Lagmental Vicfred

Primary Decomposition Splits an Operator by Coprime Factors by Vicfred

tags: mathematics, eigenvalues, canonical forms, minimal polynomials

Coprime factors of the minimal polynomial produce a direct sum of invariant kernels. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

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.

$$ m_T=f_1^{a_1}\cdots f_r^{a_r},\qquad\gcd(f_i,f_j)=1 $$

I read the first line as input and the second as output. The symbols \(\forall\) and \(\exists\) are not interchangeable, and neither may be upgraded silently to \(\Longleftrightarrow\).

$$ V=\bigoplus_{i=1}^{r}\ker f_i(T)^{a_i} $$

Stress the formula

The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.

$$ 1=\sum_i u_i(t)\prod_{j\ne i}f_j(t)^{a_j}\Longrightarrow I=\sum_i u_i(T)\prod_{j\ne i}f_j(T)^{a_j} $$

Interpretation

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

$$ \begin{aligned} \mathsf{D}\;&:\quad m_T=f_1^{a_1}\cdots f_r^{a_r},\qquad\gcd(f_i,f_j)=1,\\[5pt] \mathsf{C}\;&:\quad V=\bigoplus_{i=1}^{r}\ker f_i(T)^{a_i}. \end{aligned} $$

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.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] V=\bigoplus_{i=1}^{r}\ker f_i(T)^{a_i} \end{gathered}} $$

The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.

This article was posted on Tue 16 April 2013. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.