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

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 ‘ellipseʹ on a sphere in