  This paper proposes novel methods to test for simultaneous diagonalization of
possibly asymmetric matrices. Motivated by various applications, a two-sample
test as well as a generalization for multiple matrices are proposed. A partial
version of the test is also studied to check whether a partial set of
eigenvectors is shared across samples. Additionally, a novel algorithm for the
considered testing methods is introduced. Simulation studies demonstrate
favorable performance for all designs. Finally, the theoretical results are
utilized to decouple vector autoregression models into multiple univariate time
series, and to test for the same stationary distribution in recurrent Markov
chains. These applications are demonstrated using macroeconomic indices of 8
countries and streamflow data, respectively.
