@article{10902/3208,
year = {2010},
month = {1},
url = {http://hdl.handle.net/10902/3208},
abstract = {We define in the space of n×m matrices of rank n, n ≤ m, the condition Riemannian
structure as follows: For a given matrix A the tangent space at A is equipped with the Hermitian
inner product obtained by multiplying the usual Frobenius inner product by the inverse of the
square of the smallest singular value of A denoted σn(A). When this smallest singular value has
multiplicity 1, the function A → log(σn(A)−2) is a convex function with respect to the condition
Riemannian structure that is t → log(σn(A(t))−2) is convex, in the usual sense for any geodesic
A(t). In a more abstract setting, a function α defined on a Riemannian manifold (M,  ,  ) is said
to be self-convex when log α(γ(t)) is convex for any geodesic in (M, α  ,  ). Necessary and sufficient
conditions for self-convexity are given when α is C2. When α(x) = d(x,N)−2, where d(x,N) is the
distance from x to a C2 submanifold N ⊂Rj, we prove that α is self-convex when restricted to the
largest open set of points x where there is a unique closest point in N to x. We also show, using
this more general notion, that the square of the condition number  A F /σn(A) is self-convex in
projective space and the solution variety.},
publisher = {Society for Industrial and Applied Mathematics},
publisher = {SIAM Journal on Matrix Analysis and Applications, Vol. 31, No. 3, pp. 1491–1506},
title = {Convexity properties of the condition number},
author = {Beltrán Álvarez, Carlos and Dedieu, Jean-Pierre and Malajovich, Gregorio and Shub, Michael},
}