Visualizing Hadamard Matrices: the Propus Construction

Propus (which means twins) is a construction method for orthogonal ±1 matrices based on the propus array A B C D C D −A −B B −A −D C D −C B −A. This construction, based on circulant symmetric ±1 matrices, called propus matrices, is aimed to give aesthetically pleasing visual images (pictures) when converted using MATLAB. It gives symmetric Hadamard matrices. We give two constructions and note that using these results, we are able to find images (pictures) for propus-Hadamard matrices for orders 4n, n < 100 odd, where n is not in {11, 17, 23, 29, 33, 35, 39, 43, 47, 53, 65, 71, 73, 77, 83, 85, 93, 101, 103, 107, 109, 113, 123, 125, ...}. A computer algorithm seems to be needed to find further results. We give variations of the above array to allow for more general matrices than propus matrices, including the Goethals-Seidel-Propus matrices. We show how conference matrices can be used to find propus-Hadamard for n even. We refer the interested reader to mathscinet.ru/catalogue/propus/.