In this paper, we study the computation of the singular value decomposition of a matrix on the ILLIAC IV computer. We describe the architecture of the machine and explain why the standard Golub-Reinsch algorithm is not applicable to this problem. We then present a one-sided orthogonalization method which makes very efficient use of the parallel computing abilities of the ILLIAC machine. Our method is shown to be Jacobi-like and numerically stable. Finally, a comparison of our method on the ILLIAC IV computer with the Golub-Reinsch algorithm on a conventional machine demonstrates the great potential of parallel computers in the important area of matrix computations. Key Words and Phrases: ILLIAC IV computer, singular value decomposition, Golub-Reinsch algorithm, Jacobi-like method, parallel matrix computations.