists a doubly regular tournament of order n if and only if there exists a skew Hadamard matrix of order n + 1. In a mathematical tournament of order n, each of n players plays one match against each of the other players, each match resulting in a win for one of the players and a loss for the other. A tournament is regular if each player wins the same number of matches. A regular tournament is doubly regular if the number of opponents beaten by both of two distinct players is the same for all pairs of distinct players. Since each of the n (n−1) / 2 matches played results in a win for one of the players, each player wins (n−1) / 2 matches (and loses the same number). Since each of the (n−1) / 2 players defeated by a given player also loses to (n−3) / 2 other players, the number of player pairs (ij) such that j loses both to i and to the given player is (n−1) (n−3) / 4. The same result should be obtained if the pairs are counted differently: the given player and any of the (n−1) other players together defeat the same number of common opponents. This common number of defeated opponents must therefore be (n−3) / 4. A skew Hadamard matrix is obtained by introducing an additional player who defeats all of the original players and then forming a matrix with rows and columns labeled by players according to the rule that row i, column j contains 1 if i = j or i defeats j and −1 if j defeats i. This correspondence in reverse produces a doubly regular tournament from a skew Hadamard matrix, assuming the skew Hadamard matrix is normalized so that all elements of the first row equal 1.[11]

Regular Hadamard matrices are real Hadamard matrices whose row and column sums are all equal. A necessary condition on the existence of a regular n×n Hadamard matrix is that n be a perfect square. A circulant matrix is manifestly regular, and therefore a circulant Hadamard matrix would have to be of perfect square order. Moreover, if an n×n circulant Hadamard matrix existed with n > 1 then n would necessarily have to be of the form 4u2 with u odd.[12][13]

The circulant Hadamard matrix conjecture, however, asserts that, apart from the known 1×1 and 4×4 examples, no such matrices exist. This was verified for all but 26 values of u less than 104.[14]

Another generalization defines a complex Hadamard matrix to be a matrix in which the entries are complex numbers of unit modulus and which satisfies H H* = n In where H* is the conjugate transpose of H. Complex Hadamard matrices arise in the study of operator algebras and the theory of quantum computation. Butson-type Hadamard matrices are complex Hadamard matrices in which the entries are taken to be qth roots of unity. The term complex Hadamard matrix has been used by some authors to refer specifically to the case q = 4.