Publisher Summary In many practical applications, there is a need to transform a given sparse matrix into another matrix whose columns are orthonormal. The use of orthonormalizing codes is well known. This chapter discusses the problem of finding the optimum order in which the columns of the given sparse matrix should be orthonormalized, such that the resulting matrix is as sparse as possible. It is concerned with the Gram–Schmidt, the Householder, and the Givens methods for orthonormalization. Schmidt method involves the determination of an upper triangular matrix U such that the columns of AU are orthonormal. If A is sparse, then it is generally advantageous to find a permutation matrix Q such that A Q U and U are both sparse. A slightly modified version of the Gram-Schmidt method is discussed which is more accurate than the usual method in terms of round-off errors. It is called the Revised Gram-Schmidt method.