Jordan Canonical Form

The rational canonical form uses the invariant-factor decomposition of the $F [x]$ -module $V_{T}$ . The elementary-divisor decomposition of the same module gives a different matrix — one that is nearly diagonal, at the cost of needing the eigenvalues to lie in the field.

Throughout, $V$ is a finite-dimensional vector space over $F$ , $T : V \to V$ is a linear operator, $V_{T}$ is $V$ made into an $F [x]$ -module by $x \cdot v = T (v)$ , and $A$ is a matrix of $T$ .

Splitting into linear prime powers

The elementary divisors of $V_{T}$ are the prime-power factors of its invariant factors. Over $F [x]$ the primes are the monic irreducible polynomials, so an elementary divisor is $p (x)^{k}$ for some irreducible $p$ . To make it as simple as possible, assume every irreducible factor is linear.

Under this assumption the elementary divisors are powers $(x - λ)^{k}$ with $λ \in F$ an eigenvalue, and the structure theorem gives

V_{T} ≅ i ⨁ F [x] / ((x - λ_{i})^{k_{i}}) .

Each summand is a cyclic $F [x]$ -module, and the task is to choose a good basis for it.

The Jordan block

Fix one summand $F [x] / ((x - λ)^{k})$ . Instead of the coefficient basis $1, x, \dots, x^{k - 1}$ used for companion matrices, take the basis built from powers of $(x - λ)$ :

(x - λ)^{k - 1}, (x - λ)^{k - 2}, \dots, (x - λ), 1.

This is a basis because the change from the coefficient basis is upper triangular with $1$ 's on the diagonal. Writing $x = λ + (x - λ)$ and using $(x - λ)^{k} = 0$ in the quotient, multiplication by $x$ acts as

(x - λ)^{j} ⟼ λ (x - λ)^{j} + (x - λ)^{j + 1},

so on each basis vector $T$ returns $λ$ times itself plus the next vector up. The matrix is diagonal with an extra superdiagonal of ones.

A single Jordan block of size

k

: the eigenvalue (here drawn as

d

for

λ

) repeated on the diagonal, ones on the first superdiagonal, zeros elsewhere.

A Jordan block is the sum of the scalar $λ I$ and a nilpotent shift on the superdiagonal. When $k = 1$ the block is just the scalar $λ$ , and the shift disappears.

Assembling the form

Applying this basis to every cyclic summand assembles the whole operator.

The Jordan canonical form: one Jordan block per elementary divisor, stacked along the diagonal, zeros off it.

The blocks for a fixed eigenvalue can share that eigenvalue in different sizes, so the same $λ$ may head several blocks.

Because a Jordan block differs from a diagonal matrix only by the superdiagonal ones, the Jordan form is diagonal exactly when every block has size $1$ .

Diagonalizability is therefore a property of the minimal polynomial alone, not the characteristic polynomial: repeated eigenvalues are fine, repeated factors in $m_{A}$ are not.

Generalized eigenspaces

Grouping the Jordan blocks by eigenvalue reassembles the primary decomposition of $V_{T}$ in operator language.

On the generalized eigenspace, $T - λ$ is nilpotent, and its Jordan blocks are the shift-only matrices with zeros on the diagonal. The kernels of successive powers of $T - λ$ form an increasing flag that stabilizes at the whole generalized eigenspace.

On the generalized eigenspace for

λ

, the kernels

K_{j} = ker (T - λ)^{j}

nest and grow until they fill the space; the largest Jordan block sets how many steps this takes.

The flag makes the block structure computable. Let $r_{k} = dim (T - λ)^{k} V$ be the rank of the $k$ -th power. The count

r_{k - 1} - 2 r_{k} + r_{k + 1}

is the number of Jordan blocks of size exactly $k$ for the eigenvalue $λ$ . Taking $k = 1$ (with $r_{0} = dim V$ ) gives the total number of blocks for $λ$ , which equals the geometric multiplicity $dim ker (T - λ)$ . This gives a rank-only route to the Jordan form: compute the ranks of the powers $(A - λ I)^{k}$ and difference them.³

Rational and Jordan forms of the same operator

Both canonical forms come from the same module, through its two decompositions. Passing between them is the invariant-factor / elementary-divisor conversion of the first lesson, run on polynomials.

	Rational canonical form	Jordan canonical form
Module decomposition	invariant factors $a_{1} ∣ \dots ∣ a_{m}$	elementary divisors $(x - λ)^{k}$
Building block	companion matrix $C_{a_{i}}$	Jordan block $J_{k} (λ)$
Field requirement	any field (rational)	must contain the eigenvalues
Shape	block companion	nearly diagonal
Uniqueness	unique	unique up to block order
Diagonal case	rarely diagonal	diagonal iff $m_{A}$ squarefree

The same operator in both canonical forms. The rational form uses companion blocks of the invariant factors; the Jordan form uses blocks of the elementary divisors obtained by splitting those factors over the field.

Worked examples

The three matrices from the rational canonical form computation have eigenvalues $2$ and $3$ , both in $Q$ , so their Jordan forms exist over $Q$ :

A = 200 - 2 30 14 - 7 2, B = 200120003, C = 200220 - 1 33 .

For $3 \times 3$ matrices the characteristic and minimal polynomials determine the Jordan form. From $4 \times 4$ on they do not, and the rank counts take over.

Worked example. Find the Jordan canonical form of

D = 0010 - 1 20 - 1 - 1 020 01 - 1 0 over Q .

Expanding $det (x I - D)$ gives the characteristic polynomial $(x - 1)^{4}$ , so the only eigenvalue is $1$ and the minimal polynomial is $(x - 1)^{k}$ for some $k \leq 4$ . Compute

D - I = - 1 010 - 1 10 - 1 - 1 010 01 - 1 - 1 \neq = 0, (D - I)^{2} = 0,

so $m_{D} = (x - 1)^{2}$ . Two invariant-factor lists are consistent with $c_{D} = (x - 1)^{4}$ and $m_{D} = (x - 1)^{2}$ :

$(x - 1), (x - 1), (x - 1)^{2}$ — three blocks of sizes $1, 1, 2$ ;
$(x - 1)^{2}, (x - 1)^{2}$ — two blocks of sizes $2, 2$ .

The rank of $D - I$ decides. In $D - I$ , row $4$ is the negative of row $2$ , and row $3$ is the negative of the sum of rows $1$ and $2$ , so only two rows are independent: $r_{1} = rank (D - I) = 2$ . With $r_{0} = 4$ and $r_{2} = r_{3} = 0$ , the block counts are

blocks of size 1 r_{0} - 2 r_{1} + r_{2} = 4 - 4 + 0 = 0, blocks of size 2 r_{1} - 2 r_{2} + r_{3} = 2 - 0 + 0 = 2,

so the second list is correct: elementary divisors $(x - 1)^{2}, (x - 1)^{2}$ and

JCF (D) = 1000110000100011 .

The polynomials $c_{D}$ and $m_{D}$ alone could not separate the two candidate structures; the rank computation supplied the missing invariant.⁴

Worked example. Count the similarity classes of $4 \times 4$ matrices over $Q$ with characteristic polynomial $(x - 1)^{4}$ . Each class is a multiset of elementary divisors $(x - 1)^{k_{1}}, \dots, (x - 1)^{k_{t}}$ with $k_{1} + \dots + k_{t} = 4$ — a partition of $4$ . There are five partitions, hence five classes:

partition	Jordan blocks	$m (x)$	$rank (A - I)$
$4$	$J_{4} (1)$	$(x - 1)^{4}$	$3$
$3 + 1$	$J_{3} (1) \oplus J_{1} (1)$	$(x - 1)^{3}$	$2$
$2 + 2$	$J_{2} (1) \oplus J_{2} (1)$	$(x - 1)^{2}$	$2$
$2 + 1 + 1$	$J_{2} (1) \oplus J_{1} (1) \oplus J_{1} (1)$	$(x - 1)^{2}$	$1$
$1 + 1 + 1 + 1$	$I$	$x - 1$	$0$

The rank column is $4$ minus the number of blocks. The pairs $2 + 2$ and $2 + 1 + 1$ share the minimal polynomial $(x - 1)^{2}$ and are separated only by the rank — the same phenomenon as the matrix $D$ above.

The five similarity classes with characteristic polynomial

(x - λ)^{4}

: one Jordan structure per partition of

4

. Shaded squares are the Jordan blocks; everything off the blocks is zero.

Field dependence

The Jordan form exists only when the eigenvalues lie in $F$ , and enlarging the field can create new blocks. Consider a $6 \times 6$ operator over $Q$ with an elementary divisor $x^{2} + 1$ . Over $Q$ this factor is irreducible, so it contributes a $2 \times 2$ companion block and no Jordan block exists. Over $C$ it splits as $(x - i) (x + i)$ , adding two size- $1$ blocks with eigenvalues $\pm i$ . The rational canonical form is the same over both fields; the Jordan form appears only after adjoining the roots. This is the sense in which the rational form is rational and the Jordan form is not.⁵

When the field is not algebraically closed, the general-purpose canonical form is the rational one; the Jordan form is the specialization available once the characteristic polynomial splits.

Dummit & Foote, §12.3 — Theorems 22 and 23: existence of the Jordan canonical form for operators and matrices when $F$ contains the eigenvalues, and uniqueness up to permutation of the blocks. ↩
Dummit & Foote, §12.3 — Corollaries 24 and 25: a matrix is diagonalizable if and only if its minimal polynomial has no repeated roots, in which case the Jordan form is diagonal. ↩
Dummit & Foote, §12.3 — Exercises 29 and 30 and the definition of the generalized eigenspace as the $(x - λ)$ -primary component: the number of Jordan blocks of size $k$ for $λ$ is $r_{k - 1} - 2 r_{k} + r_{k + 1}$ with $r_{k} = dim (T - λ)^{k} V$ . ↩
Dummit & Foote, §12.3 — Examples 1–3: the Jordan forms of the matrices from Example 1 of §12.2, and a $4 \times 4$ matrix with invariant factors $(x - 1)^{2}, (x - 1)^{2}$ illustrating that the characteristic and minimal polynomials do not determine similarity for $n \geq 4$ . The matrix $D$ here is a different representative of that same similarity class, chosen so the rank computation is checkable by eye. ↩
Dummit & Foote, §12.3 — Example 4 and Corollary 18 of §12.2: over $C$ an elementary divisor $x^{2} + 1$ splits into blocks with eigenvalues $\pm i$ , while the rational canonical form is unchanged by the field extension. ↩