Abstract
We consider proof certificates for identities of rectangular matrices. To automate their construction, we introduce an algebraic framework and supply explicit algorithms relying on non-commutative Gröbner bases. We address not only verification, but also exploration and reasoning towards establishing new identities and even proving mathematical properties. Especially Gröbner-driven elimination theory navigates us to insightful conclusions. We present several applications, that is efficiently formalized proofs for important identities of matrices: cancellation properties of triple products with Moore–Penrose pseudoinverses, a generalized Sherman–Morrison–Woodbury formula and an automated derivation of feedback loops in the famous Youla controller parametrization from control theory.
For actual computations we employ the open source computer algebra system Singular which is used as a backend by systems like SAGE and OSCAR. The non-commutative extension Letterplace provides users with all the required functionality. Singular has numerous conversion tools and supports various standards. Therefore, it can facilitate the integration with existing theorem provers.
Supported by DFG TRR 195 “Symbolic Tools in Mathematics and their applications”.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Bergman, G.: The diamond lemma for ring theory. Adv. Math. 29, 178–218 (1978)
Chenavier, C., Hofstadler, C., Raab, C.G., Regensburger, G.: Compatible rewriting of noncommutative polynomials for proving operator identities. https://arxiv.org/abs/2002.03626 (2020)
Damm, T., Wimmer, H.K.: A cancellation property of the Moore-Penrose inverse of triple products. J. Aust. Math. Soc. 86(1), 33–44 (2009)
Decker, W., Greuel, G.M., Pfister, G., Schönemann, H.: Singular 4-1-3 – A computer algebra system for polynomial computations (2020). http://www.singular.uni-kl.de
Deng, C.Y.: A generalization of the Sherman-Morrison-Woodbury formula. Appl. Math. Lett. 24(9), 1561–1564 (2011)
Grégoire, B., Pottier, L., Théry, L.: Proof certificates for algebra and their application to automatic geometry theorem proving. In: Sturm, T., Zengler, C. (eds.) ADG 2008. LNCS (LNAI), vol. 6301, pp. 42–59. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-21046-4_3
Helton, J., Kronewitter, F.: Computer algebra in the control of singularly perturbed dynamical systems (1999). http://math.ucsd.edu/~ncalg/DELL/SingPert/singpertcdc99.pdf
Hofstadler, C., Raab, C.G., Regensburger, G.: Certifying operator identities via noncommutative Gröbner bases. ACM Commun. Comput. Algebra 53, 49–52 (2019)
Joswig, M., Fieker, C., Horn, M., et al.: The oscar project (2020). https://oscar.computeralgebra.de
Kronewitter, F.D.: Using noncommutative Gröbner bases in solving partially prescribed matrix inverse completion problems. Linear Algebra Appl. 338(1–3), 171–199 (2001)
Levandovskyy, V., Abou Zeid, K., Schönemann, H.: Singular: Letterplace – A singular 4-1-3 subsystem for non-commutative finitely presented algebras (2020). http://www.singular.uni-kl.de
Mora, T.: Groebner bases in non-commutative algebras. In: Gianni, P. (ed.) ISSAC 1988. LNCS, vol. 358, pp. 150–161. Springer, Heidelberg (1989). https://doi.org/10.1007/3-540-51084-2_14
Mora, T.: An introduction to commutative and non-commutative Gröbner bases. Theor. Comput. Sci. 134, 131–173 (1994)
Mora, T.: Solving Polynomial Equation Systems IV: vol. 4. Buchberger Theory and Beyond. Cambridge University Press, Cambridge (2016)
Pritchard, F.L.: The ideal membership problem in non-commutative polynomial rings. J. Symb. Comput. 22(1), 27–48 (1996)
Raab, C.G., Regensburger, G., Poor, J.H.: Formal proofs of operator identities by a single formal computation. https://arxiv.org/abs/1910.06165 (2019)
Stein, W., et al.: Sage Mathematics Software. The Sage Development Team (2020)
Wavrik, J.J.: Rewrite rules and simplification of matrix expressions. Comput. Sci. J. Moldova 4(3), 360–398 (1996)
Youla, D.C., Jabr, H.A., Bongiorno, J.J.: Modern Wiener-Hopf design of optimal controllers. II: the multivariable case. IEEE Trans. Autom. Control 21 319–338 (1976)
Zhou, K., Doyle, J.C., Glover, K.: Robust and Optimal Control. Prentice Hall, Upper Saddle River (1996)
Acknowledgements
We are grateful to Eva Zerz (Aachen) and Bernd Sturmfels (Leipzig) for fruitful discussions. We also thank Mariia Anapolska and Sven Gross for carefully reading preliminary versions of this article. The authors have been supported by Project II.6 of SFB-TRR 195 “Symbolic Tools in Mathematics and their Applications” of the German Research Foundation (DFG).
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Schmitz, L., Levandovskyy, V. (2020). Formally Verifying Proofs for Algebraic Identities of Matrices. In: Benzmüller, C., Miller, B. (eds) Intelligent Computer Mathematics. CICM 2020. Lecture Notes in Computer Science(), vol 12236. Springer, Cham. https://doi.org/10.1007/978-3-030-53518-6_14
Download citation
DOI: https://doi.org/10.1007/978-3-030-53518-6_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-53517-9
Online ISBN: 978-3-030-53518-6
eBook Packages: Computer ScienceComputer Science (R0)