Hadamard matrix

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In mathematics, a Hadamard matrix is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. This means that every two different rows in a Hadamard matrix represent two perpendicular vectors. Such matrices are used in error-correcting codes such as the Reed–Muller code.

Hadamard matrices are also used in balanced repeated replication (BRR), used by statisticians to estimate the variance of a parameter estimator. They are named after the French mathematician Jacques Hadamard.

1 Properties
2 Sylvester's construction
3 Alternative Construction
4 The Hadamard conjecture
5 Equivalence of Hadamard matrices
6 Skew Hadamard matrices
7 Generalizations and special cases
8 Practical applications
9 References and external links

[edit] Properties

It follows from the definition that a Hadamard matrix H of order n satisfies

$H^{\mathrm{T}} H = n I_n \$

where I_n is the n × n identity matrix. Thus $\det H =\pm n^{n/2}$ .

Suppose that M is a complex matrix of order n, whose entries are bounded by |M_ij| ≤1. Then Hadamard's determinant bound states that

$|\operatorname{det}(M)| \leq n^{n/2}.$

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

The order of a Hadamard matrix must be 1, 2, or a multiple of 4.

[edit] Sylvester's construction

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

$\begin{bmatrix} H & H\\ H & -H\end{bmatrix}$

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.

$H_1 = \begin{bmatrix} 1 \end{bmatrix},$

$H_2 = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix},$

and

$H_{2^k} = \begin{bmatrix} H_{2^{k-1}} & H_{2^{k-1}}\\ H_{2^{k-1}} & -H_{2^{k-1}}\end{bmatrix} = H_2\otimes H_{2^{k-1}},$

for $2 \le k \in N$ , where $\otimes$ denotes the Kronecker product.

In this manner, Sylvester constructed Hadamard matrices of order 2^k for every non-negative integer k. ^[1]

Sylvester's matrices have a number of special properties. They are symmetric and traceless. 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.

[edit] Alternative Construction

If we map the elements of the Hadamard matrix using the group homomorphism $\{1,-1,\times\}\mapsto \{0,1,\oplus\}$ , we can describe an alternative construction of Sylvester's Hadamard matrix. First consider the matrix $F n$ , the $n\times 2^n$ matrix whose columns consist of all n-bit numbers arranged in ascending counting order. We may define $F n$ recursively by

$F_1=\begin{bmatrix} 0 & 1\end{bmatrix}$

$F_n=\begin{bmatrix} 0_{1\times 2^{n-1}} & 1_{1\times 2^{n-1}} \\ F_{n-1} & F_{n-1} \end{bmatrix}.$

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

$H_{2^n}=F_n^TF_n.$

This construction demonstrates that the rows of the Hadamard matrix $H_{2^n}$ can be viewed as a length $2 n$ linear error-correcting code of rank n, and minimum distance $2 n - 1$ with generating matrix $F n .$

This code is also referred to as a Walsh code.

[edit] The Hadamard conjecture

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.

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). ^[2] Later, in 1933, Raymond Paley showed how to construct a Hadamard matrix of order q+1 where q is any prime power which is congruent to 3 modulo 4. ^[3] He also constructed matrices of order 2(q+1) for prime powers q which are congruent to 1 modulo 4. His method uses finite fields. The Hadamard conjecture should probably be attributed to Paley. 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. They used a construction, due to Williamson, that has yielded many additional orders. Many other methods for constructing Hadamard matrices are now known.

Hadi Kharaghani and Behruz Tayfeh-Rezaie announced on 21 June 2004 that they constructed a Hadamard matrix of order 428. As a result, the smallest order for which no Hadamard matrix is presently known is 668.

[edit] Equivalence of Hadamard matrices

Two Hadamard matrices are considered equivalent if one can be obtained from the other by negating rows, negating columns, interchanging rows, and interchanging columns. Up to equivalence, there is a unique Hadamard matrices in orders 1, 2, 4, 8, and 12. There are 5 inequivalent matrices in order 16, 3 in order 20, 60 in order 24, and 487 in order 28. Millions of inequivalent matrices are known in orders 32, 36, and 40.

[edit] Skew Hadamard matrices

A Hadamard matrix H is skew if $H^T + H= 2I \$ .

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.

[edit] Generalizations and special cases

There are many generalizations and special cases of Hadamard matrices investigated in the mathematical literature. One basic generalization is a weighing matrix in which one allows also zeros in the entries of the matrix and requires the number of +1, −1 to be a certain constant throughout the matrix. Another generalization are complex Hadamard matrices in which the entries are complex numbers of unit modulus and for which the matrix satisfies H H^*= n I_n 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 a special case of complex Hadamard matrices in which the entries are taken to be q^th roots of unity. The term "complex Hadamard matrix" has been used by some authors to refer specifically to the case q = 4. A special case of real Hadamard matrices are circulant Hadamard matrices. Such a square matrix is defined by its first row, and every other row is a cyclic shift of its predecessor row. Circulant Hadamard matrices of orders 1 and 4 are known and it is conjectured that no other orders exist.

[edit] Practical applications

Olivia MFSK - an amateur-radio digital protocol designed to work in difficult (low signal-to-noise ratio plus multipath propagation) conditions on shortwave bands.

[edit] References and external links

^ J.J. Sylvester. Thoughts on inverse orthogonal matrices, simultaneous sign successions, and tessellated pavements in two or more colours, with applications to Newton's rule, ornamental tile-work, and the theory of numbers. Philosophical Magazine, 34:461-475, 1867
^ J. Hadamard. Résolution d'une question relative aux déterminants. Bulletin des Sciences Mathématiques, 17:240-246, 1893
^ R. E. A. C. Paley. On orthogonal matrices. Journal of Mathematics and Physics, 12:311–320, 1933.

H. Kharaghani and B. Tayfeh-Rezaie, A Hadamard matrix of order 428, 2004.

S. Georgiou, C. Koukouvinos, J. Seberry, Hadamard matrices, orthogonal designs and construction algorithms, pp. 133-205 in DESIGNS 2002: Further computational and constructive design theory, Kluwer 2003.