Study of Generalization of Hadamard Matrix

Abstract

The main aim of this thesis is to study of generalization of Hadamard matrix. The newlineHadamard matrices have the great applications in various field, such as design of newlineexperiments, including factorial designs, for Hadamard transform spectroscopy newlineand object newlinerecognition and coding of digital signals [118). newlineThe Hadamard matrices was constructed first by Sylvester for an infinite class of newlineHadamard matrices of order 2, where n 2 0. Since in (1893), Hadamard concluded newlinehis paper on the maximal determinant problem by describing Hadamard matrices newlineof orders 12 and 20 ([92), which are not included in Sylvester construction. newlineHowever, Hadamard obtained these matrices heuristically. In (1933) Raymond newlinePaley, a young English mathematician created a new construction technique newlinethat could be used to obtain Hadamard matrices of orders 12 and 20, amongst newlinethat widely expanded the number of known Hadamard matrices and also this newlineconstruction generates new order of Hadamard matrices, which provide further newlinejustification for the validity of Hadamard conjecture. After Paley, in 1944 newlineWilliamson found an array, which is later on familiar with the name of Williamson newlinearray and a sequence of unknown Hadamard matrices was found with the help of newlineWilliamson array. After Williamson various mathematicians came and modify the newlineconditions of Williamson Hadamard matrices.