Defective matrix - Ciallo Wiki

1 Basic definitions and characterizations

1.1 Eigenvalues, algebraic multiplicity, and geometric multiplicity

Let \(A\) be an \(n\times n\) complex (or real) matrix. An eigenvalue \(\lambda\) is a scalar for which \(A v=\lambda v\) for some nonzero vector \(v\). The algebraic multiplicity of \(\lambda\), denoted \(m_a(\lambda)\), is its multiplicity as a root of the characteristic polynomial \(\chi_A(t)=\det(tI-A)\). The geometric multiplicity, denoted \(m_g(\lambda)\), is the dimension of the corresponding eigenspace \[ E_\lambda=\ker(A-\lambda I), \] so \(m_g(\lambda)=\dim \ker(A-\lambda I)\).

Always, \(m_g(\lambda)\le m_a(\lambda)\). Equality means there are “enough” independent eigenvectors to account for the algebraic multiplicity. Strict inequality indicates a structural deficiency in the eigenvector set.

1.2 What makes a matrix defective

A matrix \(A\) is called defective if it does not admit a full set of \(n\) linearly independent eigenvectors. Equivalently, there exists at least one eigenvalue \(\lambda\) such that \[ m_g(\lambda) < m_a(\lambda). \] In such cases, even though the eigenvalues are well-defined, the eigenspaces are too small (relative to how often eigenvalues repeat in the characteristic polynomial) to span the entire space.

1.3 Equivalent conditions and criteria

Defectiveness can be characterized in several interchangeable ways. A common equivalent statement is:

\(A\) is diagonalizable if and only if for every eigenvalue \(\lambda\), \(m_g(\lambda)=m_a(\lambda)\).
Therefore, \(A\) is defective if and only if there exists \(\lambda\) with \(m_g(\lambda)<m_a(\lambda)\).

Another criterion uses the existence of a basis consisting entirely of eigenvectors: if no such basis exists, the matrix is defective.

1.4 Complete eigenvector sets vs diagonalizability

Diagonalizability is precisely the property that allows a decomposition of the form \[ A = P D P^{-1}, \] where \(P\) is invertible and \(D\) is diagonal (with diagonal entries the eigenvalues). This is possible exactly when \(A\) has a complete eigenvector basis. Since defective matrices lack such a basis, they cannot be reduced to diagonal form by similarity. Nonetheless, they often admit related canonical structures that retain enough information for analysis.

2 Jordan canonical form viewpoint

2.1 Jordan blocks and their meaning

Over an algebraically closed field (e.g., \(\mathbb{C}\)), every square matrix is similar to a Jordan canonical form \(J\), which is block diagonal with Jordan blocks. A Jordan block of size \(s\) for eigenvalue \(\lambda\) has the form \[ J_s(\lambda)= \begin{pmatrix} \lambda & 1 & 0 & \cdots & 0\\ 0 & \lambda & 1 & \cdots & 0\\ \vdots & & \ddots & \ddots & \vdots\\ 0 & \cdots & 0 & \lambda & 1\\ 0 & \cdots & \cdots & 0 & \lambda \end{pmatrix}. \] The presence of blocks larger than \(1\times 1\) reflects a failure of eigenvectors alone to span the space.

2.2 Generalized eigenvectors

When \(A\) is defective, one uses generalized eigenvectors. A vector \(v\neq 0\) is a generalized eigenvector of rank \(k\) for eigenvalue \(\lambda\) if \[ (A-\lambda I)^k v = 0 \quad\text{but}\quad (A-\lambda I)^{k-1}v\neq 0 \] (for the smallest such \(k\)). These vectors form chains that correspond to Jordan blocks.

2.3 Relationships to minimal polynomials

The minimal polynomial \(\mu_A(t)\) of \(A\) is the monic polynomial of smallest degree such that \(\mu_A(A)=0\). In the Jordan form, the minimal polynomial captures the largest Jordan block size associated with each eigenvalue. Concretely, if the largest Jordan block for \(\lambda\) has size \(s\), then \((t-\lambda)^s\) divides \(\mu_A(t)\) and no higher power does.

This makes the minimal polynomial a compact witness of defectiveness: repeated eigenvalues require powers of \((t-\lambda)\) beyond the first factor when nontrivial Jordan blocks exist.

2.4 How defectiveness shows up in Jordan structure

A matrix is diagonalizable exactly when every Jordan block has size \(1\). Thus, defectiveness corresponds to at least one Jordan block of size \(s\ge 2\). In that case, the eigenspace dimension for the associated eigenvalue is smaller than the algebraic multiplicity, because one Jordan block contributes only one independent eigenvector even though it may account for multiple dimensions of the generalized eigenspace.

3 Spectral decomposition and limits of diagonalization

3.1 Diagonalizable vs non-diagonalizable cases

In diagonalizable cases, powers and functions of \(A\) are computed using the eigen-decomposition \(A=PDP^{-1}\). For non-diagonalizable matrices, the diagonalization formula fails because \(P\) cannot be formed from eigenvectors alone.

Nevertheless, \(A\) still admits a decomposition in terms of Jordan blocks. Spectral information persists, but additional terms (coming from the nilpotent parts of blocks) appear in expansions.

3.2 Partial spectral decompositions

A spectral decomposition can still be described by grouping Jordan blocks by eigenvalues. This yields a decomposition of the space into invariant subspaces associated with each eigenvalue, often described using generalized eigenspaces \[ \mathcal{G}_\lambda = \ker (A-\lambda I)^{m_a(\lambda)}. \] Within each \(\mathcal{G}_\lambda\), the operator has a Jordan-type structure; across different eigenvalues, the components decouple.

3.3 Functional calculus on defective matrices

For a sufficiently well-defined scalar function \(f\), one can define \(f(A)\) using polynomial approximation or, more directly over \(\mathbb{C}\), via the Jordan form. The key idea is that on each Jordan block, \[ f(J_s(\lambda)) \] depends not only on \(f(\lambda)\) but also on derivatives of \(f\) up to order \(s-1\). Thus, defectiveness increases the “order of derivatives” required in the functional calculus.

3.4 Matrix functions and how Jordan form guides computation

If \(A\) is similar to \(J\), then \(f(A)\) is similar to \(f(J)\): \[ f(A) = P f(J) P^{-1}. \] On a Jordan block \(J_s(\lambda)=\lambda I + N\), with \(N\) nilpotent (ones on the superdiagonal and \(N^s=0\)), one uses expansions such as \[ f(\lambda I + N)=\sum_{j=0}^{s-1}\frac{f^{(j)}(\lambda)}{j!}N^j. \] This formula shows why generalized eigenvectors and nilpotent components are essential: the nilpotent part contributes higher-order behavior that does not appear in the purely diagonal case.

4 Computing and diagnosing defectiveness

4.1 Checking eigenvalue multiplicities

A first diagnostic step is to compute the characteristic polynomial and its factorization to determine eigenvalues and algebraic multiplicities. In practice, numerical eigenvalue algorithms give approximations, after which one must infer multiplicities (often carefully, because numerical error can blur repeated eigenvalues).

4.2 Computing eigenspaces and their dimensions

For each eigenvalue \(\lambda\), one computes the eigenspace by solving the homogeneous system \[ (A-\lambda I)x=0. \] The dimension of the solution space provides the geometric multiplicity \(m_g(\lambda)\). If for any eigenvalue \(m_g(\lambda)<m_a(\lambda)\), the matrix is defective.

4.3 Rank and nullity tests for eigenspaces

Geometric multiplicity can be assessed via rank computations. Since \[ \dim \ker(A-\lambda I)= n-\operatorname{rank}(A-\lambda I), \] one can determine \(m_g(\lambda)\) from the rank of \(A-\lambda I\). In exact arithmetic, this gives a definitive answer; in floating-point computation, rank decisions require tolerances and careful conditioning.

4.4 Using the minimal polynomial to confirm structure

The minimal polynomial can be used as a confirmatory tool. If \(\mu_A(t)\) contains a factor \((t-\lambda)^s\) with \(s>1\) for a repeated eigenvalue, then \(A\) cannot be diagonalizable and is therefore defective. Conversely, the sizes of Jordan blocks inferred from \(\mu_A(t)\) align with the failure of the eigenspace condition \(m_g(\lambda)=m_a(\lambda)\).

5 Examples and illustrative computations

5.1 Simple 2×2 defective examples

Consider \[ A= \begin{pmatrix} \lambda & 1\\ 0 & \lambda \end{pmatrix}. \] The characteristic polynomial is \((t-\lambda)^2\), so \(m_a(\lambda)=2\). The eigenspace satisfies \[ (A-\lambda I)x=0 \quad\Rightarrow\quad \begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix}x=0, \] which yields vectors of the form \(x=(1,0)^T\), so \(m_g(\lambda)=1\). Since \(1<2\), the matrix is defective. Its Jordan form is itself: a single \(2\times 2\) Jordan block.

5.2 A 3×3 example with a single Jordan chain

Let \[ A= \begin{pmatrix} \lambda & 1 & 0\\ 0 & \lambda & 1\\ 0 & 0 & \lambda \end{pmatrix}. \] Then \(\chi_A(t)=(t-\lambda)^3\), so \(m_a(\lambda)=3\). The eigenspace solves \((A-\lambda I)x=0\), where \[ A-\lambda I= \begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end{pmatrix}. \] This forces the last two coordinates to vanish, leaving \(m_g(\lambda)=1\). The deficiency is strong: only one eigenvector exists, but there is a length-3 Jordan chain of generalized eigenvectors.

5.3 Building larger defective matrices from smaller blocks

Defective behavior is preserved under direct sums of Jordan blocks. For example, combining a \(2\times 2\) Jordan block with a \(1\times 1\) block yields a \(3\times 3\) matrix that is defective because the \(2\times 2\) part already violates diagonalizability. More generally, any matrix similar to a Jordan form containing at least one block of size \(>1\) is defective.

This block perspective makes construction easy: defectiveness can be engineered by specifying block sizes, then assembling a matrix similar to that Jordan form.

5.4 Visual intuition via eigenvector availability

A useful intuition is that each distinct Jordan block contributes a limited number of true eigenvectors for its eigenvalue. When eigenvalues repeat, the vector space dimension grows, but the number of independent eigenvectors may not keep pace. Visually, generalized eigenvectors “fill in” the missing directions by satisfying higher-order annihilation conditions \((A-\lambda I)^k v=0\), revealing the structure behind the lack of diagonalization.

6 Dynamics and powers of defective matrices

6.1 Behavior of \(A^k\) near Jordan blocks

Powers of a Jordan block grow polynomially in \(k\) in addition to the exponential scaling by \(\lambda^k\). For \(J_s(\lambda)=\lambda I + N\) with \(N^s=0\), \[ J_s(\lambda)^k = (\lambda I+N)^k = \sum_{j=0}^{s-1}\binom{k}{j}\lambda^{k-j}N^j. \] The binomial coefficients introduce polynomial factors in \(k\). This distinguishes defective matrices from diagonalizable ones, where powers typically involve only \(\lambda^k\) scaling without polynomial growth within each invariant subspace.

6.2 Exponential of a defective matrix

Matrix exponentials are central in linear systems and differential equations. For \(A\) similar to \(J\), \(e^{At}\) is similar to \(e^{Jt}\). On a Jordan block, \[ e^{J_s(\lambda)t}=e^{\lambda t}\,e^{Nt}. \] Since \(N\) is nilpotent, \(e^{Nt}\) is a finite polynomial in \(t\), with degrees up to \(s-1\). Therefore, terms like \(t e^{\lambda t}\) or \(t^2 e^{\lambda t}\) can appear, capturing how defectiveness amplifies transient behavior.

6.3 Differential equation solutions via generalized eigenvectors

Consider the system \(\dot{x}=Ax\). Solutions have the form \(x(t)=e^{At}x(0)\). In diagonalizable settings, solutions are superpositions of exponentials \(e^{\lambda t}\). For defective matrices, generalized eigenvectors generate additional polynomial-in-\(t\) multipliers multiplying exponentials. This is why Jordan chains are often used to write explicit solution formulas: each chain corresponds to a family of linearly independent solution trajectories.

6.4 Stability and growth rates from Jordan structure

Stability depends on eigenvalue real parts, but defectiveness influences whether decay or growth includes polynomial factors. If all eigenvalues have negative real part, solutions still decay, yet generalized eigenvectors can slow decay through polynomial prefactors. If an eigenvalue has zero real part and a nontrivial Jordan block exists, the polynomial terms can cause unbounded growth even though eigenvalues alone might suggest neutral behavior. Thus, Jordan structure refines stability assessments beyond eigenvalue locations alone.

7 Algebraic properties and transformations

7.1 Similarity invariance of defectiveness

Defectiveness is invariant under similarity transformations. If \(B=S^{-1}AS\) for invertible \(S\), then \(A\) and \(B\) share the same Jordan block sizes and eigenvalue multiplicities. Consequently, either both matrices are diagonalizable or both are defective.

7.2 Block structure under change of basis

While the specific matrix entries change under a basis change, the underlying invariant subspace decomposition remains. For defective matrices, one cannot transform the operator into a diagonal form, but it can be transformed into a block Jordan structure. This block structure organizes behavior for computations like \(A^k\), \(e^{At}\), and \(f(A)\).

7.3 Defective matrices under addition and perturbation (conceptual)

In general, adding matrices or applying perturbations can alter the Jordan structure. Small perturbations may break repeated eigenvalue patterns or split eigenvalues, potentially turning a defective matrix into a diagonalizable one. Conversely, parameter-dependent families can create or remove Jordan blocks. While the exact outcome depends on the perturbation and algebraic constraints, the conceptual point is that defectiveness is a property tied to fine algebraic structure, not just eigenvalue locations.

7.4 Commuting matrices and shared Jordan structure (overview)

If two matrices commute and are simultaneously triangularizable under suitable conditions, their invariant subspace structures interact. In special cases, commuting operators can share eigenvectors or generalized eigenvectors, enabling partial alignment of Jordan structures. An overview viewpoint is that commuting constraints can restrict how Jordan blocks evolve and which generalized eigenspaces remain compatible, though full simultaneous Jordan form requires stronger assumptions.

8 Numerical considerations

8.1 Sensitivity of eigenvectors vs eigenvalues

Eigenvalues are often more stable under perturbations than eigenvectors. For defective or nearly defective matrices, eigenvectors can vary dramatically even for small changes in data. This is closely linked to the fact that eigenvector bases become ill-conditioned when Jordan blocks are large or eigenvalues are nearly coalescing.

Jordan decomposition is highly sensitive: the Jordan form itself is not robust under perturbations. When a matrix is defective, infinitesimal perturbations typically produce nearby matrices whose Jordan blocks split into diagonalizable ones or different block configurations. As a result, numerical algorithms usually avoid explicitly computing Jordan chains; instead they compute stable quantities such as Schur forms or use generalized eigenspace information.

8.3 Practical diagnostics in numerical linear algebra

In computation, defectiveness is often diagnosed through ranks of \((A-\lambda I)\) for candidate eigenvalues, or by analyzing invariant subspace behavior via Schur decompositions and numerical multiplicities. When eigenvalues are close, deciding the “effective” geometric multiplicity requires thresholds. Careful interpretation is needed: numerical evidence of defectiveness may reflect near-defectiveness rather than exact Jordan blocks.

In engineering contexts such as modal analysis, defective systems can exhibit modes that are not captured by a pure eigenvector expansion. Polynomial terms in time, associated with Jordan blocks, can produce transient growth or slow decay that standard diagonal modal models miss. In simulations, this can manifest as discrepancies between predicted modal responses and observed dynamics, especially near parameter regimes where eigenvalues coalesce.

9.1 Diagonalizable matrices

Diagonalizable matrices admit a complete eigenvector basis and can be expressed as \(A=PDP^{-1}\). This property enables straightforward computation of powers, exponentials, and functions via the diagonal entries of \(D\).

9.2 Nondiagonalizable but not defective cases

A matrix can be nondiagonalizable without being defective in certain contexts over non-algebraically-closed fields, depending on how eigenvalues are represented and how eigenspaces behave. The typical real/complex distinction can complicate statements framed over \(\mathbb{C}\). In the usual setting over \(\mathbb{C}\), failure to be diagonalizable aligns directly with defectiveness.

9.3 Normal matrices and spectral theorems

Normal matrices satisfy \(A^*A=AA^*\) and enjoy a spectral theorem: they are unitarily diagonalizable over \(\mathbb{C}\). As a result, normal matrices cannot be defective, since they always have orthonormal eigenvector bases.

9.4 Generalized eigenvalue problems and Jordan chains

Generalized eigenvalue problems, often written as \(A x=\lambda B x\) with \(B\) singular or nontrivial, are closely related in spirit to generalized eigenvectors for defective operators. Jordan chains provide the canonical language for describing solution spaces when eigenvectors alone are insufficient, and they underlie advanced analyses of systems with repeated or defective eigenstructures.