🤖 AI Summary
This paper addresses the statistical inference problem of simultaneous diagonalization for asymmetric matrix families. Methodologically, it integrates matrix perturbation theory, subspace distance metrics, decoupled vector autoregressive (VAR) modeling, and Markov chain stationarity testing to develop hypothesis tests applicable to two-sample, multi-sample, and partial eigenvector-sharing settings, accompanied by efficient computational algorithms. Contributions include: (i) the first formal, statistically grounded framework for testing simultaneous diagonalizability of asymmetric matrices—overcoming the restrictive symmetry assumptions prevalent in prior work; and (ii) a novel identifiability criterion and testable condition for partial eigenvector sharing. Simulation studies demonstrate robust finite-sample performance. Empirical applications include decoupling macroeconomic VAR systems across eight countries and assessing stationarity of river runoff Markov chains, confirming both statistical power and practical interpretability.
📝 Abstract
Abstract 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.
