An open letter concerning Subspaces that Minimize the Condition Number of a Matrix

We define the condition number of a nonsingular matrix on a subspace, and consider the problem of finding a subspace of given dimension that minimizes the condition number of a given matrix. We give a general solution to this problem, and show in particular that when the given dimension is less than half the dimension of the matrix, a subspace can be found on which the condition number of the matrix is one. 1 The problem Suppose A ∈ Rn×n and V ⊆ R is a subspace with dimV = k ≥ 1. We define the maximum gain (minimum gain) of A on V , as Gmax = sup x∈V, x6=0 ‖Ax‖ ‖x‖ , Gmin = inf x∈V, x6=0 ‖Ax‖ ‖x‖ , respectively, where ‖ ‖ denotes the Euclidean norm. When A is nonsingular, we define its condition number on the subspace V as