Testing Simultaneous Diagonalizability

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.

KEYWORDS:

• Common eigenvectors
• Dimension reduction
• Joint diagonalization
• Likelihood ratio test
• Markov chain
• Partially common eigenvectors
• Vector autoregression
• Wald test

Supplementary Materials

The supplementary material includes all appendices. In particular, it includes an alternative approach for the two-sample test, all proofs, and some additional discussions.

Disclosure Statement

The authors report there are no competing interests to declare.

Acknowledgments

The authors thank the anonymous Reviewer and the Associate editor for their comments which helped improving the article.

Additional information

Funding

The authors gratefully acknowledge financial support from a Xerox PARC Faculty Research Award, National Science Foundation Awards 1455172, 1934985, 1940124, and 1940276, USAID, and Cornell University Atkinson Center for a Sustainable Future. This research was conducted with support from the Cornell University Center for Advanced Computing, which receives funding from Cornell University, the National Science Foundation, and members of its Partner Program.