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The product of the invariant factors is the characteristic polynomial

Statement

If $f_{1}, \dots, f_{r}$ are the invariant factors of an endomorphism $T$ , then the product of the invariant factors is the characteristic polynomial:

$χ_{T} (x) = \prod_{i = 1}^{r} f_{i} (x) .$

On the zero space this is the empty product $1$ .

Facts & Assumptions

Given: The characteristic polynomial of The basis-independent characteristic polynomial $χ_{T}$ of an endomorphism of a finite-dimensional space, including $χ_{T} = 1$ in dimension zero, the companion-matrix convention of The companion matrix of a monic polynomial, and multiplication of characteristic polynomials across block triangular matrices (The characteristic polynomial of a block upper- or lower-triangular matrix is the product of the characteristic polynomials of its diagonal blocks).

[L1]

In rational canonical form, the blocks are the companion matrices of the invariant factors (Existence and uniqueness of rational canonical form).

Proof

technique · direct

1.1

Expanding $\det (x I - C (f_{i}))$ along the companion rows gives the monic polynomial $f_{i} (x)$ ; this includes zero coefficients and linear companion matrices.

givenalgebra

2.1

By [L1], $T$ is similar to the block diagonal matrix with blocks $C (f_{i})$ . Characteristic polynomials are similarity-invariant and multiply over block diagonals, so step 1.1 gives $χ_{T} = \prod_{i} f_{i}$ . With no blocks, the determinant and product are both $1$ .

step 1.1L1given∎

Depends on

Used by

Rational canonical form of an explicit four-by-four matrix Example
FALSE: characteristic and minimal polynomials determine similarity False statement
Cayley-Hamilton by the PID-module structure theorem Theorem
Jordan canonical form from the elementary divisors of V_T Theorem

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

A. Apisa, Wisconsin Math 542, Lemma 33 (standard reference, not scraped)
M. Brussel, Finitely Generated Modules over a PID, Section 5.2 (standard reference, not scraped)