Lagmental Vicfred

The Minimal Polynomial Has the Largest Jordan Exponents by Vicfred

Last updated: Sun 25 April 2021

tags: mathematics, eigenvalues, canonical forms, minimal polynomials

For each eigenvalue, the exponent in the minimal polynomial is the size of the largest Jordan block. I want the notation, the mechanism, and the failure mode visible at the same time.

Set-up

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.

$$ m_T(t)=\prod_{\lambda}(t-\lambda)^{s_\lambda} $$

The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.

$$ s_\lambda=\max\{\text{size of a Jordan block for }\lambda\} $$

The calculation

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

$$ J=\begin{pmatrix}\lambda&1&0\\0&\lambda&1\\0&0&\lambda\end{pmatrix}\Longrightarrow(J-\lambda I)^3=0,\quad(J-\lambda I)^2\ne0 $$

What survives abstraction

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

$$ \begin{aligned} \mathsf{D}\;&:\quad m_T(t)=\prod_{\lambda}(t-\lambda)^{s_\lambda},\\[5pt] \mathsf{C}\;&:\quad s_\lambda=\max\{\text{size of a Jordan block for }\lambda\}. \end{aligned} $$

The boundary

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] s_\lambda=\max\{\text{size of a Jordan block for }\lambda\} \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Tue 03 July 2012. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.