1 Background and Definitions

1.1 Eigenvalues and multiplicities

Let \(A\) be a linear transformation on a finite-dimensional vector space over a field (typically \(\mathbb{C}\) or \(\mathbb{R}\) when eigenvalues exist). A scalar \(\lambda\) is an eigenvalue of \(A\) if there exists a nonzero vector \(v\) such that \(Av=\lambda v\). The algebraic multiplicity of \(\lambda\) is the multiplicity of \(\lambda\) as a root of the characteristic polynomial \[ \chi_A(t)=\det(tI-A). \] Algebraic multiplicity counts how many times the factor \((t-\lambda)\) appears in \(\chi_A(t)\).

1.2 Geometric versus algebraic multiplicity

The geometric multiplicity of \(\lambda\) is the dimension of the eigenspace \[ E_\lambda=\ker(A-\lambda I), \] which equals the number of linearly independent eigenvectors associated with \(\lambda\). By general theory, \[ 1 \le \dim E_\lambda \le \text{algebraic multiplicity of } \lambda. \] Equality holds exactly when enough eigenvectors exist to match the algebraic count.

1.3 Defective eigenvalue: formal characterization

An eigenvalue \(\lambda\) is called defective (or that the matrix is non-diagonalizable because of \(\lambda\)) when its geometric multiplicity is strictly less than its algebraic multiplicity. Equivalently, \[ \dim \ker(A-\lambda I) < m_{\text{alg}}(\lambda). \] A matrix is diagonalizable over the chosen field precisely when every eigenvalue has geometric multiplicity equal to its algebraic multiplicity; otherwise, some eigenvalue(s) are defective.

1.4 Generalized eigenvectors and their role

When eigenvectors are insufficient, one enlarges the notion of eigenspace using generalized eigenvectors. For a defective eigenvalue \(\lambda\), vectors satisfying \[ (A-\lambda I)^k v = 0 \] for some positive integer \(k\) but not necessarily for \(k=1\) are generalized eigenvectors. They fill the gap between the ordinary eigenspace and the size implied by algebraic multiplicity, and they are the building blocks for Jordan canonical form and related constructions.

2 Detecting Defective Eigenvalues

2.1 Using eigenspace dimensions

2.1.1 Computing nullspaces for \(A-\lambda I\)

A direct approach is to compute \(\ker(A-\lambda I)\). If \(\lambda\) is known, one forms the matrix \(A-\lambda I\) and determines the nullspace dimension. This dimension is the geometric multiplicity.

In practice, one may row-reduce \(A-\lambda I\) to obtain the rank and then use nullity to find the eigenspace dimension.

2.1.2 Rank-nullity checks

Over a vector space of dimension \(n\), \[ \dim \ker(A-\lambda I) = n - \operatorname{rank}(A-\lambda I). \] Comparing this quantity with the algebraic multiplicity of \(\lambda\) extracted from \(\chi_A(t)\) gives a test: if \[ n-\operatorname{rank}(A-\lambda I) < m_{\text{alg}}(\lambda), \] then \(\lambda\) is defective.

2.2 Characteristic and minimal polynomials

The minimal polynomial \(\mu_A(t)\) is the monic polynomial of least degree such that \(\mu_A(A)=0\). Defectiveness is often reflected in the structure of \(\mu_A\): if \(\mu_A(t)\) contains a squared factor \((t-\lambda)^2\), then \(A\) is not diagonalizable, and \(\lambda\) is involved in nontrivial Jordan structure.

More precisely:

  • \(A\) is diagonalizable over the field if and only if \(\mu_A(t)\) has no repeated roots (equivalently, \(\mu_A\) is square-free).
  • Repeated factors in \(\mu_A\) indicate that some eigenvalue has a Jordan block of size greater than \(1\), which corresponds to a defect in eigenvector count.

2.3 Jordan block viewpoint (intuition)

Each Jordan block of size \(k\ge 2\) associated with \(\lambda\) contributes only one eigenvector but accounts for \(k\) toward algebraic multiplicity. Thus, the presence of a Jordan block larger than \(1\) is another way to detect a defective eigenvalue. Even without explicitly forming Jordan form, repeated factors in \(\mu_A\) or a smaller-than-expected eigenspace dimension reveal that not all algebraic multiplicity can be realized by genuine eigenvectors.

2.4 Examples of detection workflows

A common workflow:

  1. Compute \(\chi_A(t)\) and factor it to find candidate eigenvalues and their algebraic multiplicities.
  2. For each candidate \(\lambda\), compute \(\ker(A-\lambda I)\) and obtain geometric multiplicity.
  3. Compare multiplicities; whenever geometric multiplicity is smaller, the eigenvalue is defective.
  4. Optionally, compute or infer \(\mu_A(t)\) to confirm whether Jordan block sizes exceed \(1\).

This sequence is often used in symbolic settings and in classroom computations, while numerical settings may use additional diagnostics like spectral projectors or algorithms that detect non-diagonalizable behavior.

3 Jordan Canonical Form and Structure

3.1 Jordan blocks and multiplicities

Over an algebraically closed field, any square matrix \(A\) is similar to a block diagonal matrix composed of Jordan blocks. For an eigenvalue \(\lambda\), a Jordan block of size \(k\) has the form \[ J_k(\lambda)= \begin{pmatrix} \lambda &amp; 1 &amp; 0 &amp; \cdots &amp; 0\\ 0 &amp; \lambda &amp; 1 &amp; \cdots &amp; 0\\ \vdots &amp; &amp; \ddots &amp; \ddots &amp; \vdots\\ 0 &amp; \cdots &amp; 0 &amp; \lambda &amp; 1\\ 0 &amp; \cdots &amp; \cdots &amp; 0 &amp; \lambda \end{pmatrix}. \] The algebraic multiplicity of \(\lambda\) equals the total size of all Jordan blocks for \(\lambda\).

3.2 Relationship between defect and Jordan chains

Defectiveness is exactly the failure of eigenvectors to cover the algebraic multiplicity. In the Jordan picture, a Jordan block of size \(k\) contributes:

  • \(1\) linearly independent eigenvector (one “head” of the chain),
  • but \(k\) dimensions to the generalized eigenspace.

Therefore, if the eigenspace dimension is smaller than algebraic multiplicity, it means some Jordan block(s) have size greater than \(1\), creating nontrivial Jordan chains.

3.3 Size of Jordan blocks and missing eigenvectors

If \(\lambda\) has algebraic multiplicity \(m\) and there are \(r\) Jordan blocks for \(\lambda\), then the geometric multiplicity is \(r\). Defect occurs when \(r&lt;m\). The “missing” eigenvectors correspond to the number and sizes of blocks exceeding size \(1\). Larger blocks imply longer generalized chains, which in turn influence how matrix functions behave (e.g., powers and exponentials develop polynomial factors).

3.4 Diagonalizable versus non-diagonalizable cases

A matrix is diagonalizable if and only if every Jordan block has size \(1\). In that situation, generalized eigenvectors collapse to ordinary eigenvectors, and the Jordan canonical form becomes a diagonal matrix. Non-diagonalizable behavior is localized to eigenvalues with at least one Jordan block of size \(&gt;1\), precisely the eigenvalues that are defective.

4 Consequences for Linear Systems

4.1 Matrix powers with defective eigenvalues

Consider powers \(A^n\). Using Jordan decomposition \(A = PJP^{-1}\), one can compute \[ A^n = PJ^nP^{-1}. \] For a Jordan block \(J_k(\lambda)=\lambda I + N\) with nilpotent \(N\) (where \(N^k=0\) and \(N^{k-1}\ne 0\)), one obtains \[ J_k(\lambda)^n = (\lambda I + N)^n = \sum_{j=0}^{k-1} \binom{n}{j}\lambda^{n-j} N^j. \] The key feature is the appearance of binomial coefficients \(\binom{n}{j}\), which create growth patterns that differ from the pure exponential \(\lambda^n\) behavior seen in diagonalizable cases.

4.2 Exponential of a matrix (matrix \(e^{tA}\))

For a linear system \(\dot{x}=Ax\), solutions involve \(e^{tA}\). With Jordan form, \[ e^{tA}=Pe^{tJ}P^{-1}. \] On a Jordan block, \[ e^{tJ_k(\lambda)} = e^{t\lambda} e^{tN} = e^{t\lambda}\sum_{j=0}^{k-1}\frac{t^j}{j!}N^j. \] Thus, defective eigenvalues introduce polynomial-in-\(t\) factors multiplying \(e^{t\lambda}\). These terms are absent when the matrix is diagonalizable.

4.3 Transient growth from Jordan blocks

Even if all eigenvalues suggest decay (for instance, eigenvalues with negative real part), polynomial factors can produce pronounced transient behavior. Large Jordan blocks can create growth in norms over intermediate times before asymptotic decay dominates (when decay is ensured by eigenvalues). This “non-normality” effect is structurally tied to the presence of Jordan blocks larger than \(1\).

4.4 Stability and long-term behavior (structural perspective)

Long-term behavior is governed primarily by eigenvalues with maximal real part. However, defectiveness modifies the approach to the long-term regime by adding polynomial prefactors. Structurally:

  • Repeated eigenvalues with nontrivial Jordan blocks can slow decay or alter transient magnitude.
  • If eigenvalues have zero real part, polynomial growth can lead to unbounded behavior even when the spectrum is not strictly unstable in the naive exponential sense.

This perspective emphasizes that stability is not determined by eigenvalues alone when non-diagonalizable structure is present.

5 Generalized Eigenvector Chains

5.1 Jordan chains: definition and construction

A Jordan chain for eigenvalue \(\lambda\) of length \(k\) is a sequence of vectors \((v_1,\dots,v_k)\) such that \[ (A-\lambda I)v_1=0,\quad (A-\lambda I)v_2=v_1,\quad \dots,\quad (A-\lambda I)v_k=v_{k-1}. \] Here, \(v_1\) is an ordinary eigenvector, and \(v_k\) is a generalized eigenvector at level \(k\). The chain length corresponds to the size of the associated Jordan block.

5.2 Building bases from generalized eigenvectors

For each eigenvalue, generalized eigenvectors can be assembled into a basis by taking a collection of Jordan chains whose total lengths match the algebraic multiplicities. When such chains are chosen appropriately, their union forms a basis in which the matrix takes Jordan canonical form. Defective behavior manifests through chains longer than \(1\).

5.3 Resolvent behavior near defective eigenvalues

The resolvent \((A-\lambda I)^{-1}\) (when defined) reveals spectral structure. Near a defective eigenvalue, the resolvent can develop higher-order singularities. Intuitively, the nilpotent parts in Jordan blocks contribute terms that behave like \((z-\lambda)^{-j}\) for \(j\) up to the maximum Jordan block size. This explains why non-diagonalizable matrices can exhibit more severe sensitivity in spectral computations and perturbations.

5.4 Practical computation strategies

When computing generalized eigenvectors directly:

  • First determine \(\lambda\) and compute eigenspace \(E_\lambda\).
  • Then seek vectors \(v\) solving \((A-\lambda I)v = u\) for previously found eigenvectors \(u\).
  • Repeatedly apply this lifting process to build longer chains.

Verification is essential: each candidate vector must satisfy the correct chain equations up to the intended level, and the constructed set must be linearly independent.

6 Algebraic and Computational Considerations

6.1 Symbolic vs numerical methods

Symbolic methods (exact arithmetic, rational/algebraic field manipulations) can reveal Jordan structure exactly for matrices with suitable entries, but may be computationally expensive as dimension grows or as algebraic numbers appear.

Numerical approaches typically avoid explicit Jordan form because it can be extremely sensitive to perturbations. Instead, they may detect diagonalizability indirectly (e.g., via ranks of \(A-\lambda I\) and \( (A-\lambda I)^k\), or via invariant subspace computations).

6.2 Conditioning and perturbation effects (conceptual)

Jordan form is notoriously ill-conditioned: small perturbations of \(A\) can dramatically change the Jordan structure, especially when eigenvalues coalesce. As a result:

  • A matrix that is exactly defective may appear nearly diagonalizable under perturbation.
  • Conversely, numerically observed near-defectiveness may not correspond to an exact Jordan block structure unless verified with exact arithmetic or robust criteria.

Common computational workflows include:

  • Computing eigenvalues and then verifying eigenspace dimensions by nullspace/rank computations of \(A-\lambda I\).
  • Estimating minimal polynomial factors using controllable/observable structure or polynomial basis methods.
  • In specialized settings, computing Jordan form (or Schur form with further refinement) and then extracting Jordan block sizes through chain construction and verification.

Even when software can output Jordan form, independent checks (chain relations, ranks of \((A-\lambda I)^k\)) are recommended.

6.4 Common pitfalls and verification steps

Typical issues include:

  • Mistaking approximate eigenvalues from floating-point computation for exact algebraic multiplicities.
  • Using an eigenspace computed numerically without confirming its dimension against theoretical multiplicity.
  • Assuming that repeated eigenvalues imply defect; multiplicities alone do not guarantee non-diagonalizability.

A robust verification approach compares geometric multiplicity with algebraic multiplicity, and, when needed, examines whether \((A-\lambda I)^k\) introduces additional nullity consistent with larger Jordan blocks.

7 Worked Examples

7.1 \(2\times 2\) defective eigenvalue example

Let \[ A=\begin{pmatrix} 1 &amp; 1\\ 0 &amp; 1 \end{pmatrix}. \] The characteristic polynomial is \(\chi_A(t)=(t-1)^2\), so \(\lambda=1\) has algebraic multiplicity \(2\). Compute \[ A-I=\begin{pmatrix} 0 &amp; 1\\ 0 &amp; 0 \end{pmatrix}. \] Then \(\ker(A-I)\) consists of vectors \((x,0)^\top\), which has dimension \(1\). Hence geometric multiplicity is \(1\), which is smaller than \(2\), so \(1\) is defective.

One can also note that \(A\) has a single Jordan block of size \(2\) for \(\lambda=1\), which accounts for the missing eigenvector.

7.2 \(3\times 3\) Jordan block example

Consider \[ A= \begin{pmatrix} 2 &amp; 1 &amp; 0\\ 0 &amp; 2 &amp; 1\\ 0 &amp; 0 &amp; 2 \end{pmatrix} =J_3(2). \] Here the characteristic polynomial is \((t-2)^3\), so the algebraic multiplicity of \(\lambda=2\) is \(3\). Compute \[ A-2I= \begin{pmatrix} 0 &amp; 1 &amp; 0\\ 0 &amp; 0 &amp; 1\\ 0 &amp; 0 &amp; 0 \end{pmatrix}. \] The nullspace has dimension \(1\) (eigenvectors are scalar multiples of \((1,0,0)^\top\)), so the geometric multiplicity is \(1&lt;3\). The eigenvalue \(2\) is therefore defective, and the Jordan structure is a single chain of length \(3\).

7.3 Comparing diagonalizable and defective cases

Take \[ B= \begin{pmatrix} 2 &amp; 0 &amp; 0\\ 0 &amp; 2 &amp; 0\\ 0 &amp; 0 &amp; 3 \end{pmatrix}. \] The eigenvalue \(2\) has algebraic multiplicity \(2\). The eigenspace \(\ker(B-2I)\) is spanned by the first two coordinate vectors, so its dimension is \(2\). Therefore \(2\) is not defective, and \(B\) is diagonalizable.

Comparing \(B\) with the previous Jordan block example illustrates that repeated eigenvalues can be either diagonalizable or defective depending on eigenspace dimension and Jordan block sizes.

7.4 Interpreting results in terms of eigenspaces

In defective cases, the eigenspace dimension underestimates the dimension of the generalized eigenspace. For instance, in the \(3\times 3\) Jordan block \(J_3(2)\), ordinary eigenvectors account for only one dimension, while generalized eigenvectors fill the full three-dimensional invariant subspace. This interpretation clarifies why matrix functions exhibit additional polynomial factors: the system “needs” generalized eigenvectors to represent its dynamics in a basis compatible with \(A\).

8.1 Diagonalizability criteria

A matrix \(A\) over a field is diagonalizable if and only if it has a basis of eigenvectors. Equivalently, for every eigenvalue \(\lambda\), the geometric multiplicity equals the algebraic multiplicity. Defective eigenvalues are precisely those for which this equality fails.

8.2 Minimal polynomial

The minimal polynomial \(\mu_A(t)\) provides an efficient diagnostic for diagonalizability. If \(\mu_A\) has repeated roots, then \(A\) is not diagonalizable. For defective eigenvalues, \(\mu_A\) includes higher powers \((t-\lambda)^k\) with \(k\ge 2\), reflecting the existence of Jordan blocks larger than \(1\).

8.3 Eigenspace decomposition vs Jordan decomposition

Eigenspace decomposition expresses a diagonalizable matrix as a direct sum of eigenspaces. For non-diagonalizable matrices, eigenspace decomposition is insufficient: generalized eigenspaces and Jordan chains replace the role of ordinary eigenspaces. Jordan decomposition organizes the action of \(A\) into block structures that capture both eigenvalue scaling and nilpotent coupling.

8.4 Connections to polynomial functional calculus

For a polynomial \(p(t)\), applying \(p(A)\) corresponds to substituting \(A\) into the polynomial. In the diagonalizable case, \(p(A)\) acts by scaling eigenvectors with \(p(\lambda)\). In the defective case, nilpotent parts contribute derivatives-like behavior in the Jordan block setting, leading to nontrivial off-diagonal effects inside \(p(J)\). This explains why defectiveness influences not only dynamics (powers and exponentials) but also general polynomial transformations.