Lagmental Vicfred

Rational Canonical Form Works without Splitting Polynomials by Vicfred

tags: mathematics, eigenvalues, canonical forms, minimal polynomials

Companion matrices of invariant factors give a canonical form over any field. I want the notation, the mechanism, and the failure mode visible at the same time.

Notation

An endomorphism \(T\in\operatorname{End}(V)\) carries two canonical polynomials: the characteristic polynomial \(\chi_T\) and minimal polynomial \(m_T\). Their factorizations control invariant subspaces.

$$ f_1\mid f_2\mid\cdots\mid f_r $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ A\sim C(f_1)\oplus\cdots\oplus C(f_r) $$

Stress the formula

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ f(t)=t^n+a_{n-1}t^{n-1}+\cdots+a_0,\qquad C(f)=\begin{pmatrix}0&0&\cdots&-a_0\\1&0&\cdots&-a_1\\0&1&\cdots&-a_2\\\vdots&\ddots&\ddots&\vdots\\0&\cdots&1&-a_{n-1}\end{pmatrix} $$

Interpretation

A good test for understanding is to change the presentation while keeping the invariant fixed. The aligned form makes that comparison unusually easy.

$$ \begin{aligned} \mathsf{D}\;&:\quad f_1\mid f_2\mid\cdots\mid f_r,\\[5pt] \mathsf{C}\;&:\quad A\sim C(f_1)\oplus\cdots\oplus C(f_r). \end{aligned} $$

Limit of the argument

An eigenvalue list does not determine a matrix up to similarity. Jordan block sizes or invariant factors contain the missing data.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] A\sim C(f_1)\oplus\cdots\oplus C(f_r) \end{gathered}} $$

The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.

This article was posted on Thu 07 May 2015. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.