A class of Square Matrix invented by Sylvester (1867) under the name of Anallagmatic Pavement. A Hadamard matrix is a Square Matrix containing only 1s and s such that when any two columns or rows are placed side by side, Half the adjacent cells are the same Sign and half the other (excepting from the count an -shaped half-frame'' bordering the matrix on two sides which is composed entirely of 1s). When viewed as pavements, cells with 1s are colored black and those with s are colored white. Therefore, the Hadamard matrix must have white squares (s) and black squares (1s).

This is equivalent to the definition

 (1)

where is the Identity Matrix. A Hadamard matrix of order corresponds to a Hadamard Design (, , ).

Paley's Theorem guarantees that there always exists a Hadamard matrix when is divisible by 4 and of the form , where is an Odd Prime. In such cases, the Matrices can be constructed using a Paley Construction. The Paley Class is undefined for the following values of : 92, 116, 156, 172, 184, 188, 232, 236, 260, 268, 292, 324, 356, 372, 376, 404, 412, 428, 436, 452, 472, 476, 508, 520, 532, 536, 584, 596, 604, 612, 652, 668, 712, 716, 732, 756, 764, 772, 808, 836, 852, 856, 872, 876, 892, 904, 932, 940, 944, 952, 956, 964, 980, 988, 996.

Sawade (1985) constructed . It is conjectured (and verified up to ) that exists for all Divisible by 4 (van Lint and Wilson 1993). However, the proof of this Conjecture remains an important problem in Coding Theory. The number of Hadamard matrices of order are 1, 1, 1, 5, 3, 60, 487, ... (Sloane's A007299).

If and are known, then can be obtained by replacing all 1s in by and all s by . For , Hadamard matrices with , 20, 28, 36, 44, 52, 60, 68, 76, 84, 92, and 100 cannot be built up from lower order Hadamard matrices.

 (2) (3)

can be similarly generated from . Hadamard matrices can also be expressed in terms of the Walsh Functions Cal and Sal
 (4)

Hadamard matrices can be used to make Error-Correcting Codes.

References

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