The strong Kronecker product has proved a powerful new multiplication tool for orthogonal matrices. This paper obtains algebraic structure theorems and properties for this new product. The results are then applied to give new multiplication theorems for Hadamard matrices, complex Hadamard matrices and other related orthogonal matrices. We obtain complex Hadamard matrices of order 8abcd from complex Hadamard matrices of order 2a, 2b, 2c, and 2d, and complex Hadamard matrices of order 32abcdef from Hadamard matrices of orders 4a, 4b, 4c, 4d, 4e, and 4f. We also obtain a pair of disjoint amicable OD(8hn; 2hn, 2hn)s from Hadamard matrices of orders 4h and 4n, and Plotkin's result that a pair of amicable OD(4h; 2h, 2h)s and an OD(8h; 2h, 2h, 2h, 2h) can be constructed from an Hadamard matrix of order 4h as a corollary. © 1994.