Sensing with Optimal Matrices

We consider the problem of designing optimal M × N (M ≤ N ) sensing matrices which minimize the maximum condition number of all the submatrices of K columns. Such matrices minimize the worst-case estimation errors when only K sensors out of N sensors are available for sensing at a given time. For M = 2 and matrices with unit-normed columns, this problem is equivalent to the problem of maximizing the minimum singular value among all the submatrices of K columns. For M = 2, we are able to give a closed form formula for the condition number of the submatrices. When M = 2 and K = 3, for an arbitrary N ≥ 3, we derive the optimal matrices which minimize the maximum condition number of all the submatrices of K columns. Surprisingly, a uniformly distributed design is often not the optimal design minimizing the maximum condition number.