Lagmental Vicfred

Cayley--Hamilton Substitutes a Matrix into Its Characteristic Polynomial by Vicfred

tags: mathematics, eigenvalues, canonical forms, minimal polynomials

Every square matrix annihilates itself under its own characteristic polynomial. I will separate the object being defined from the consequence being claimed.

Statement

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.

$$ \chi_A(t)=\det(tI-A) $$

There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.

$$ \chi_A(A)=0 $$

Worked algebra

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

$$ A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\Longrightarrow A^2-(a+d)A+(ad-bc)I=0 $$

Conceptual compression

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.

$$ \begin{aligned} \mathsf{D}\;&:\quad \chi_A(t)=\det(tI-A),\\[5pt] \mathsf{C}\;&:\quad \chi_A(A)=0. \end{aligned} $$

Caveat

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \chi_A(A)=0 \end{gathered}} $$

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 Sun 05 April 2009. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.