The distribution of the maximum condition number on great circles through a fixed 2 × 2 real matrix

Abstract

If PA(χ) denotes the probability that the maximum condition number along a great circle passing through a matrix A in the unit sphere in the space of 2 × 2 matrices is less than χ, then PA(χ) always attains its maximum at the normalized identity matrix. This result is the first nontrivial case of a linear algebra version of a conjecture formulated in Shub and Smale (M. Shub and S. Smale, Theoretical Computer Science 113 (1994) 141-164) for homotopies of systems of homogeneous equations. The Hopf fibration is used to relate the probability PA(χ) to the area of an 'ellipse' on a sphere in ℝ3. © 1999 Elsevier Science Inc. All rights reserved.