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 ∈ R 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, x 6=0 ‖Ax‖ ‖x‖ , Gmin = inf x∈V, x 6=0 ‖Ax‖ ‖x‖ , respectively, where ‖ ‖ denotes the Euclidean norm. When A is nonsingular, we define its condition number on the subspace V as κV(A) = Gmax/Gmin. The condition number of A on any one-dimensional subspace is 1, and its condition number on V = R is the (usual) condition number of A, which we denote κ(A). The condition number on any subspace is between 1 and κ(A). If κV(A) = 1, we say that A is isotropic on V , since its gain ‖Ax‖/‖x‖ is the same for any nonzero vector x ∈ V. In this note we address the following problem: Given a nonsingular matrix A ∈ R, and k ∈ {1, . . . , n}, find a subspace V ⊆ R of dimension k which minimizes κV(A). The number κV(A) is a measure of the anisotropy of the linear function induced by A, restricted to the subspace V , so our problem is to find a subspace of dimension k on which A is maximally isotropic.