Lagmental Vicfred

Jordan Form Separates Eigenvalues from Nilpotent Drift by Vicfred

tags: mathematics, eigenvalues, canonical forms, minimal polynomials

Over an algebraically closed field, every operator is similar to blocks lambda I plus a nilpotent shift. This is a compact note, but the quantifiers and hypotheses stay on the page.

The data

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.

$$ J_r(\lambda)=\begin{pmatrix}\lambda&1&&0\\&\lambda&\ddots&\\&&\ddots&1\\0&&&\lambda\end{pmatrix} $$

The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.

$$ A=PJP^{-1} $$

Derivation

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ J_r(\lambda)^n=\sum_{k=0}^{\min(n,r-1)}\binom nk\lambda^{n-k}N^k,\qquad N=J_r(0),\quad N^r=0 $$

Invariant content

The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.

$$ \begin{aligned} \mathsf{D}\;&:\quad J_r(\lambda)=\begin{pmatrix}\lambda&1&&0\\&\lambda&\ddots&\\&&\ddots&1\\0&&&\lambda\end{pmatrix},\\[5pt] \mathsf{C}\;&:\quad A=PJP^{-1}. \end{aligned} $$

Scope

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] A=PJP^{-1} \end{gathered}} $$

A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.

This article was posted on Mon 20 May 2024. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.