k was developed in his snappily titled 1867 paper 'Thoughts on Inverse Orthogonal Matrices, Simul- taneous Sign-successions and Tesselated Pavements in two or more colours, with applications to Newton's Rule, Ornamental Tile Work and the Theory of Numbers'. This paper established a new level of understanding about pattern behaviour in matrices (using ornamental tiles as an example). In between times, Sylvester gave maths tuition to Florence Nightingale who in turn revolutionised health care through her understanding and application of the statistical analysis of illness and associated treatment options, yet another example of applied maths. At which point we can introduce Jacques Hadamard. Jacques Hadamard and his family survived the Prussian siege of Paris in 1870 by eating elephant meat. Hadamard obtained his doctorate in 1892 with a thesis on analytic theory and related work on determinant equality, the property that allows matrices to be used as a reversible transform. Hadamard also produced pioneering work on boundary-value problems and functional analysis and is generally seen as 'the founding father' of modern coding theory. Hadamard's work was developed by Hans Rademacher, particularly in the area of orthogonal functions now known as Rademacher functions that appeared in a paper published in 1922 and was the forerunner of pioneering work in analytic number theory. Hans Rademacher's work was contemporaneous with the work of Joseph Leonard Walsh known as 'Joe' to his friends. This included a publication in 1923 on orthogonal expansions, later called 'Walsh functions'. Joe Walsh became a full professor at Harvard in 1935 and produced pioneering work on the relationship of maths and discrete harmonic analysis. 3.2.2 The Hadamard Matrix and the Hadamard Transform A transform changes something into something else. The process is most useful when it is reversible/bidirectional and the purpose is generally to make a particular process easier to achieve. In our context of interest, we want to take a string of numb