# Compute rational canonical form of a given matrix

I feel like I am going around in circles in trying to find out HOW to compute the rational canonical form. Everywhere I look has gaps in explanation, making it impossible to follow. I am working out of Dummit & Foote:

RCF is defined as the matrix with the companion matrices of each invariant factor going down the diagonal blocks. So, I look up what invariant factors are defined as.

Definition The invariant factors of an $n \times n$ matrix over a field $F$ are the invariant factors of its rational canonical form.

I hope you can see my confusion/frustration from this, and I was hoping someone could help explain how to compute rational canonical form, or at least how to compute the invariant factors.

## 1 Answer

Finding the rational form is the same as finding the invariant factors. This is done by taking the matrix $xI-A$ and preforming a series of row and column operations to reduce it to Smith normal form, the invariant factors are then the diagonal entries.

You can look in Hoffman&Kunze or Jacobson, Basic Algebra I.