Testing Simultaneous Diagonalizability

TL;DR

This paper introduces new statistical tests and algorithms for assessing whether multiple matrices can be simultaneously diagonalized, with applications in time series analysis and Markov chain comparison.

Contribution

It develops novel methods and algorithms for testing simultaneous diagonalizability, including partial tests and applications to VAR models and Markov chains.

Findings

Simulation results show high accuracy of the tests.

Applications successfully decouple VAR models into univariate series.

Tests effectively determine shared eigenvectors across samples.

Abstract

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.