1 Definition and Basic Properties
1.1 Similarity and Canonical Forms
For a square matrix \(A\) over a field \(\mathbb{F}\), the Jordan (canonical) form is a matrix \(J\) such that \(A\) and \(J\) are similar: there exists an invertible matrix \(P\) with \[ A = PJP^{-1}. \] The purpose of this construction is to encode the intrinsic linear-algebraic structure of the linear map \(x \mapsto Ax\) in a standardized block-diagonal arrangement. Similarity invariants—quantities unchanged under \(A \mapsto P^{-1}AP\)—are therefore reflected directly in \(J\).
A key point is that the Jordan form is “canonical” in the sense that, once the sizes of the Jordan blocks for each eigenvalue are fixed, the resulting matrix is determined up to a prescribed ambiguity (block ordering).
1.2 Jordan Blocks and Their Structure
A Jordan block associated with an eigenvalue \(\lambda \in \mathbb{F}\) of size \(m\) is the \(m\times m\) matrix \[ J_m(\lambda)= \begin{pmatrix} \lambda & 1 & 0 & \cdots & 0\\ 0 & \lambda & 1 & \cdots & 0\\ \vdots & \ddots & \ddots & \ddots & \vdots\\ 0 & \cdots & 0 & \lambda & 1\\ 0 & \cdots & \cdots & 0 & \lambda \end{pmatrix}. \] Equivalently, it is the sum of \(\lambda I_m\) and the nilpotent Jordan block \(N_m\) (ones on the superdiagonal, zeros elsewhere). The superdiagonal ones represent the failure of the matrix to be fully reducible into independent eigenvectors, and they govern how generalized eigenvectors link together.
A Jordan form \(J\) is then a block diagonal matrix whose blocks are Jordan blocks for various eigenvalues and sizes.
1.3 Generalized Eigenspaces and Jordan Chains
While ordinary eigenvectors form the eigenspace \(\ker(A-\lambda I)\), Jordan form also uses generalized eigenvectors. For \(k\ge 1\), the generalized eigenspace of order \(k\) is \[ \ker\bigl((A-\lambda I)^k\bigr). \] Jordan chains provide an explicit organizational picture: a Jordan chain of length \(m\) for \(\lambda\) is a sequence of vectors \(v_1,\dots,v_m\) such that \[ (A-\lambda I)v_1=0,\quad (A-\lambda I)v_2=v_1,\quad \dots,\quad (A-\lambda I)v_m=v_{m-1}. \] The length of the chain matches the size of a corresponding Jordan block. Collecting one chain basis for each Jordan block yields a basis in which the matrix takes Jordan block diagonal form.
1.4 Uniqueness Up to Block Ordering
Over a given algebraically closed field, the multiset of Jordan block sizes associated with each eigenvalue is uniquely determined by \(A\). Changing the basis produces only reordering among blocks with the same eigenvalue (and potentially different arrangement across eigenvalues in the block diagonal). Thus, Jordan forms of \(A\) are unique up to permutation of the Jordan blocks.
More precisely, if the Jordan form is written with blocks grouped by eigenvalue, then for each eigenvalue \(\lambda\), the list of block sizes for that \(\lambda\) is an invariant of \(A\).
2 Existence of Jordan Form
2.1 Eigenvalues Over an Algebraically Closed Field
Jordan form is guaranteed to exist when the matrix entries lie in a field over which the characteristic polynomial splits completely, i.e., when every eigenvalue lies in the field. A standard setting is an algebraically closed field such as \(\mathbb{C}\). Under this condition, \(A\) admits a decomposition into Jordan blocks with eigenvalues taken from the field.
If the field is not algebraically closed, some eigenvalues may not exist as elements of the field, preventing the construction of Jordan blocks with those eigenvalues in that field.
2.2 Minimal Polynomials and Factorization
The minimal polynomial of \(A\), denoted \(m_A(t)\), is the monic polynomial of least degree such that \(m_A(A)=0\). When the characteristic polynomial splits, \(m_A\) also factors into linear terms (possibly with multiplicities). These factorizations influence the lengths of Jordan chains: higher powers in \(m_A\) correspond to larger Jordan blocks.
In effect, Jordan form can be understood as a refinement of the information in \(m_A\): the minimal polynomial records the largest Jordan block size for each eigenvalue, while the full Jordan structure records all block sizes.
2.3 Rational Canonical Form Connection (Conceptual Bridge)
Rational canonical form (RCF) exists over any field and uses invariant factors rather than eigenvalues. Conceptually, RCF and Jordan form are linked through the way they organize similarity classes. When the field is extended so that the characteristic polynomial splits, the invariant factors further decompose, and the Jordan block structure can be recovered.
Thus, RCF serves as a more general “starting point,” while Jordan form is the “refined” version available when eigenvalues can be expressed within the field.
2.4 Conditions for Existence Over Real Numbers
Over \(\mathbb{R}\), Jordan form in the strict sense (with real eigenvalues as block diagonals) exists when all eigenvalues are real. If \(A\) has complex conjugate eigenpairs, one cannot form Jordan blocks with those complex eigenvalues while staying inside \(\mathbb{R}\). In that case, a real canonical structure uses real block forms related to those complex eigenvalues rather than purely real Jordan blocks.
Practically, many real problems reduce to a complexification step, where Jordan form is constructed over \(\mathbb{C}\) and then interpreted for real data.
3 Constructing Jordan Form
3.1 From Characteristic Polynomial to Eigenvalues
A standard starting point is computing the characteristic polynomial \(\chi_A(t)=\det(tI-A)\). Its roots (in the chosen field) identify the eigenvalues \(\lambda\). Their multiplicities in \(\chi_A\) provide the algebraic multiplicities, which must match the total sizes of the Jordan blocks for each eigenvalue.
This step does not yet determine block sizes; it only partitions the matrix’s structure by eigenvalue.
3.2 Computing Eigenspaces
For each eigenvalue \(\lambda\), compute the eigenspace \[ E_\lambda=\ker(A-\lambda I). \] Its dimension is the geometric multiplicity. While geometric multiplicity does not determine the full Jordan structure alone, it constrains it: the number of Jordan blocks associated with \(\lambda\) equals \(\dim E_\lambda\).
Thus, eigenspaces determine how many blocks there are for each eigenvalue, but not how large each block becomes.
3.3 Computing Generalized Eigenspaces
To refine the block sizes, one examines the growth of kernels \[ \ker\bigl((A-\lambda I)^k\bigr)\quad \text{as } k=1,2,3,\dots \] The eventual stabilization occurs once \(k\) exceeds the size of the largest Jordan block for \(\lambda\). These dimensions reveal how many generalized eigenvectors of each order are present, which in turn determines the distribution of block sizes.
Operationally, one often computes \(\dim \ker((A-\lambda I)^k)\) for successive \(k\) until the dimensions stop increasing.
3.4 Determining Block Sizes via Ranks of Powers
Jordan block sizes can be derived from the sequence of kernel dimensions or, equivalently, from ranks of powers of \(A-\lambda I\). If one defines \[ d_k=\dim \ker\bigl((A-\lambda I)^k\bigr), \] then the increments \(d_k-d_{k-1}\) measure how many new vectors appear when moving from order \(k-1\) to \(k\). Those increments can be translated into counts of Jordan blocks of size at least \(k\).
A typical workflow is:
- compute \(d_k\) for increasing \(k\),
- determine how many blocks have size \(\ge k\),
- deduce the exact multiplicity of each block size by subtraction across successive \(k\).
Once block sizes are known, one constructs a Jordan basis using generalized eigenvector chains.
4 Theorems and Relationships
4.1 Algebraic Multiplicity vs. Geometric Multiplicity
For an eigenvalue \(\lambda\), the algebraic multiplicity is its multiplicity as a root of \(\chi_A(t)\). The geometric multiplicity is \(\dim \ker(A-\lambda I)\). The general inequality \[ 1 \le \dim \ker(A-\lambda I) \le \text{(algebraic multiplicity)} \] always holds. Equality of the two implies that the corresponding Jordan blocks are all size \(1\), meaning the eigenvalue is not defective.
When geometric multiplicity is smaller, it signals the presence of Jordan blocks larger than \(1\times 1\), i.e., generalized eigenvectors are needed beyond ordinary eigenvectors.
4.2 Jordan Block Sizes and Invariant Factors
The invariant factors used in rational canonical form can be related to Jordan data by factoring them into primary components. Over a splitting field, invariant factors decompose into powers of \((t-\lambda)\), and the exponents reflect the largest Jordan block sizes for each eigenvalue. More detailed information about all block sizes corresponds to finer invariants obtainable from the structure of \(\ker((A-\lambda I)^k)\) for each \(k\).
In short, invariant factors track the “maximum chain lengths,” while Jordan blocks provide the “full chain-length distribution.”
4.3 Minimal Polynomial and Largest Jordan Blocks
The minimal polynomial factors as \[ m_A(t)=\prod_\lambda (t-\lambda)^{s_\lambda}, \] where \(s_\lambda\) equals the size of the largest Jordan block for eigenvalue \(\lambda\). This theorem provides a direct bridge from a polynomial invariant to the extremal part of Jordan structure.
Consequently, even without knowing all block sizes, one can read off the biggest Jordan blocks from \(m_A\).
4.4 Companion Matrices and Jordan Structure
Companion matrices represent cyclic modules over \(\mathbb{F}[t]\), making them central to rational canonical form. For a polynomial \(p(t)\), the companion matrix \(C(p)\) satisfies \(p(C(p))=0\), and its similarity class is determined by invariant factors.
When \(p(t)\) splits into linear factors, the Jordan structure of the companion matrix can be analyzed by how those factors repeat and by how the module decomposes into primary components. This provides a structural pathway connecting polynomial data to Jordan block patterns.
5 Computation Methods
5.1 Using Kernel Dimensions of (A − λI)^k
A direct computational approach begins with the eigenvalue \(\lambda\) and computes the dimensions \[ \dim \ker\bigl((A-\lambda I)^k\bigr) \] for increasing \(k\). The resulting sequence determines, for each \(k\), how many Jordan blocks have size at least \(k\). From there, the exact count of blocks of each size follows by standard difference relations.
This method is robust because it relies on linear-algebraic computations (solving homogeneous systems) rather than explicitly finding Jordan chains at first.
5.2 Practical Algorithm Outline
A practical algorithm for constructing Jordan form typically includes:
- Find \(\chi_A(t)\) and its roots to identify eigenvalues.
- For each eigenvalue \(\lambda\), compute \(d_k=\dim\ker((A-\lambda I)^k)\) until stabilization.
- Determine the partition of the algebraic multiplicity into Jordan block sizes.
- Build generalized eigenspaces \(\ker((A-\lambda I)^k)\) and then extract Jordan chains to produce a basis \(P\).
- Assemble \(J\) as the block diagonal matrix of the Jordan blocks and verify \(A=PJP^{-1}\).
Symbolic computations often require careful pivoting and basis management, while numerical computations require additional caution.
5.3 Handling Multiple Eigenvalues
When several eigenvalues are present, the decomposition splits across eigenvalues: Jordan blocks for distinct eigenvalues occupy separate invariant subspaces. In calculations, one treats each eigenvalue independently by using \((A-\lambda I)^k\) and computing kernels within that eigenvalue’s primary component.
This separation prevents interference between eigenstructures and reduces the complexity of basis construction.
5.4 Symbolic vs. Numerical Considerations
Jordan form is sensitive: small perturbations in \(A\) can change Jordan block sizes dramatically, especially in defective cases. As a result, numerical eigenvalue routines may produce misleading approximations of Jordan structure, even when eigenvalues appear stable.
Symbolic methods (exact arithmetic, rational functions) are typically used when precise Jordan blocks are required, while numerical methods are more appropriate for approximate spectral behavior, not exact chain lengths.
6 Jordan Form Applications
6.1 Powers and Polynomials of Matrices
Once the Jordan form \(J\) is known, computing \(p(A)\) for a polynomial \(p\) becomes simpler because \[ p(A)=P\,p(J)\,P^{-1}. \] Moreover, \(p(J)\) is block diagonal with blocks \(p(J_m(\lambda))\). For a Jordan block, \(p(J_m(\lambda))\) can be expressed using derivatives of \(p\) evaluated at \(\lambda\), reflecting how the polynomial interacts with the nilpotent part \(N_m\).
Thus, Jordan form converts matrix polynomial evaluation into manageable block computations.
6.2 Matrix Exponential and Differential Equations
The matrix exponential \(e^{tA}\) is central in solving systems of linear differential equations \(x'(t)=Ax(t)\). Using similarity, \[ e^{tA}=P\,e^{tJ}\,P^{-1}, \] and \(e^{tJ}\) splits into exponentials of Jordan blocks. For a Jordan block \(J_m(\lambda)=\lambda I_m+N_m\) with \(N_m\) nilpotent, \[ e^{tJ_m(\lambda)}=e^{\lambda t}\,e^{tN_m}, \] and \(e^{tN_m}\) becomes a finite polynomial in \(t\) (because \(N_m^m=0\)). This yields explicit formulas combining exponentials with polynomial factors, a hallmark of repeated eigenvalues.
6.3 Solving Linear Systems with Repeated Eigenvalues
Systems involving \((A-\lambda I)\) or \((A-\lambda I)^k\) often arise in steady-state and resonance-type linear algebra problems. Jordan form clarifies the solution structure by showing when solutions involve only eigenmodes versus when generalized modes contribute polynomial terms in time or in parameter expansions.
In algebraic terms, repeated eigenvalues correspond to nontrivial nilpotent parts, which produce additional degrees of freedom in solutions.
6.4 Stability Analysis via Jordan Structure
For dynamical systems \(x'(t)=Ax(t)\), stability depends on eigenvalues: modes with negative real parts decay, while those with positive real parts grow. Jordan form refines this by describing how decay or growth can be multiplied by polynomial factors when nontrivial Jordan blocks occur.
For example, if an eigenvalue has real part zero, nontrivial Jordan blocks can create polynomial growth even when pure eigenvectors would suggest neutral behavior. Therefore, Jordan structure can indicate whether “marginal” eigenvalues produce lasting effects.
7 Special Cases and Examples
7.1 Diagonalizable Matrices (Jordan Form with 1×1 Blocks)
A matrix \(A\) is diagonalizable over \(\mathbb{F}\) precisely when every eigenvalue has enough eigenvectors so that the geometric multiplicity equals the algebraic multiplicity. In that case, all Jordan blocks are size \(1\), and the Jordan form equals a diagonal matrix of eigenvalues.
Diagonalizable matrices thus represent the simplest Jordan scenario: generalized eigenspaces collapse to ordinary eigenspaces.
7.2 Nilpotent Matrices (All Eigenvalues Zero)
A matrix is nilpotent if \(A^k=0\) for some \(k\). Then its only eigenvalue is \(0\), and its Jordan form consists solely of nilpotent Jordan blocks \(J_m(0)\). The index of nilpotency—the smallest \(k\) with \(A^k=0\)—equals the size of the largest Jordan block.
This case makes the relation between powers of \(A\) and block sizes particularly transparent.
7.3 Defective Eigenvalues and Nontrivial Jordan Chains
An eigenvalue \(\lambda\) is defective if its geometric multiplicity is strictly less than its algebraic multiplicity. In Jordan form this appears as at least one Jordan block of size larger than \(1\). Such blocks correspond to Jordan chains of length greater than \(1\), meaning that not all eigenvectors are sufficient to form a basis.
Defectiveness is the algebraic source of polynomial terms in exponentials and of increased sensitivity under perturbations.
7.4 Worked Example: Building Blocks Step-by-Step
Consider a \(3\times 3\) matrix \(A\) over \(\mathbb{C}\) whose characteristic polynomial is \((t-\lambda)^3\) for some \(\lambda\), and suppose \[ \dim\ker(A-\lambda I)=1,\qquad \dim\ker\bigl((A-\lambda I)^2\bigr)=2. \] Because the algebraic multiplicity is \(3\), the Jordan blocks for \(\lambda\) must sum to size \(3\). The geometric multiplicity \(1\) implies there is exactly one Jordan block. Since the dimension after squaring is \(2\), one infers that the largest chain has length \(2\) or \(3\), and the only partition consistent with a single block and the kernel growth is a single block of size \(3\). Thus the Jordan form is \[ J= \begin{pmatrix} \lambda & 1 & 0\\ 0 & \lambda & 1\\ 0 & 0 & \lambda \end{pmatrix}. \] A Jordan basis can then be constructed by choosing a vector \(v_1\) spanning \(\ker(A-\lambda I)\), finding \(v_2\) with \((A-\lambda I)v_2=v_1\), and then finding \(v_3\) with \((A-\lambda I)v_3=v_2\). With the basis \(P=[v_3\, v_2\, v_1]\) arranged appropriately, one obtains \(A=PJP^{-1}\).