1 Definition and Basic Structure

1.1 Jordan block form

A Jordan block is an \(n\times n\) matrix associated with a single eigenvalue \(\lambda\) that encodes how a linear transformation acts on a chain of generalized eigenvectors. In its standard form, it has \(\lambda\) on the diagonal, ones on the superdiagonal, and zeros elsewhere. For a block of size \(n\), \[ J_n(\lambda)= \begin{pmatrix} \lambda & 1 & 0 & \cdots & 0\\ 0 & \lambda & 1 & \ddots & \vdots\\ \vdots & \ddots & \ddots & \ddots & 0\\ 0 & \cdots & 0 & \lambda & 1\\ 0 & \cdots & \cdots & 0 & \lambda \end{pmatrix}. \] The superdiagonal ones are what prevent the block from being diagonal while still keeping its eigenvalue constant along the diagonal.

1.2 Notation and parameter choices

Common notation writes the block as \(J_n(\lambda)\), emphasizing both its size \(n\) and its eigenvalue \(\lambda\). Other conventions may use \(J(\lambda,n)\) or describe the block as \(\lambda I + N\), where \(N\) is the nilpotent Jordan block with ones on the superdiagonal and zeros on the diagonal. The parameter \(\lambda\) can be any element of the base field (or splitting field) in which the eigenvalues are defined.

1.3 Eigenvalues and size of the block

Every Jordan block \(J_n(\lambda)\) has exactly one eigenvalue, namely \(\lambda\), with algebraic multiplicity \(n\). Its size \(n\) is the length of the Jordan chain it supports: larger blocks correspond to longer chains of generalized eigenvectors and thus to a “more nondiagonalizable” behavior.

1.4 Relation to nilpotent Jordan blocks

Writing \(J_n(\lambda)=\lambda I + N_n\), the matrix \(N_n\) is nilpotent and has ones on the superdiagonal. The nilpotent part captures the deviation from diagonal action: when \(N_n=0\) (which occurs only for \(n=1\)), the block becomes diagonal and the transformation acts semisimply on that invariant subspace.

2 Algebraic Properties

2.1 Matrix powers of a Jordan block

Powers of \(J_n(\lambda)\) can be expressed using the binomial expansion for commuting matrices: \[ J_n(\lambda)^m=(\lambda I+N_n)^m=\sum_{k=0}^{m}\binom{m}{k}\lambda^{m-k}N_n^k. \] Since \(N_n^n=0\), the sum effectively truncates at \(k=n-1\). This yields a structured pattern: entries along the diagonal behave like \(\lambda^m\), while higher superdiagonals involve binomial coefficients times powers of \(\lambda\).

2.2 Rank, nullity, and characteristic spaces

Because \(N_n^k\) shifts basis vectors upward by \(k\) superdiagonal steps, its rank and nullity are determined by \(k\). For \(k\ge 1\),

  • \(\operatorname{rank}(N_n^k)=\max(n-k,0)\),
  • \(\dim\ker(N_n^k)=\min(k,n)\).

For the Jordan block, kernels of \((J_n(\lambda)-\lambda I)^k=(N_n)^k\) describe the “growth” of generalized eigenvectors of increasing order.

2.3 Minimal polynomial and its degree

For a single Jordan block \(J_n(\lambda)\), the minimal polynomial is \((x-\lambda)^n\). For a direct sum of Jordan blocks, the minimal polynomial is the least common multiple of the individual block minimal polynomials, so the exponent of \((x-\lambda)\) in the minimal polynomial equals the largest block size corresponding to \(\lambda\).

2.4 Generalized eigenspaces

The generalized eigenspace for eigenvalue \(\lambda\) at level \(k\) is \[ \ker\bigl((A-\lambda I)^k\bigr). \] For a Jordan block \(J_n(\lambda)\), the sequence \(\ker(N_n^k)\) grows strictly until \(k=n\), after which it stabilizes to the whole space. In a block diagonal Jordan decomposition, the generalized eigenspace is the direct sum over blocks for that eigenvalue, with dimensions determined by how many chains of each length occur.

3 Jordan Canonical Form

3.1 Existence and uniqueness (block decomposition)

Over an algebraically closed field (or an appropriate splitting field), every square matrix is similar to a direct sum of Jordan blocks. The decomposition is “unique up to ordering of blocks,” and the multiset of block sizes for each eigenvalue is an invariant of the similarity class.

3.2 Jordan normal form via similarity

If \(A\) is similar to its Jordan normal form \(J\), then \(A=PJP^{-1}\) for some invertible \(P\). The similarity transformation transfers structural information: properties preserved under similarity (such as eigenvalues, minimal polynomial, and ranks of \((A-\lambda I)^k\)) can be analyzed by examining the corresponding Jordan blocks in \(J\).

3.3 Building the full Jordan form from blocks

The Jordan normal form is constructed by grouping generalized eigenspaces by eigenvalue and then decomposing each into Jordan chains. Each chain corresponds to one Jordan block, and the collection of chains yields a block diagonal matrix where blocks for distinct eigenvalues appear along the diagonal.

3.4 Geometric vs. algebraic multiplicity

For an eigenvalue \(\lambda\), the algebraic multiplicity is the total number of times \(\lambda\) appears as a diagonal eigenvalue count (the sum of Jordan block sizes). The geometric multiplicity equals the dimension of the ordinary eigenspace \(\ker(A-\lambda I)\), which is the number of Jordan blocks for \(\lambda\). Thus, when several blocks for the same \(\lambda\) are present, the geometric multiplicity is larger; a single block of size \(n\) yields geometric multiplicity \(1\).

4 Generalized Eigenvectors and Jordan Chains

4.1 Definitions of generalized eigenvectors

A generalized eigenvector of \(A\) for eigenvalue \(\lambda\) is a nonzero vector \(v\) such that \((A-\lambda I)^k v=0\) for some positive integer \(k\). The smallest such \(k\) is called the order of \(v\) (relative to \(\lambda\)). When \(k=1\), the vector is an ordinary eigenvector.

4.2 Jordan chains and their length

A Jordan chain is a sequence of vectors \((v_1,\dots,v_r)\) satisfying \[ (A-\lambda I)v_1=0,\quad (A-\lambda I)v_{i+1}=v_i\ \text{for } i=1,\dots,r-1. \] The length \(r\) of the chain corresponds to the size of the associated Jordan block. Longer chains correspond to higher-order nilpotent behavior in the generalized eigenspace.

4.3 Constructing a Jordan basis

To form a Jordan basis, one selects Jordan chains that span the generalized eigenspace for each eigenvalue and then combines these bases across eigenvalues. Practically, the process can be organized by identifying kernels \(\ker(A-\lambda I)^k\) and extending bases step-by-step so that chain relations hold. The resulting basis puts \(A\) into Jordan form under change-of-basis.

4.4 Change of basis and similarity transformation

Once a Jordan basis is chosen, the change-of-basis matrix \(P\) transforms \(A\) into its Jordan normal form: \[ P^{-1}AP = J. \] In this framework, each Jordan chain becomes a column block of \(P\), and the matrix \(A\)’s action on that chain reproduces exactly the superdiagonal “shift” behavior seen in the Jordan block.

5 Functions of Jordan Blocks

5.1 Exponential of a Jordan block

The matrix exponential for a Jordan block can be computed from the decomposition \(J_n(\lambda)=\lambda I + N_n\): \[ e^{tJ_n(\lambda)} = e^{t\lambda}e^{tN_n}. \] Because \(N_n\) is nilpotent, \(e^{tN_n}\) is a finite sum: \[ e^{tN_n}=\sum_{k=0}^{n-1}\frac{t^k}{k!}N_n^k. \] This yields a polynomial-in-\(t\) factor multiplying \(e^{t\lambda}\) along superdiagonals, reflecting the generalized (non-eigenvector) directions.

5.2 Polynomial and analytic functions (functional calculus)

For polynomials \(p\), one defines \(p(J_n(\lambda))\) by substituting the matrix \(J_n(\lambda)\) into the polynomial. More generally, for analytic functions \(f\) (over suitable domains), one can define \(f(J_n(\lambda))\) via the same principle used for diagonalizable matrices, but with the Jordan block requiring derivatives. The result depends only on values of \(f\) and its derivatives at \(\lambda\).

5.3 Computing f(J) using series expansion

A useful formula comes from expanding around \(\lambda\): \[ f(J_n(\lambda)) = \sum_{k=0}^{n-1}\frac{f^{(k)}(\lambda)}{k!}N_n^k, \] since \(N_n^n=0\) eliminates higher terms. This shows that the upper superdiagonals of \(f(J_n(\lambda))\) encode successive derivatives of \(f\) at the eigenvalue.

5.4 Applications to differential equations

Jordan blocks arise naturally when solving linear systems of differential equations. For example, for \(\dot{x}=Ax\), the solution is \(x(t)=e^{tA}x(0)\). Under Jordan decomposition, \(e^{tA}\) splits into block exponentials; Jordan blocks contribute terms of the form \(e^{t\lambda}\) times polynomials in \(t\). These polynomial factors correspond to the presence of generalized eigenvectors and lead to transient growth even when eigenvalues have the same real part.

6 Similarity and Decomposition Techniques

6.1 Determining Jordan block sizes

Jordan block sizes can be deduced from the dimensions of kernels \(\ker((A-\lambda I)^k)\). For a fixed eigenvalue \(\lambda\), the number of Jordan blocks and their sizes can be recovered by examining how these kernel dimensions increase as \(k\) grows. The largest \(k\) for which growth continues indicates the maximum Jordan block size.

6.2 Using kernels of (A−λI)^k

The invariant data is captured by the sequence \[ d_k=\dim\ker((A-\lambda I)^k),\quad k=1,2,\dots \] For a direct sum of Jordan blocks, each block of size \(n\) contributes \(\min(k,n)\) to \(d_k\). Differences \(d_k-d_{k-1}\) reflect how many blocks have size at least \(k\), making these kernels a practical tool for reconstructing the block partition.

6.3 Invariants extracted from ranks

Closely related invariants can be obtained from ranks: \[ \operatorname{rank}((A-\lambda I)^k). \] Since \(\dim\ker(M)=n-\operatorname{rank}(M)\) for an \(n\times n\) matrix, rank information encodes the same kernel dimensions. In computations, ranks may be easier to evaluate reliably than explicit basis construction.

6.4 Examples of decomposition workflow

A typical workflow begins by finding eigenvalues and factoring characteristic polynomials, then computing \(\dim\ker((A-\lambda I)^k)\) for increasing \(k\) to determine the Jordan block sizes for each eigenvalue. Finally, one constructs Jordan chains by selecting vectors in successive kernel layers and enforcing the chain relation \((A-\lambda I)v_{i+1}=v_i\). The assembled chains yield a similarity transform to the Jordan normal form.

7.1 Diagonalizable case (1×1 blocks)

If a matrix is diagonalizable over the field, every Jordan block has size \(1\). In that situation, each \(J_1(\lambda)\) is simply \([\lambda]\), and the Jordan normal form becomes a diagonal matrix with eigenvalues along the diagonal. The generalized eigenvectors reduce to ordinary eigenvectors, and functions like \(e^{tA}\) have no polynomial-in-\(t\) factors.

7.2 Nilpotent Jordan blocks

When \(\lambda=0\), the Jordan block \(J_n(0)\) is nilpotent. Such blocks satisfy \(N_n^n=0\) but \(N_n^{n-1}\ne 0\). Nilpotent blocks are central in the study of Jordan form because they represent the purely non-diagonalizable part of an operator after shifting by \(\lambda I\).

7.3 Connection to companion matrices (conceptual comparison)

Companion matrices are another family of matrices used to represent polynomial data. While companion matrices are closely tied to the structure of a polynomial (especially the characteristic polynomial or minimal polynomial), Jordan blocks are tied to the factorization of that polynomial into linear factors and the organization of generalized eigenspaces. Conceptually, both approaches encode algebraic information into a matrix form, but they organize it differently: companion matrices emphasize a polynomial’s coefficients, whereas Jordan blocks emphasize eigenvalue multiplicities and chain lengths.

7.4 Jordan form over different fields

Jordan decomposition depends on having eigenvalues available in the base field. If the field does not contain all eigenvalues, the matrix may not admit a Jordan form over that field; instead, one may use rational canonical form or other decompositions. Over an extension field where the characteristic polynomial splits, Jordan blocks can be defined and the operator can be fully reduced to Jordan normal form.