The present work proposes tests for reduced rank in multivariate regression coefficient matrices, under rather general conditions. A heuristic approach is to first estimate the regressions via standard methods, then compare the coefficient matrix rows (or columns) to assess their redundancy. A formal version of this approach utilizes the distance between an unrestricted coefficient matrix estimate and an estimate restricted by reduced rank. Two distance minimization problems emerge, based on equivalent formulations of the null hypothesis. For each method we derive estimators and tests, and their asymptotic distributions. We examine test performance in simulation, and give some numerical examples.