Convexity properties of the condition number

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 σ n(A). When this smallest singular value has multiplicity 1, the function A → log(σ n(A) -2) is a convex function with respect to the condition Riemannian structure that is t → log(σ n(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 C 2. When α(x) = d(x,N) -2, where d(x,N) is the distance from x to a C 2 submanifold N ⊂R j, 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 /σ n(A) is self-convex in projective space and the solution variety.