Abstract
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.
Highlights
A matrix is similar to Jordan canonical form over the complex field and the rational canonical form over a number field, respectively
We further study the rational canonical form over any number field
Similar to Jordan canonical forms of a matrix over the complex field, if we find all elementary divisors of a matrix over a number field and rational blocks of these elementary divisors, the direct sum of these rational blocks is precisely the quasi-rational canonical form of the matrix
Summary
A matrix is similar to Jordan canonical form over the complex field and the rational canonical form over a number field, respectively. Jordan and the rational canonical forms of a matrix over the complex field are similar. Li [4] discussed the property of the rational canonical form of a matrix, Liu [5] gave out a constructive proof of existence theorem for rational form, and Radjabalipour [6] investigated the rational canonical form via the splitting field. We further study the rational canonical form over any number field. We give the quasi-rational canonical forms of a matrix by combining Jordan and the rational canonical forms. We show that a matrix is similar to its quasi-rational canonical forms over any number field
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