2009/10/30 by Beltrán, Carlos, Dedieu, Jean-Pierre, Malajovich, Gregorio +1
#15A12 #53C23 #65F35 #Differential Geometry (math.DG) #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.0910.5936
In our previous paper [SIMAX 31 n.3 1491-1506(2010)], we studied the condition metric in the space of maximal rank matrices. Here, we show that this condition metric induces a Lipschitz-Riemann 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 (Theorem 1). We also show that a similar result holds for the solution variety of linear systems (Theorem 31). Some of our intermediate results, such as Theorem 12, on the second covariant derivative or Hessian of a function with symmetries on a manifold, and Theorem 29 on piecewise self-convex functions, are of independent interest. Those results were motivated by our investigations on the com- plexity of path-following algorithms for solving polynomial systems.