On the distribution of a scaled condition number

Author:
Alan Edelman

Journal:
Math. Comp. **58** (1992), 185-190

MSC:
Primary 15A52; Secondary 15A12, 62H10, 65F99, 65U05

DOI:
https://doi.org/10.1090/S0025-5718-1992-1106966-2

MathSciNet review:
1106966

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this note, we give the exact distribution of a scaled condition number used by Demmel to model the probability that matrix inversion is difficult. Specifically, consider a random matrix *A* and the scaled condition number . Demmel provided bounds for the condition number distribution when *A* has real or complex normally distributed elements. Here, we give the exact formula.

**[1]**M. Abramowitz and I. A. Stegun, eds.,*Handbook of mathematical functions*, Dover, New York, 1970.**[2]**T. W. Anderson,*An introduction to multivariate statistical analysis*, Wiley, New York, 1958. MR**0091588 (19:992a)****[3]**A. W. Davis,*On the ratios of the individual latent roots to the trace of a Wishart matrix*, J. Multivariate Anal.**2**(1972), 440-443. MR**0324834 (48:3183)****[4]**J. W. Demmel,*The probability that a numerical analysis problem is difficult*, Math. Comp.**50**(1988), 449-480. MR**929546 (89g:65062)****[5]**A. Edelman,*Eigenvalues and condition numbers of random matrices*, SIAM J. Matrix Anal. Appl.**9**(1988), 543-560. MR**964668 (89j:15039)****[6]**-,*Eigenvalues and condition numbers of random matrices*, Ph.D. thesis, Dept. of Math., M.I.T., 1989.**[7]**I. S. Gradshteyn and I. W. Ryzhik,*Table of integrals, series, and products*, 6th ed., Academic Press, New York, 1980.**[8]**P. R. Krishnaiah and F. J. Schuurmann,*On the evaluation of some distributions that arise in simultaneous tests for the equality of the latent roots of the covariance matrix*, J. Multivariate Anal.**4**(1974), 265-282. MR**0359178 (50:11633)****[9]**F. J. Schuurmann, P. R. Krishnaiah, and A. K. Chattopadhyay,*On the distributions of the ratios of the extreme roots to the trace of the Wishart matrix*, J. Multivariate Anal.**3**(1973), 445-453. MR**0331644 (48:9976)**

Retrieve articles in *Mathematics of Computation*
with MSC:
15A52,
15A12,
62H10,
65F99,
65U05

Retrieve articles in all journals with MSC: 15A52, 15A12, 62H10, 65F99, 65U05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1992-1106966-2

Keywords:
Condition number,
ill-conditioning,
multivariate statistics,
numerical analysis,
random matrix

Article copyright:
© Copyright 1992
American Mathematical Society