Nikita 

Universitas Gadjah Mada, Sleman, Yogyakarta

Ari Suparwanto

Universitas Gadjah Mada, Sleman, Yogyakarta

Sutopo

Universitas Gadjah Mada, Sleman, Yogyakarta

DOI: https://doi.org/10.19184/mims.v24i2.35918

ABSTRACT 

Generally, discussion about diagonalization of matrices in linear algebra is a matrix over the field. This research presents the diagonalization of matrices over commutative rings. Previous studies have explained the diagonalization of the matrix over a commutative ring, but there are some shortcomings in it. Therefore, this paper will present a matrix diagonalization process that could overcome these shortcomings. This research proposes a method for diagonalization matrices where the characteristic polynomial splits completely over the image of a ring homomorphism. Furthermore, the diagonalization is done over ring localization, so that there are more commutative ring matrices which can be diagonalized in this way. Meanwhile, the sufficient condition for a matrix which can be diagonalized in this thesis is when the determinant of the matrix whose columns are the eigenvectors is regular. Furthermore, to show this diagonalization method applies in general, given a special matrix n × n which satisfies the sufficient condition.

Keywords: Matrices, diagonalization, eigenvector, determinant, localization

MSC2020: 15A09, 15A18, 15A20,13B05,13B20