Hadamard matrix

In mathematics, an 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 every two different rows in a Hadamard matrix represent two perpendicular vectors, while in combinatorial terms, it means that every two different rows have 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.

Contents

Properties

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

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

where In is the n × n identity matrix and HT is the transpose of H. Consequently the determinant of H equals ±nn/2.

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

|\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.

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 \left.\otimes\right. denotes the Kronecker product.

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

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

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}.

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

H_{2^n}=F_n^{\rm T}F_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. The Hadamard code, by contrast, is constructed from the Hadamard matrix H_{2^n} by a slightly different procedure.

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.

A generalization of Sylvester's construction proves that if H_n and H_m are Hadamard matrices of orders n and m respectively, then \scriptstyle H_n \otimes H_m is an Hadamard matrix of order nm. This result is used to produce Hadamard matrices of higher order once those of smaller orders are known.

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] In 1933 Raymond Paley discovered a construction that 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.[3] 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.[4] They used a construction, due to Williamson,[5] 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.[6] 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.[7] They are: 668, 716, 892, 1004, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, 1964.

Equivalence of Hadamard matrices

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[8] .

Skew Hadamard matrices

A Hadamard matrix H is skew if H^{\rm T} %2B 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.

Generalizations and special cases

Many generalizations and special cases of Hadamard matrices have been investigated in the mathematical literature. One basic generalization is the weighing matrix, a square matrix in which entries may also be zero and which satisfies WW^{T}=wI for some w, its weight. A weighing matrix with its weight equal to its order is a Hadamard matrix.

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.

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.[9][10]

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.[11]

Practical applications

See also

Notes

  1. ^ 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
  2. ^ Hadamard, J. (1893). "Résolution d'une question relative aux déterminants". Bulletin des Sciences Mathématiques 17: 240–246. 
  3. ^ Paley, R. E. A. C. (1933). "On orthogonal matrices". Journal of Mathematics and Physics 12: 311–320. 
  4. ^ Baumert, L.; Golomb, S. W.; Hall, M., Jr. (1962). "Discovery of an Hadamard Matrix of Order 92". Bulletin of the American Mathematical Society 68 (3): 237–238. doi:10.1090/S0002-9904-1962-10761-7. MR0148686. 
  5. ^ Williamson, J. (1944). "Hadamard’s determinant theorem and the sum of four squares". Duke Mathematical Journal 11 (1): 65–81. doi:10.1215/S0012-7094-44-01108-7. MR0009590. 
  6. ^ Kharaghani, H.; Tayfeh-Rezaie, B. (2005). "A Hadamard matrix of order 428". Journal of Combinatorial Designs 13 (6): 435–440. doi:10.1002/jcd.20043. 
  7. ^ Đoković, Dragomir Ž (2008). "Hadamard matrices of order 764 exist". Combinatorica 28 (4): 487–489. doi:10.1007/s00493-008-2384-z. 
  8. ^ Wanless, I.M. (2005). "Permanents of matrices of signed ones". Linear and Multilinear Algebra 53: 427–433. 
  9. ^ Turyn, R. J. (1965). "Character sums and difference sets". Pacific Journal of Mathematics 15 (1): 319–346. MR0179098. 
  10. ^ Turyn, R. J. (1969). "Sequences with small correlation". In Mann, H. B.. Error Correcting Codes. New York: Wiley. pp. 195–228. 
  11. ^ Schmidt, B. (1999). "Cyclotomic integers and finite geometry". Journal of the American Mathematical Society 12 (4): 929–952. JSTOR 2646093. 

Further reading

External links