The Jordan normal form

Theorem

The Jordan normal form

T-049Home MU-202Threads structure

Statement

Let \( V \) be a nonzero finite-dimensional vector space over \( \mathbb{C} \) (more generally, over any field \( F \) such that the characteristic polynomial of \( T \) splits into linear factors over \( F \)), and let \( T : V \to V \) be a linear operator. Then there exists a basis of \( V \) in which the matrix of \( T \) is a direct sum of Jordan blocks \[ J_k(\lambda) \;=\; \begin{pmatrix} \lambda & 1 & & \\ & \lambda & \ddots & \\ & & \ddots & 1 \\ & & & \lambda \end{pmatrix} \in M_k(\mathbb{C}), \] i.e. \( [T] = J_{k_1}(\lambda_1) \oplus J_{k_2}(\lambda_2) \oplus \cdots \oplus J_{k_r}(\lambda_r) \), where the \( \lambda_j \) are eigenvalues of \( T \) (repetitions allowed). Moreover the multiset of blocks \( \{ (k_j, \lambda_j) \} \) is uniquely determined by \( T \): two operators (equivalently, two matrices in \( M_n(\mathbb{C}) \)) are similar if and only if they have the same Jordan blocks up to reordering.

Why it matters

The Jordan normal form is the complete classification of linear operators on a finite-dimensional complex vector space up to similarity. Diagonalisation fails precisely when eigenvectors run out; the Jordan form quantifies that failure exactly, replacing "diagonalisable" with the universally valid "diagonalisable plus nilpotent, in commuting pieces". Every similarity-invariant question about a single operator — minimal polynomial, ranks of powers of \( T - \lambda \), matrix functions \( f(T) \), solutions of \( \dot{x} = Ax \) — can be answered by inspecting the block data.

It is also the first structure theorem most students meet: the statement "a finitely generated torsion module over a PID is a direct sum of cyclic modules" specialises, for the ring \( \mathbb{C}[x] \) acting on \( V \) via \( T \), to exactly this theorem. Understanding the Jordan form well is a rehearsal for module theory, representation theory, and the study of unipotent and nilpotent elements in Lie theory.

Hypotheses

Hypothesis.\( V \) is finite-dimensional. Sketch of failure: on \( V = \mathbb{C}[x] \) the multiplication operator \( M : f \mapsto x f \) is injective but has no eigenvalue at all (\( x f = \lambda f \) forces \( f = 0 \) by comparing degrees), so no basis of generalized eigenvectors can exist; likewise the right shift on \( \mathbb{C}^{\mathbb{N}} \) has empty point spectrum. No decomposition into Jordan blocks, finite or infinite, represents these operators. Hypothesis.The scalar field is \( \mathbb{C} \) — or at least the characteristic polynomial of \( T \) splits over the field. Sketch of failure: the rotation \( \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \) over \( \mathbb{R} \) has characteristic polynomial \( x^2 + 1 \) with no real roots; it has no real eigenvector, hence is similar to no real upper-triangular matrix, let alone a Jordan form. Over \( \mathbb{R} \) one must settle for the real Jordan form with \( 2 \times 2 \) rotation-scaling blocks. Hypothesis.\( T \) is a single linear operator, i.e. \( V \) is a module over the PID \( \mathbb{C}[x] \), not over a larger ring. Sketch of failure: for two commuting operators \( S, T \) (a \( \mathbb{C}[x,y] \)-module; \( \mathbb{C}[x,y] \) is not a PID) there is in general no basis putting both in Jordan form simultaneously with matching blocks — e.g. \( S = J_2(0) \) and \( T = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \) do not commute, and even commuting pairs such as \( S = J_2(0), \, T = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} + I \) admit no canonical joint block classification of this type: the classification of pairs of commuting matrices up to simultaneous similarity is a wild problem.

Proof

Write \( n = \dim V \geq 1 \). The proof has three movements: (I) split \( V \) into generalized eigenspaces (steps 1–6); (II) prove the structure theorem for nilpotent operators by induction (steps 7–11); (III) assemble, and prove uniqueness (steps 12–13).

\[ \chi_T(x) \;=\; \det\left( x\,\mathrm{id} - T \right) \;=\; \prod_{i=1}^{r} (x - \lambda_i)^{m_i}, \qquad \lambda_1, \dots, \lambda_r \in \mathbb{C} \text{ distinct}, \quad \sum_{i=1}^r m_i = n. \]

The characteristic polynomial is monic of degree \( n \geq 1 \); the Fundamental Theorem of Algebra factors it completely over \( \mathbb{C} \). Each \( \lambda_i \) is an eigenvalue since \( \det(\lambda_i \,\mathrm{id} - T) = 0 \). A

Define the generalized eigenspaces \( V_i := \ker\left( (T - \lambda_i \,\mathrm{id})^{m_i} \right) \). Each \( V_i \) is \( T \)-invariant.

\( T \) commutes with every polynomial in \( T \), in particular with \( (T - \lambda_i)^{m_i} \); the kernel of an operator commuting with \( T \) is \( T \)-invariant: if \( (T-\lambda_i)^{m_i} v = 0 \) then \( (T-\lambda_i)^{m_i} T v = T (T-\lambda_i)^{m_i} v = 0 \). A

Set \( p_i(x) := \dfrac{\chi_T(x)}{(x - \lambda_i)^{m_i}} = \prod_{j \neq i} (x - \lambda_j)^{m_j} \). Then \( \gcd(p_1, \dots, p_r) = 1 \), so there exist \( q_1, \dots, q_r \in \mathbb{C}[x] \) with \[ \sum_{i=1}^r q_i(x)\, p_i(x) = 1. \]

No \( \lambda_j \) is a root of every \( p_i \) (indeed \( p_j(\lambda_j) \neq 0 \)), so the \( p_i \) have no common irreducible factor. Bézout’s identity in the Euclidean domain \( \mathbb{C}[x] \) (extended to \( r \) polynomials by induction on \( r \)) produces the \( q_i \). This is the key arithmetic input. B

Define \( E_i := q_i(T)\, p_i(T) \in \mathrm{End}(V) \). Then \( \sum_{i=1}^r E_i = \mathrm{id}_V \) and \( \operatorname{im} E_i \subseteq V_i \), hence \( V = V_1 + \cdots + V_r \).

Evaluating the Bézout identity of step 3 at \( T \) (the evaluation map \( \mathbb{C}[x] \to \mathrm{End}(V) \) is a ring homomorphism) gives \( \sum E_i = \mathrm{id}_V \). For any \( v \in V \): \( (T - \lambda_i)^{m_i} E_i v = q_i(T)\, \chi_T(T)\, v = 0 \) by the Cayley–Hamilton theorem (proved earlier in MU-202: \( \chi_T(T) = 0 \)), so \( E_i v \in V_i \). Then \( v = \sum_i E_i v \in \sum_i V_i \). C

The sum is direct: \( V = V_1 \oplus \cdots \oplus V_r \), and \( E_i \) is the projection onto \( V_i \) killing every \( V_j \), \( j \neq i \).

First, \( E_i|_{V_j} = 0 \) for \( j \neq i \): if \( v \in V_j \) then \( (x - \lambda_j)^{m_j} \) divides \( p_i \), so \( p_i(T) v = 0 \), hence \( E_i v = 0 \). Second, for \( v \in V_i \): \( v = \sum_j E_j v = E_i v \) since all other terms vanish. Now suppose \( 0 = v_1 + \cdots + v_r \) with \( v_i \in V_i \); applying \( E_i \) gives \( 0 = E_i(v_1 + \cdots + v_r) = v_i \) for each \( i \). Directness follows from the definition of internal direct sum. C

On each summand, \( N_i := (T - \lambda_i \,\mathrm{id})\big|_{V_i} \) is a nilpotent operator on \( V_i \): \( N_i^{m_i} = 0 \). Thus \( T|_{V_i} = \lambda_i \,\mathrm{id}_{V_i} + N_i \), and the theorem reduces to: every nilpotent operator on a nonzero finite-dimensional space admits a basis that is a disjoint union of Jordan chains.

\( N_i^{m_i} = 0 \) on \( V_i \) is precisely the definition \( V_i = \ker (T-\lambda_i)^{m_i} \), and \( N_i \) maps \( V_i \) into itself by step 2. If \( N_i \) is a direct sum of nilpotent Jordan blocks \( J_k(0) \) in some basis, then \( T|_{V_i} = \lambda_i I + N_i \) is the direct sum of the blocks \( J_k(\lambda_i) \) in the same basis, and \( V = \bigoplus V_i \) concatenates these bases. B

Nilpotent structure lemma, setup. Let \( N : W \to W \) be nilpotent, \( 0 \lt \dim W \lt \infty \). A Jordan chain of length \( k \) is a list \( N^{k-1}w, \, N^{k-2}w, \dots, Nw, \, w \) with \( N^k w = 0 \) and \( N^{k-1} w \neq 0 \). We prove by strong induction on \( \dim W \) that \( W \) has a basis which is a disjoint union of Jordan chains. If \( N = 0 \), any basis is a union of chains of length \( 1 \); done. Otherwise set \( U := N(W) \neq 0 \); then \( U \) is \( N \)-invariant and \[ \dim U = \dim W - \dim \ker N \lt \dim W, \]

\( N(U) \subseteq N(W) = U \) gives invariance. The dimension formula is rank–nullity; the strict inequality holds because \( \ker N \neq 0 \): a nilpotent operator on a nonzero space kills some nonzero vector (take \( v \neq 0 \) and the last nonzero vector among \( v, Nv, N^2 v, \dots \)). A

Inductive hypothesis: \( N|_U \) (nilpotent on \( U \), \( \dim U \lt \dim W \)) has a basis of Jordan chains. Say the chains are generated by \( u_1, \dots, u_s \in U \) with lengths \( k_1, \dots, k_s \geq 1 \): the vectors \[ \left\{ N^{a} u_j : 1 \leq j \leq s, \; 0 \leq a \leq k_j - 1 \right\} \] form a basis of \( U \), with \( N^{k_j} u_j = 0 \).

Direct application of the strong induction hypothesis to the operator \( N|_U \) on the strictly smaller space \( U \) (using step 7; if \( U = 0 \) we were already done). A

Lifting: since \( u_j \in U = N(W) \), choose \( w_j \in W \) with \( N w_j = u_j \). Each chain of \( U \) extends to a chain of \( W \) of length \( k_j + 1 \): \[ N^{k_j} w_j = N^{k_j - 1} u_j, \;\; \dots, \;\; N w_j = u_j, \;\; w_j . \]

Surjectivity of \( N : W \to U \) onto its image is the definition of \( U \); the displayed list is a Jordan chain because \( N^{k_j + 1} w_j = N^{k_j} u_j = 0 \) while \( N^{k_j} w_j = N^{k_j-1} u_j \neq 0 \) (it is a basis vector of \( U \)). This lifting is the key idea of the whole construction. B

The chain bottoms \( b_j := N^{k_j - 1} u_j \) \( (j = 1, \dots, s) \) form a basis of \( \ker N \cap U \).

They lie in \( \ker N \cap U \) and are linearly independent (they are distinct members of the basis of \( U \) from step 8). Spanning: let \( v \in \ker N \cap U \) and expand \( v = \sum_{j,a} c_{j,a} N^a u_j \) in the chain basis of \( U \). Apply \( N \): the map \( N \) sends each non-bottom basis vector \( N^a u_j \) (\( a \leq k_j - 2 \)) to the basis vector \( N^{a+1} u_j \), and these images are pairwise distinct elements of the basis of \( U \); the bottoms are sent to \( 0 \). Hence \( 0 = Nv = \sum_{j} \sum_{a \leq k_j - 2} c_{j,a} N^{a+1} u_j \) is a linear relation among distinct basis vectors, forcing \( c_{j,a} = 0 \) for all \( a \leq k_j - 2 \). So \( v = \sum_j c_{j, k_j - 1} b_j \), as required. C

Extend \( \{ b_1, \dots, b_s \} \) to a basis \( \{ b_1, \dots, b_s, z_1, \dots, z_t \} \) of \( \ker N \) (each \( z_l \) is a chain of length \( 1 \)). Claim: \[ \mathcal{B} \;=\; \left\{ N^a w_j : 1 \leq j \leq s, \; 0 \leq a \leq k_j \right\} \cup \{ z_1, \dots, z_t \} \] is a basis of \( W \) consisting of Jordan chains. This closes the induction.

Count: \( |\mathcal{B}| = \sum_j (k_j + 1) + t = \dim U + s + t = \dim U + \dim \ker N = \dim W \), using step 10 (so \( \dim \ker N = s + t \)) and rank–nullity. It therefore suffices to prove linear independence. Suppose \( \sum_{j} \sum_{a=0}^{k_j} c_{j,a} N^a w_j + \sum_l d_l z_l = 0 \). Apply \( N \): using \( N^{a+1} w_j = N^a u_j \) and \( N z_l = 0 \), \( N^{k_j+1} w_j = 0 \), we get \( \sum_j \sum_{a=0}^{k_j - 1} c_{j,a} N^{a} u_j = 0 \), a relation among the basis vectors of \( U \) from step 8; hence \( c_{j,a} = 0 \) for all \( 0 \leq a \leq k_j - 1 \). The original relation collapses to \( \sum_j c_{j,k_j} b_j + \sum_l d_l z_l = 0 \), a relation among the basis of \( \ker N \), so all remaining coefficients vanish. Independence plus the dimension count makes \( \mathcal{B} \) a basis (a linearly independent set of size \( \dim W \) spans). C

Assembly. Order each chain bottom-to-top: \( e_1 = N^{k-1}w, \dots, e_k = w \). Then \( N e_1 = 0 \) and \( N e_{a} = e_{a-1} \) for \( a \geq 2 \), so the matrix of \( N \) on the chain’s span is exactly \( J_k(0) \). Applying the lemma to each \( N_i \) on \( V_i \) (step 6) and concatenating bases over \( i = 1, \dots, r \) yields a basis of \( V \) in which \[ [T] \;=\; \bigoplus_{i=1}^{r} \bigoplus_{\text{chains in } V_i} J_{k}(\lambda_i). \] Existence is proved.

Reading off matrix columns: the \( a \)-th column of \( [N] \) records \( N e_a \), giving ones on the superdiagonal and zeros elsewhere, i.e. \( J_k(0) \); then \( [T|_{V_i}] = \lambda_i I + [N_i] \) turns each \( J_k(0) \) into \( J_k(\lambda_i) \). The direct sum decomposition of step 5 makes the union of the bases a basis of \( V \) and gives \( [T] \) block-diagonal form. A

Uniqueness. For \( \lambda \in \mathbb{C} \) and \( p \geq 0 \) write \( r_p(\lambda) := \operatorname{rank}\left( (T - \lambda\,\mathrm{id})^p \right) \). Then for every \( k \geq 1 \), \[ \#\{\text{blocks } J_k(\lambda) \text{ in any Jordan form of } T\} \;=\; r_{k-1}(\lambda) - 2\,r_k(\lambda) + r_{k+1}(\lambda). \] Since each \( r_p(\lambda) \) is defined from \( T \) alone (and is a similarity invariant), the multiset of blocks is uniquely determined by \( T \). Conversely, matrices with the same block multiset are similar, both being similar to the common block-diagonal matrix.

Ranks add over direct sums, so it suffices to compute on a single block. For \( J_k(\mu) \): if \( \mu \neq \lambda \) then \( J_k(\mu) - \lambda I \) is invertible and contributes \( k \) to every \( r_p(\lambda) \), hence \( 0 \) to the alternating combination. If \( \mu = \lambda \), then \( (J_k(\lambda) - \lambda I)^p = J_k(0)^p \) has rank \( \max(k - p, 0) \) (it shifts basis vectors \( p \) places down a chain). So a block \( J_m(\lambda) \) contributes \( \max(m-k+1,0) - 2\max(m-k,0) + \max(m-k-1,0) \), which equals \( 1 \) if \( m = k \) and \( 0 \) otherwise (check the three cases \( m \lt k \), \( m = k \), \( m \gt k \)). Summing over blocks gives the formula. Similar matrices have equal \( r_p(\lambda) \) because \( \operatorname{rank}\left( P^{-1} A P - \lambda I \right)^p = \operatorname{rank}\left( P^{-1} (A - \lambda I)^p P \right) = \operatorname{rank}(A - \lambda I)^p \). C

Result

\[ T \in \mathrm{End}_{\mathbb{C}}(V), \;\; \dim V \lt \infty \;\; \Longrightarrow \;\; [T] \sim \bigoplus_{j=1}^{r} J_{k_j}(\lambda_j), \quad \text{blocks unique up to order.} \]

Reading. Choose coordinates wisely and any complex square matrix becomes almost diagonal: eigenvalues down the diagonal, and a few unavoidable \( 1 \)’s just above it recording exactly how far the operator is from diagonalisable. The sizes and eigenvalues of the blocks are a complete fingerprint: two matrices are similar precisely when their fingerprints agree.

Scope. Applies to every linear operator on a nonzero finite-dimensional vector space over \( \mathbb{C} \), and over any field \( F \) provided the characteristic polynomial splits over \( F \) (e.g. always over an algebraically closed field; over \( \mathbb{R} \) only when all eigenvalues are real). It fails in infinite dimensions and gives no canonical form for families of non-commuting operators. Over a non-splitting field the correct substitute is the rational canonical form.

Corollaries & converses

Minimal polynomial read off. \( \mu_T(x) = \prod_i (x - \lambda_i)^{s_i} \) where \( s_i \) is the size of the largest block with eigenvalue \( \lambda_i \). In particular \( T \) is diagonalisable \( \iff \) \( \mu_T \) has simple roots \( \iff \) all blocks have size \( 1 \).
Jordan–Chevalley decomposition. \( T = D + N \) with \( D \) diagonalisable, \( N \) nilpotent, \( DN = ND \); moreover \( D \) and \( N \) are polynomials in \( T \) (take \( D = \sum_i \lambda_i E_i \) with the projections \( E_i = q_i(T)p_i(T) \) of step 4), and this decomposition is unique.
Similarity classification. The similarity classes in \( M_n(\mathbb{C}) \) are in bijection with functions assigning to each eigenvalue a partition (its block sizes). E.g. nilpotent \( n \times n \) classes \( \leftrightarrow \) partitions of \( n \).
Every complex matrix is similar to its transpose (conjugate each block by the reversal permutation; see Problem 3).
Matrix functions and ODEs. \( e^{tJ_k(\lambda)} = e^{t\lambda} \sum_{a=0}^{k-1} \frac{t^a}{a!} J_k(0)^a \), giving the \( t^a e^{\lambda t} \) solution basis of \( \dot{x} = Ax \).
Converse direction. "Same Jordan form \( \Rightarrow \) similar" holds trivially. But weaker invariants do not classify: \( J_2(0) \) and the zero \( 2 \times 2 \) matrix share characteristic polynomial \( x^2 \), trace, determinant and eigenvalues, yet are not similar (different ranks). Even characteristic and minimal polynomial together fail from size \( 4 \): \( J_2(0) \oplus J_2(0) \) versus \( J_2(0) \oplus J_1(0) \oplus J_1(0) \) share \( \chi = x^4, \mu = x^2 \) but differ in rank.

Fails without

Drop algebraic closure (splitting). \( A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \in M_2(\mathbb{R}) \): \( \chi_A = x^2 + 1 \) has no real root, so \( A \) has no real eigenvector and is similar over \( \mathbb{R} \) to no triangular, hence no Jordan, matrix. (Over \( \mathbb{C} \) it becomes \( \mathrm{diag}(i, -i) \).) Similarly \( \begin{pmatrix} 0 & 2 \\ 1 & 0 \end{pmatrix} \) over \( \mathbb{Q} \) fails because \( x^2 - 2 \) does not split in \( \mathbb{Q}[x] \).
Drop finite-dimensionality. On \( V = \mathbb{C}[x] \), the operator \( M : f \mapsto xf \) has no eigenvalues (degree count), so \( V \) has no generalized eigenspace decomposition whatsoever. Worse, the differentiation operator \( D : f \mapsto f' \) on \( \mathbb{C}[x] \) satisfies \( \bigcup_k \ker D^k = V \) yet admits no basis of Jordan chains for a direct-sum-of-blocks matrix: \( D \) is surjective, while a direct sum of finite nilpotent blocks \( J_k(0) \) is never surjective on a nonzero space unless some chain is infinite — the block-decomposition dichotomy of the finite theorem simply breaks.
Drop "vector space" for "module". Over \( \mathbb{Z} \), multiplication by \( 2 \) on \( \mathbb{Z}^2 \) is "diagonal", but the classification-by-blocks machinery needs division: \( \begin{pmatrix} 0 & 2 \\ 0 & 0 \end{pmatrix} \) and \( \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \) are conjugate over \( \mathbb{Q} \) but not over \( \mathbb{Z} \) (any \( P \in GL_2(\mathbb{Z}) \) conjugating one to the other would force \( 2 \mid \det P \cdot \text{unit} \), impossible); canonical forms over rings require the Smith normal form theory instead.

Common errors

Geometric multiplicity tells you everything. It only counts the number of blocks for \( \lambda \) (\( = \dim\ker(A - \lambda I) \)), not their sizes. For sizes you need the ranks \( r_p(\lambda) \) of powers (step 13). A \( 4\times 4 \) nilpotent with \( \dim\ker N = 2 \) could be \( J_3 \oplus J_1 \) or \( J_2 \oplus J_2 \).
Solving the chain top-down with a random eigenvector. Picking any eigenvector \( v \) and then trying to solve \( (A - \lambda I)w = v \) may be inconsistent: \( v \) must be chosen inside \( \operatorname{im}(A - \lambda I) \). Build chains from the top (a vector of maximal height) downward, or solve the system for \( v \) and \( w \) together.
Conflating exponents. Algebraic multiplicity \( m_i \) = sum of block sizes for \( \lambda_i \); the exponent \( s_i \) in the minimal polynomial = largest block size. These agree only when there is a single block.
Believing the Jordan basis is unique. Only the block multiset is unique; the basis (and the change-of-basis matrix \( P \)) is far from unique — e.g. any nonzero scalar rescales a whole chain.
"Same characteristic polynomial \( \Rightarrow \) similar." False: \( J_2(0) \) vs \( 0_{2\times 2} \).
Using JNF in floating-point computation. The Jordan form is a discontinuous function of the matrix: \( \begin{pmatrix} 0 & 1 \\ \varepsilon & 0 \end{pmatrix} \) is diagonalisable for every \( \varepsilon \neq 0 \) but is a \( J_2(0) \) at \( \varepsilon = 0 \). Numerically one uses the Schur form instead.

Discussion

Historically the theorem crystallised in the 1868–1874 period. Weierstrass’s 1868 theory of elementary divisors of matrix pencils contains the classification in equivalent form; Camille Jordan published the block form in his 1870 Traité des substitutions et des équations algébriques — notably working over the finite prime fields \( \mathbb{Z}/p\mathbb{Z} \), in the service of group theory, rather than over \( \mathbb{C} \). A sharp priority dispute with Kronecker followed; the modern verdict is that Weierstrass had the invariants first and Jordan the normal form. The "generalized eigenvector" language and the projection-based proof via Bézout are twentieth-century repackagings.

The right conceptual frame is module theory: \( T \) makes \( V \) a finitely generated torsion module over the PID \( \mathbb{C}[x] \), with \( x \) acting as \( T \). The structure theorem for such modules decomposes \( V \) as \( \bigoplus_j \mathbb{C}[x] / \left( (x - \lambda_j)^{k_j} \right) \), and each cyclic summand, in the basis \( \overline{(x-\lambda_j)^{k_j - 1}}, \dots, \overline{(x - \lambda_j)}, \overline{1} \), is exactly one Jordan block. Uniqueness of elementary divisors is uniqueness of blocks. Over a non-algebraically-closed field the same machine outputs the rational canonical form, whose companion blocks need no eigenvalues; the Jordan form is the special case where every irreducible factor of \( \chi_T \) is linear.

Analytically, the Jordan form is the finite-dimensional germ of spectral theory: the decomposition \( V = \bigoplus V_i \) with projections \( E_i \) that are polynomials in \( T \) prefigures the holomorphic functional calculus, where \( E_i = \frac{1}{2\pi i} \oint_{\gamma_i} (z - T)^{-1} \, dz \) around each eigenvalue. The nilpotent parts measure the failure of \( T \) to be normal-like; for normal operators (\( T T^* = T^* T \)) the spectral theorem forces every block to be \( 1 \times 1 \), which is why Jordan blocks never appear in the Hermitian theory. In applications the form is a theoretical instrument — it proves that solutions of \( \dot{x} = Ax \) are sums of \( t^a e^{\lambda t} \) terms, that \( \rho(A) \lt 1 \) forces \( A^m \to 0 \) — while actual computation routes through the numerically stable Schur triangularisation (Golub–Wilkinson analysed the ill-posedness of numerical JNF in 1976).

Deeper still, the nilpotent classification by partitions is the entrance to a rich geometry: the set of nilpotent matrices in \( M_n(\mathbb{C}) \) is an algebraic variety (the nilpotent cone), stratified by the \( GL_n \)-orbits that this theorem enumerates. Orbit closures are described by dominance order on partitions (Gerstenhaber–Hesselink): \( \overline{\mathcal{O}_{\mu}} \supseteq \mathcal{O}_{\nu} \iff \mu \trianglerighteq \nu \), the same order governing degenerations \( J_2 \rightsquigarrow J_1 \oplus J_1 \) under perturbation — the source of the discontinuity noted above. In Lie theory the Jordan–Chevalley decomposition generalises to semisimple algebraic groups, and the partition combinatorics reappears in Springer theory and the representation theory of \( S_n \). Common misconceptions. The Jordan form is not "just diagonalisation with extra steps": the set of non-diagonalisable matrices has measure zero, but it is exactly where the interesting degeneration phenomena (resonance in ODEs, defective eigenproblems, sensitivity of eigenvalues of order \( \varepsilon^{1/k} \)) live. And "generalized eigenvector" does not mean any solution of \( (A - \lambda I)^p v = 0 \) for huge \( p \) gives new information: the kernels stabilise at \( p = s_i \), the largest block size, never later.

Worked examples

Example 1. Find the Jordan normal form and a Jordan basis for \( A = \begin{pmatrix} 2 & 1 & 1 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{pmatrix} \in M_3(\mathbb{C}) \).

\( \chi_A(x) = (x-2)^3 \): the single eigenvalue is \( \lambda = 2 \), so \( A = 2I + N \) with \( N = A - 2I = \begin{pmatrix} 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \).

\( A \) is upper triangular, so \( \chi_A \) is the product of the diagonal terms \( (x - 2) \); the whole space is one generalized eigenspace. A

\( \operatorname{rank} N = 1 \), so the number of blocks is \( \dim \ker N = 3 - 1 = 2 \); and \( N^2 = 0 \), so every block has size \( \leq 2 \). The only partition of \( 3 \) into \( 2 \) parts of size \( \leq 2 \) is \( 2 + 1 \): \[ \text{JNF}(A) = J_2(2) \oplus J_1(2). \]

Block count = geometric multiplicity (each block contributes one kernel vector); largest block size = nilpotency index (uniqueness data of step 13 of the theorem, in low-dimensional shorthand). B

Chain of length 2: take \( w = e_2 \); then \( N w = e_1 \neq 0 \), \( N^2 w = 0 \). Chain: \( \{ e_1, e_2 \} \). Complete \( \ker N = \{ x : x_2 + x_3 = 0 \} \): its basis extending the chain bottom \( e_1 \) is \( \{ e_1, \; z = e_2 - e_3 \} \). Jordan basis: \( (e_1, \, e_2, \, e_2 - e_3) \).

This is steps 9–11 of the theorem in miniature: lift a top vector, then pad the kernel. Check \( z \in \ker N \): coordinates \( (0,1,-1) \) satisfy \( x_2 + x_3 = 0 \). Independence of the three vectors is immediate (triangular pattern). A

With \( P = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & -1 \end{pmatrix} \) (columns \( e_1, e_2, e_2 - e_3 \)): verify \( A e_1 = 2e_1 \), \( A e_2 = e_1 + 2 e_2 \), \( A(e_2 - e_3) = (1,2,0)^{\mathsf T} - (1,0,2)^{\mathsf T} = 2(e_2 - e_3) \).

Column-by-column verification that \( P^{-1} A P \) has first column \( (2,0,0)^{\mathsf T} \), second \( (1,2,0)^{\mathsf T} \), third \( (0,0,2)^{\mathsf T} \). A

\[ P^{-1} A P \;=\; \begin{pmatrix} 2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{pmatrix} \;=\; J_2(2) \oplus J_1(2). \]

Reading. One genuine \( 2 \)-chain and one loose eigenvector: \( A \) is defective with a one-dimensional defect.

Scope. Any \( 3 \times 3 \) with a triple eigenvalue is settled the same way by the pair \( \left( \operatorname{rank}(A - \lambda I), \operatorname{rank}(A-\lambda I)^2 \right) \).

Example 2. Find the Jordan normal form and a Jordan basis for \( A = \begin{pmatrix} 1 & 1 & 0 \\ -1 & 3 & 0 \\ -1 & 1 & 1 \end{pmatrix} \in M_3(\mathbb{C}) \).

\[ \chi_A(x) = \det(xI - A) : \quad \det(A - xI) = (1-x)\det\begin{pmatrix} 1-x & 1 \\ -1 & 3-x \end{pmatrix} = (1-x)\left( (1-x)(3-x) + 1 \right) = (1-x)(x-2)^2 . \] Eigenvalues: \( 2 \) (algebraic multiplicity \( 2 \)), \( 1 \) (simple).

Expand along the third column, whose only nonzero entry is \( a_{33} - x = 1 - x \); then \( (1-x)(3-x) + 1 = x^2 - 4x + 4 = (x-2)^2 \). A

\( A - 2I = \begin{pmatrix} -1 & 1 & 0 \\ -1 & 1 & 0 \\ -1 & 1 & -1 \end{pmatrix} \) has rank \( 2 \) (rows 1 and 3 independent), so \( \dim\ker(A - 2I) = 1 \): one block for \( \lambda = 2 \), necessarily of size \( 2 \). Hence \[ \text{JNF}(A) = J_2(2) \oplus J_1(1). \]

Geometric multiplicity \( 1 \) with algebraic multiplicity \( 2 \) forces a single block of size \( 2 \) (partition of \( 2 \) into \( 1 \) part). The simple eigenvalue \( 1 \) always yields \( J_1(1) \). B

Chain for \( \lambda = 2 \): solve \( (A - 2I)v = 0 \): \( -v_1 + v_2 = 0, \; -v_1 + v_2 - v_3 = 0 \Rightarrow v = (1,1,0)^{\mathsf T} \). Solve \( (A - 2I) w = v \): \( -w_1 + w_2 = 1 \) and \( -w_1 + w_2 - w_3 = 0 \Rightarrow w_3 = 1 \); take \( w = (0,1,1)^{\mathsf T} \). Eigenvector for \( \lambda = 1 \): \( (A - I)u = 0 \) gives \( u_2 = 0, -u_1 + 2u_2 = 0 \Rightarrow u = (0,0,1)^{\mathsf T} \).

Consistency of \( (A-2I)w = v \) is guaranteed here because \( v \) spans the whole (one-dimensional) eigenspace and the block has size \( 2 \), so \( v \in \operatorname{im}(A - 2I) \); the check is that rows 1 and 2 of the augmented system agree. B

Set \( P = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{pmatrix} \) (columns \( v, w, u \)). Verify: \( Av = (2,2,0)^{\mathsf T} = 2v \); \( Aw = (1,3,2)^{\mathsf T} = v + 2w \); \( Au = (0,0,1)^{\mathsf T} = u \).

Direct multiplication; the three verified columns are exactly the columns of \( P \cdot \left( J_2(2) \oplus J_1(1) \right) \). \( \det P = 1 \neq 0 \), so \( P \) is invertible. A

\[ P^{-1} A P = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix} = J_2(2) \oplus J_1(1). \]

Reading. The repeated eigenvalue \( 2 \) is defective (one eigenvector short); the simple eigenvalue \( 1 \) contributes a trivial block. General solution of \( \dot{x} = Ax \): \( x(t) = (c_1 + c_2 t) e^{2t} v + c_2 e^{2t} w + c_3 e^{t} u \).

Scope. The method (char. poly \( \to \) ranks \( \to \) chains, always solving for the eigenvector first inside \( \operatorname{im}(A - \lambda I) \) when several choices exist) works verbatim for any matrix whose eigenvalues are known exactly.

Problems

Find the Jordan normal form of \( A = \begin{pmatrix} 1 & 1 \\ -1 & 3 \end{pmatrix} \) and an explicit \( P \) with \( P^{-1} A P \) in Jordan form.
Solution
\( \chi_A(x) = x^2 - 4x + 4 = (x-2)^2 \). \( A - 2I = \begin{pmatrix} -1 & 1 \\ -1 & 1 \end{pmatrix} \neq 0 \) has rank \( 1 \), so \( A \) is not diagonalisable and \( \text{JNF}(A) = J_2(2) \). Eigenvector: \( (A-2I)v = 0 \Rightarrow v = (1,1)^{\mathsf T} \). Generalized: \( (A - 2I)w = v \Rightarrow -w_1 + w_2 = 1 \); take \( w = (0,1)^{\mathsf T} \). Then \( P = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \), and one checks \( Av = 2v \), \( Aw = (1,3)^{\mathsf T} = v + 2w \), so \( P^{-1} A P = \begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix} \).
Let \( N \in M_4(\mathbb{C}) \) be nilpotent with \( N^2 = 0 \) and \( \operatorname{rank} N = 2 \). Determine the Jordan normal form of \( N \), and prove there is no other possibility.
Solution
All eigenvalues are \( 0 \) (nilpotency), so the JNF is \( \bigoplus J_{k_j}(0) \) with \( \sum k_j = 4 \). \( N^2 = 0 \) forces every \( k_j \leq 2 \) (a block \( J_k(0) \) satisfies \( J_k(0)^2 \neq 0 \) when \( k \geq 3 \), since it shifts a chain by two places and \( k - 2 \geq 1 \)). The number of blocks is \( \dim \ker N = 4 - \operatorname{rank} N = 2 \). A partition of \( 4 \) into exactly \( 2 \) parts, each \( \leq 2 \), must be \( 2 + 2 \). Hence \( \text{JNF}(N) = J_2(0) \oplus J_2(0) \), uniquely: any other partition either has the wrong number of parts (\( 4 = 2+1+1 \) has \( 3 \), \( 4 = 4 \) has \( 1 \)) or a part exceeding \( 2 \) (\( 3 + 1 \)).
Prove that every \( A \in M_n(\mathbb{C}) \) is similar to its transpose \( A^{\mathsf T} \).
Solution
Step 1: it suffices to prove it for a single Jordan block. If \( A = P J P^{-1} \) with \( J = \bigoplus_j J_{k_j}(\lambda_j) \), then \( A^{\mathsf T} = (P^{-1})^{\mathsf T} J^{\mathsf T} P^{\mathsf T} \), so \( A^{\mathsf T} \sim J^{\mathsf T} = \bigoplus_j J_{k_j}(\lambda_j)^{\mathsf T} \). If each \( J_k(\lambda)^{\mathsf T} \sim J_k(\lambda) \), then (conjugating blockwise by a block-diagonal matrix) \( J^{\mathsf T} \sim J \), whence \( A^{\mathsf T} \sim J \sim A \). Step 2: single block. Let \( R \in M_k \) be the reversal matrix, \( R_{ab} = 1 \) if \( a + b = k+1 \), else \( 0 \); note \( R^{-1} = R \). For any \( B \), \( (R B R)_{ab} = B_{k+1-a, \, k+1-b} \), i.e. conjugation by \( R \) rotates the matrix by half a turn. Applying this to \( B = J_k(\lambda) \): the diagonal \( (a = b) \) maps to itself, and the superdiagonal entry at \( (a, a+1) \) maps to position \( (k+1-a, k-a) \), which is the subdiagonal — exactly \( J_k(\lambda)^{\mathsf T} \). Hence \( R J_k(\lambda) R^{-1} = J_k(\lambda)^{\mathsf T} \). (Remark: the statement is true over every field via the rational canonical form; the JNF proof needs \( \mathbb{C} \) only because it invokes the theorem above.)
How many similarity classes of nilpotent matrices are there in \( M_5(\mathbb{C}) \)? List them.
Solution
A nilpotent matrix has all eigenvalues \( 0 \), so by existence and uniqueness of the JNF its similarity class is exactly the multiset of block sizes: a partition of \( 5 \). Conversely every partition occurs (write down the block-diagonal matrix). The partitions of \( 5 \) are: \( 5;\; 4+1;\; 3+2;\; 3+1+1;\; 2+2+1;\; 2+1+1+1;\; 1+1+1+1+1 \) — seven in total. So there are exactly \( 7 \) classes, from the regular nilpotent \( J_5(0) \) down to the zero matrix \( J_1(0)^{\oplus 5} \). (The count of classes of nilpotents in \( M_n \) is the partition function \( p(n) \).)
Let \( A \in M_n(\mathbb{C}) \) satisfy \( A^m = I \) for some integer \( m \geq 1 \). Prove that \( A \) is diagonalisable, and show by counterexample that the conclusion fails over a field of positive characteristic.
Solution
By the theorem, \( A \sim J = \bigoplus_j J_{k_j}(\lambda_j) \), and \( A^m = I \iff J^m = I \). Since \( A \) is invertible (\( A \cdot A^{m-1} = I \)), every \( \lambda_j \neq 0 \). Suppose some block has \( k := k_j \geq 2 \). Write \( J_k(\lambda) = \lambda I + N \) with \( N = J_k(0) \), \( N^k = 0 \), and \( \lambda I \) commuting with \( N \); the binomial theorem (valid for commuting matrices) gives \[ J_k(\lambda)^m = \sum_{a=0}^{m} \binom{m}{a} \lambda^{m-a} N^a = \lambda^m I + m \lambda^{m-1} N + \binom{m}{2}\lambda^{m-2} N^2 + \cdots \] The \( (1,2) \) entry of this matrix is \( m \lambda^{m-1} \) (only the \( N^1 \) term contributes to the first superdiagonal). In characteristic zero \( m \lambda^{m-1} \neq 0 \) since \( m \geq 1 \) and \( \lambda \neq 0 \) — contradicting \( J_k(\lambda)^m = I_k \), whose \( (1,2) \) entry is \( 0 \). Hence every block has size \( 1 \) and \( J \) is diagonal, i.e. \( A \) is diagonalisable. (Equivalently: the minimal polynomial divides \( x^m - 1 \), which has \( m \) distinct roots over \( \mathbb{C} \), so all blocks have size \( 1 \).) Counterexample in characteristic \( p \): over \( \mathbb{F}_p \) take \( A = J_2(1) = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \). Then \( A^p = (I + N)^p = I + pN + \binom{p}{2} N^2 = I \) since \( p \equiv 0 \) and \( N^2 = 0 \); but \( A \neq I \) is a nontrivial Jordan block, hence not diagonalisable (its only eigenvalue is \( 1 \), and \( \ker(A - I) \) is one-dimensional). Here \( x^p - 1 = (x-1)^p \) has a repeated root, which is exactly where the characteristic-zero argument breaks.