A matrix is similar to Jordan canonical form over the complex field and the rational canonical form over a number field, respectively. In this paper, we further study the rational canonical form of a matrix over any number field. We firstly discuss the elementary divisors of a matrix over a number field. Then, we give the quasi-rational canonical forms of a matrix by combining Jordan and the rational canonical forms. Finally, we show that a matrix is similar to its quasi-rational canonical forms over a number field.
Barone, M., Lima, J.B. and Campello de Souza, R.M. (2016) The Eigenstructure and Jordan Form of the Fourier Transform over Fields of Characteristic 2 and a Generalized Vandermonde-Type Formula. Linear Algebra and Its Applications, 494, 245-262. https://doi.org/10.1016/j.laa.2015.12.021