Matrix design for optimal sensing

We design optimal 2 × N (2 <; N) matrices, with unit columns, so that the maximum condition number of all the submatrices comprising 3 columns is minimized. The problem has two applications. When estimating a 2-dimensional signal by using only three of N observations at a given time, this minimizes the worst-case achievable estimation error. It also captures the problem of optimum sensor placement for monitoring a source located in a plane, when only a minimum number of required sensors are active at any given time. For arbitrary N ≥ 3, we derive the optimal matrices which minimize the maximum condition number of all the submatrices of three columns. Surprisingly, a uniform distribution of the columns is not the optimal design for odd N ≥ 7.