Recipe

Build the rational canonical form from the characteristic and minimal polynomials

Use when: you know $T$ 's characteristic polynomial $p_T$ and minimal polynomial $m_T$ and need $T$ 's invariant factors, elementary divisors, or its rational canonical form

Goal: the full list of invariant factors, assembled into companion-matrix blocks that build the rational canonical form

Taught in Rational canonical form, invariant factors, and elementary divisors

Quick recipe

Set the last invariant factor equal to the minimal polynomial
Recover the earlier invariant factors from $p_T=f_1f_2\cdots f_r$
Confirm the divisibility chain $f_1\mid f_2\mid\cdots\mid f_r$
Build one companion matrix per invariant factor
Assemble the rational canonical form, and factor for elementary divisors if needed

1
The invariant factors satisfy $f_1\mid f_2\mid\cdots\mid f_r$ with $f_r=m_T$ always; write this one down first.

Example

$T$ is $3\times3$ with $m_T(x)=x^2+x-2$ and $p_T(x)=(x-1)^2(x+2)$ , so set $f_2=x^2+x-2$ .

Why: the minimal polynomial is, by construction, the largest-degree invariant factor of $T$ .
2
Divide the characteristic polynomial by the already-known invariant factor(s); what remains splits into the rest, largest known first.

Example

$p_T/f_2=(x-1)^2(x+2)/(x^2+x-2)$ . Since $x^2+x-2=(x-1)(x+2)$ , the quotient is $(x-1)$ , so $f_1=x-1$ .
- exactly two invariant factors remain (the common case) — the leftover quotient IS $f_1$ directly, with no further splitting needed.
- three or more invariant factors remain — split the leftover quotient further, keeping every prime factor's exponent chain non-decreasing from $f_1$ up to $f_r$ .
Why: the invariant factors multiply back to exactly the characteristic polynomial, so each earlier factor is forced once the later ones are already known.
3
Check that each invariant factor genuinely divides the next one in the list before trusting the split.

Example

$f_1=x-1$ divides $f_2=(x-1)(x+2)$ exactly, with quotient $x+2$ .

Why: a genuine invariant-factor list is defined by exactly this divisibility chain, so skipping the check risks building a canonical form for a factor list that no real operator actually has.
4
For each invariant factor $f_i(x)=x^n+c_{n-1}x^{n-1}+\cdots+c_0$ , form its companion matrix with $1$ 's on the subdiagonal and $-c_0,-c_1,\dots,-c_{n-1}$ down the last column.

Example

The companion matrix of $f_1=x-1$ is the $1\times1$ block $(1)$ ; the companion matrix of $f_2=x^2+x-2$ is $\begin{pmatrix}0&2\\1&-1\end{pmatrix}$ .

Why: a companion matrix's own characteristic and minimal polynomial are always exactly the polynomial it was built from, the exact property this construction needs block by block.
5
Stack the companion matrices block-diagonally for the rational canonical form; over an algebraically closed field, factor each $f_i$ into irreducibles and group same-eigenvalue divisors into Jordan blocks.

Example

$\text{RCF}=\begin{pmatrix}1&0&0\\0&0&2\\0&1&-1\end{pmatrix}$ . Over $\mathbb C$ , the elementary divisors are $(x-1),(x-1),(x+2)$ , giving Jordan blocks of size $1,1,1$ : $J=\text{diag}(1,1,-2)$ .

⚡ Shortcut

If $m_T$ already factors into distinct linear factors, $T$ is diagonalizable: skip straight to one elementary divisor per eigenvalue's own multiplicity, with no companion-matrix bookkeeping needed at all.

Why: this final grouping step is what recovers the familiar Jordan form once the field actually contains every eigenvalue.

Common mistakes

Assuming $p_T/m_T$ directly gives the missing invariant factors without checking the divisibility chain holds.
Confusing an elementary divisor with a full invariant factor; elementary divisors are the prime-power pieces of the invariant factors, not equal to them in general.
Building a companion matrix with the wrong sign convention, forgetting the coefficients must be negated down the final column.
Assuming every $3\times3$ matrix has exactly two invariant factors; the count depends entirely on how many distinct primes appear and with what multiplicities.
Skipping the elementary-divisor grouping step and reporting the wrong number of Jordan blocks for a repeated eigenvalue.