This paper surveys the contributions of five mathematicians,---,Eugenio Beltrami (1835--1899), Camille Jordan (1838--1921), James Joseph Sylvester (1814--1897), Erhard Schmidt (1876--1959), and Hermann Weyl (1885--1955),---,who were responsible for establishing the existence of the singular value decomposition and developing its theory. (Also cross-referenced as UMIACS-TR-92-31)