✦ LIBER ✦
The distribution of the maximum condition number on great circles through a fixed 2×2 real matrix
✍ Scribed by Debra Lewis; Mike Shub
- Publisher
- Elsevier Science
- Year
- 1999
- Tongue
- English
- Weight
- 92 KB
- Volume
- 297
- Category
- Article
- ISSN
- 0024-3795
No coin nor oath required. For personal study only.
✦ Synopsis
If P A (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 P A (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 P A (χ) to the area of an 'ellipse' on a sphere in R 3 .