1 Definition and statement

The Jordan canonical form is a normal form for square matrices under similarity. It expresses a matrix as a block diagonal matrix whose blocks, called Jordan blocks, correspond to eigenvalues and describe how far the matrix is from being diagonalizable. Over an algebraically closed field, every square matrix is similar to a Jordan matrix.

1.1 Similarity to Jordan form

Two matrices are similar when one can be obtained from the other by a change of basis. Jordan canonical form classifies linear transformations up to this equivalence. If a matrix can be transformed into Jordan form, then the form records the essential algebraic data of the transformation while suppressing basis-dependent details.

1.2 Jordan blocks

A Jordan block is a square matrix associated with a single eigenvalue. It is the basic building unit of Jordan canonical form and has the eigenvalue repeated along the main diagonal, with ones immediately above the diagonal and zeros elsewhere.

1.2.1 Diagonal entries

Each diagonal entry of a Jordan block equals the same eigenvalue. This repetition reflects the algebraic multiplicity of that eigenvalue within the block structure.

1.2.2 Superdiagonal entries

The entries directly above the main diagonal are equal to 1 within a Jordan block, while all other off-diagonal entries are 0. These superdiagonal ones encode the coupling between generalized eigenvectors and distinguish a Jordan block from a purely diagonal matrix.

1.3 Existence and uniqueness

Jordan canonical form exists under suitable field assumptions and is unique up to the order of the blocks. This makes it a reliable classification tool for matrices over fields where all characteristic polynomials split into linear factors.

1.3.1 Algebraic closure assumptions

The standard form is guaranteed over an algebraically closed field, such as the complex numbers. In that setting, every matrix has enough eigenvalues to be decomposed completely into Jordan blocks.

1.3.2 Uniqueness up to block ordering

Although the order of Jordan blocks on the diagonal may vary, the multiset of blocks is uniquely determined by the matrix. The sizes of the blocks and the eigenvalues they carry are invariants of similarity.

2 Eigenvalues and generalized eigenvectors

Jordan form refines the ordinary eigenspace decomposition by incorporating generalized eigenvectors. These vectors extend the notion of eigenvectors and reveal the internal layered structure associated with repeated eigenvalues.

2.1 Generalized eigenspaces

For a given eigenvalue, the generalized eigenspace consists of all vectors annihilated by some power of the shifted matrix. It contains the ordinary eigenspace and captures every vector associated with that eigenvalue in the Jordan decomposition.

2.2 Chains of generalized eigenvectors

Generalized eigenvectors often appear in linked sequences rather than as isolated vectors. These sequences organize the basis vectors within each Jordan block.

2.2.1 Jordan chains

A Jordan chain is an ordered sequence of generalized eigenvectors in which each vector is mapped to the previous one by the shifted matrix, except for the first vector, which is an eigenvector. Such chains correspond directly to individual Jordan blocks.

2.2.2 Chain length and block size

The length of a Jordan chain equals the size of the corresponding Jordan block. Longer chains indicate a larger nilpotent component and a greater departure from diagonalizability.

2.3 Minimal polynomial and characteristic polynomial

The characteristic polynomial records the eigenvalues and their algebraic multiplicities. The minimal polynomial records the largest Jordan block size for each eigenvalue. Together, they provide strong constraints on the possible Jordan form of a matrix.

3 Construction of Jordan canonical form

Constructing Jordan form involves identifying eigenvalues, determining generalized eigenspaces, and assembling a basis from Jordan chains. The process is theoretically precise, though it can be numerically delicate.

3.1 From eigenspaces

The first step is to compute the eigenspaces of the matrix. These spaces reveal the number of Jordan chains beginning at each eigenvalue and provide the starting vectors for the block structure.

3.2 Determining block sizes

The sizes of the Jordan blocks are determined by analyzing the ranks or nullities of powers of shifted matrices. This information distinguishes between multiple small blocks and fewer large ones.

3.2.1 Rank and nullity methods

By examining the nullity of successive powers of the matrix minus an eigenvalue times the identity, one can infer the number and lengths of Jordan blocks. Differences between successive nullities are especially informative.

3.2.2 Powers of shifted matrices

The kernels of powers of the shifted matrix grow as the power increases. Stabilization of this growth indicates that all blocks for that eigenvalue have been fully detected, and the largest power needed reflects the largest block size.

3.3 Basis selection

Once the block sizes are known, a basis must be chosen to realize the Jordan structure. This basis is built from generalized eigenvectors arranged into chains.

3.3.1 Jordan basis

A Jordan basis is a basis in which the matrix takes Jordan canonical form. Its vectors are grouped into chains, and each chain corresponds to one block in the decomposition.

3.3.2 Change of basis matrix

The change of basis matrix is formed by the Jordan basis vectors written in the original coordinate system. Conjugating the original matrix by this matrix produces the Jordan form.

4 Properties and interpretation

Jordan canonical form clarifies the algebraic behavior of a matrix by separating diagonal behavior from nilpotent behavior. It is especially useful for understanding invariant structure and matrix functions.

4.1 Diagonalizable matrices as a special case

A matrix is diagonalizable precisely when all its Jordan blocks have size 1. In that case, the Jordan form reduces to a diagonal matrix, and generalized eigenvectors are unnecessary.

4.2 Nilpotent matrices

A nilpotent matrix has only the eigenvalue 0. Its Jordan form consists entirely of nilpotent Jordan blocks, and the block sizes determine the index of nilpotency.

4.3 Commuting with polynomial functions

Jordan form simplifies the evaluation of polynomial expressions in matrices. Since polynomials act block by block, the structure of each Jordan block determines the result.

4.3.1 Evaluating polynomials of matrices

For a polynomial p, the matrix p(A) can be computed by applying the polynomial to each Jordan block. The nilpotent part of a block produces truncated Taylor-like expansions around the eigenvalue.

4.3.2 Matrix exponentials and functions

Analytic functions of matrices, including the exponential, can be evaluated using Jordan form when the function is defined on the spectrum. This is particularly useful in solving systems of differential equations and in describing continuous-time evolution.

5 Applications

Jordan canonical form appears in many areas where linear transformations must be classified or computed explicitly. Its value lies in converting abstract similarity questions into concrete block data.

5.1 Linear differential equations

In systems of linear differential equations with constant coefficients, Jordan form helps compute the matrix exponential and thus the general solution. Repeated eigenvalues and nontrivial blocks lead to polynomial factors multiplying exponentials.

5.2 Dynamical systems

For discrete and continuous linear dynamical systems, Jordan form clarifies long-term behavior by separating pure scaling from generalized modes. It is useful in analyzing resonance, growth, and the presence of nontrivial invariant subspaces.

5.3 Canonical classification of linear operators

Jordan form provides a classification scheme for linear operators on finite-dimensional spaces over suitable fields. Operators with the same Jordan form are similar, so the form serves as a compact invariant description.

5.4 Computational considerations

Although Jordan form is theoretically fundamental, it is often numerically unstable to compute directly. Small perturbations can change block structure, so practical algorithms usually prefer Schur or related decompositions for floating-point work.

Several other canonical or nearly canonical forms are connected to Jordan form. These alternatives are useful when the field is not algebraically closed, when real matrices are considered, or when numerical stability is important.

6.1 Rational canonical form

The rational canonical form is another similarity invariant that exists over any field. It replaces Jordan blocks with companion matrices and does not require eigenvalues to lie in the field.

6.2 Real Jordan form

For real matrices, the real Jordan form adapts the complex Jordan structure to real coordinates. Complex conjugate eigenpairs are represented by real block structures rather than by complex-valued blocks.

6.3 Schur decomposition

The Schur decomposition writes a matrix as unitarily similar to an upper triangular matrix. It is numerically stable and often used as a practical precursor to spectral analysis, even though it does not provide the same algebraic detail as Jordan form.

6.4 Jordan decomposition in Lie theory

In Lie theory, Jordan decomposition refers to a related splitting of an element into semisimple and nilpotent parts. This usage is conceptually connected to the matrix Jordan form through the separation of diagonalizable and nilpotent behavior.