n properties, later described as 'determinants' that are revealed when the numbers in the matrix are added or multiplied either vertically by row, horizontally or diagonally. This understanding was documented in 'Nine Chapters on the Mathematic Art' put together around 100 AD and was the product of a period now often described as 'The First Golden Age of Chinese Mathematics' (the second golden age was in the thirteenth and fourteenth century). Historically, it is important to realise that the development of matrix theory in China was contemporaneous with the work of Archimedes on curves and circles and Pi between 287 and 212 BC. Thus, the origins of the Fourier transform and the Hadamard transform can both be traced back to the pre-Christian era. As with the science of geometry, it took over 1000 years before much else meaningful happened in matrix theory. In 1683, the Japanese mathematician Takakazu Seki wrote the 'Method of solving dissim- ulated problems' that precisely described how the earlier Chinese matrix methods had been constructed. At the same time, in Europe, Gottfried Leibniz was producing work on determinants that he called 'resultants'. This became the basis for a body of work to which mathematicians like Cramer, Maclaurin, Bezout and Laplace all contributed throughout the eighteenth century. The term 'determinant' was used by Gauss in 1801 in his 'Disquisitiones arithmeticae' side by side with his work on the coefficients of quadratic forms in rectangular arrays and matrix Making Telecoms Work multiplication. This became the basis for a body of work to which other mathematicians such as Cauchy (who proved that every real symmetric matrix is diagonisable), Sturm, Cayley and Eisenstein contributed in the first half of the nineteenth century. However, it was JJ Sylvester who is first credited with using the term 'matrix' in 1850. Sylvester defined a matrix to be 'an oblong arrangement of terms which could be used to discover various 'determinants' from the square arrays contained within it.' This wor