ON NUMERICAL MEASURES OF SINGULARITY OF A MATRIX

Abstract

Three measures for the singularity of near singular matrices have been suggested. One based on the basic principle of linear dependence of tbe column (or row) vectors constituting the matrix, while the other two on methods of Orthonormalization" and Orthogonatization", respectively. The advantage of the first measure is the computational ease was well as greater accuracy in the case of unsymmetric matrics), for instance, in comparison to Neumaon-Goldstine's measure. The important feature of the remaining two measurers is that they are part of the inversion proces itself in contrast to those suggested by Turing, Neumanm "and Goldstine all of which involve calculations additional to the task of inversion. Moreover these measures. unlike the suggested measure of Tuling, Neumann and Goldstine, do not assume any knowledge of inverse or characteristic values of matrix and also take into cosideration the degree of accurancy of the elements of the matrix. However in the case or a large matrix, of 100 or more order, the second measure based on Orthonormalization method may be impractical and sometimes even not feasible, but the third one based on Orthogonalization method is immune to such difficulty and is, moreover, superior to other measures as actually proved by Numerical examples.