@article{10902/3206,
year = {2012},
month = {8},
url = {http://hdl.handle.net/10902/3206},
abstract = {In our previous paper [SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1491–1506],
we studied the condition metric in the space of maximal rank n × m matrices. Here, we show that
this condition metric induces a Lipschitz Riemannian structure on that space. After investigating
geodesics in such a nonsmooth structure, we show that the inverse of the smallest singular value
of a matrix is a log-convex function along geodesics. We also show that a similar result holds for
the solution variety of linear systems. Some of our intermediate results such as those on the second
covariant derivative or Hessian of a function with symmetries on a manifold, and those on piecewise
self-convex functions, are of independent interest. Those results were motivated by our investigations
on the complexity of path-following algorithms for solving polynomial systems.},
publisher = {Society for Industrial and Applied Mathematics},
publisher = {SIAM Journal on Matrix Analysis and Applications, Vol. 33, No. 3, pp. 905–939},
title = {Convexity properties of the condition number II},
author = {Beltrán Álvarez, Carlos and Dedieu, Jean-Pierre and Malajovich, Gregorio and Shub, Michael},
}