Testing Simultaneous Diagonalizability

This article 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 used to decouple multiple vector autoregression models into univariate time series, and to test for the same stationary distribution in recurrent Markov chains. These applications are demonstrated using macroeconomic indices of eight countries and streamflow data, respectively. Supplementary materials for this article are available online.