1 Diagonalizability and Its Criteria

1.1 Definitions: diagonalizable vs. non-diagonalizable

Let \(A\) be an \(n\times n\) matrix over a field (typically \(\mathbb{R}\) or \(\mathbb{C}\)). The matrix \(A\) is diagonalizable if there exists an invertible matrix \(P\) and a diagonal matrix \(D\) such that \[ A=PDP^{-1}. \] If no such representation exists, \(A\) is non-diagonalizable. A diagonalizable matrix has a basis of eigenvectors, while a non-diagonalizable matrix cannot be spanned by eigenvectors alone.

1.2 Eigenvectors, eigenvalues, and spanning conditions

If \(A\) is diagonalizable, its eigenvalues determine the diagonal entries of \(D\), and the columns of \(P\) form a basis of eigenvectors of \(A\). Conversely, if there are \(n\) linearly independent eigenvectors (equivalently, if eigenvectors span \(\mathbb{F}^n\)), then \(A\) is diagonalizable. Failure of this spanning condition is the practical hallmark of non-diagonalizability.

1.3 Algebraic vs. geometric multiplicity

For an eigenvalue \(\lambda\), the algebraic multiplicity \(m_a(\lambda)\) is its multiplicity as a root of the characteristic polynomial. The geometric multiplicity \(m_g(\lambda)\) is the dimension of the eigenspace \[ E_\lambda=\ker(A-\lambda I). \] Always \(1 \le m_g(\lambda)\le m_a(\lambda)\). A matrix is diagonalizable precisely when, for every eigenvalue \(\lambda\), one has \(m_g(\lambda)=m_a(\lambda)\). If there exists an eigenvalue where this equality fails, \(A\) is non-diagonalizable.

1.4 Minimal polynomial and diagonalizability tests

The minimal polynomial \(m_A(t)\) is the monic polynomial of least degree such that \(m_A(A)=0\). Its structure controls diagonalizability: a matrix over an algebraically closed field is diagonalizable if and only if its minimal polynomial splits into distinct linear factors, meaning it has no repeated roots. Equivalently, the largest power of \((t-\lambda)\) appearing in \(m_A(t)\) has exponent \(1\) for every eigenvalue \(\lambda\). If some \((t-\lambda)\) appears with exponent greater than \(1\), non-diagonalizability follows.

2 Structural Causes of Non-diagonalizability

2.1 Defective eigenvalues and their interpretation

An eigenvalue \(\lambda\) is sometimes called defective when its eigenspace is too small, i.e., \(m_g(\lambda) < m_a(\lambda)\). Intuitively, the transformation acts like it “wants” to have enough independent directions associated with \(\lambda\), but the geometry of the null space prevents it. This deficit forces generalized eigenvectors to appear.

2.2 Invariant subspaces and eigenvector deficits

Even when eigenvectors do not fully span the space, there are still invariant subspaces generated by the action of \(A\). For each eigenvalue \(\lambda\), the generalized eigenspace \[ G_\lambda=\ker(A-\lambda I)^{m_a(\lambda)} \] is invariant under \(A\). Non-diagonalizability reflects that the invariant structure is richer than eigenspaces alone; Jordan chains organize how vectors “shift” under repeated application of \(A-\lambda I\).

2.3 Examples from characteristic polynomials

A common source of non-diagonalizability is a characteristic polynomial with repeated roots. Repetition in the characteristic polynomial does not automatically imply non-diagonalizability, but it creates the possibility that \(m_g(\lambda)\) is strictly smaller than \(m_a(\lambda)\). For instance, a characteristic polynomial containing \((t-\lambda)^k\) may still yield a full eigenspace (diagonalizable case) or may fall short (non-diagonalizable case). The decision is made by geometric multiplicity or, equivalently, by the minimal polynomial.

2.4 What changes when eigenvectors are insufficient

When eigenvectors are insufficient, several structural consequences occur:

  • The space cannot be decomposed into a direct sum of eigenspaces.
  • Iterates \(A^k\) may involve polynomial factors in \(k\) multiplying the usual exponential behavior from eigenvalues.
  • Canonical descriptions require additional data beyond eigenvalues and eigenspaces, such as Jordan block sizes or generalized eigenvector chains.

3 Jordan Normal Form for Non-diagonalizable Matrices

3.1 Overview of Jordan form

Over \(\mathbb{C}\) (or any field where eigenvalues split), every square matrix \(A\) is similar to a Jordan normal form \(J\), meaning \(A = PJP^{-1}\) for some invertible \(P\). The Jordan form is block diagonal, with each block corresponding to an eigenvalue \(\lambda\) and having a nearly diagonal structure. If \(A\) is diagonalizable, all Jordan blocks are \(1\times 1\).

3.2 Jordan blocks and their sizes

A Jordan block of size \(r\) for eigenvalue \(\lambda\) has the form \[ J_r(\lambda)= \begin{pmatrix} \lambda & 1 & 0 & \cdots & 0\\ 0 & \lambda & 1 & \cdots & 0\\ \vdots & & \ddots & \ddots & \vdots\\ 0& \cdots & 0 & \lambda & 1\\ 0& \cdots & \cdots & 0 & \lambda \end{pmatrix}. \] The off-diagonal ones indicate how generalized eigenvectors are linked. Block sizes encode how many steps are needed for \((A-\lambda I)^k\) to annihilate vectors within that block.

3.3 How Jordan form reveals diagonalizability failure

Diagonalizability fails exactly when at least one Jordan block has size greater than \(1\). In that case, the eigenspace corresponding to \(\lambda\) has lower dimension than the algebraic multiplicity, and no similarity transformation can remove the superdiagonal ones while keeping the matrix similar. Thus Jordan form makes the eigenvector deficit explicit.

3.4 Constructing Jordan form from chains

Jordan form can be constructed by identifying, for each eigenvalue \(\lambda\), a set of Jordan chains (ordered vectors related by \((A-\lambda I)\)). The lengths of the chains determine the Jordan block sizes. Vectors from all chains are assembled into the similarity matrix \(P\), turning \(A\) into its Jordan block decomposition.

4 Generalized Eigenvectors and Jordan Chains

4.1 Generalized eigenvectors: definition and motivation

A vector \(v\neq 0\) is a generalized eigenvector of \(A\) for eigenvalue \(\lambda\) if it satisfies \[ (A-\lambda I)^k v = 0 \] for some positive integer \(k\). Unlike ordinary eigenvectors, generalized eigenvectors allow higher powers of \(A-\lambda I\) to annihilate the vector, capturing the missing directions when the eigenspace is too small.

4.2 Eigenvector chains (Jordan chains)

A Jordan chain for eigenvalue \(\lambda\) is a sequence of vectors \(v_1,\dots,v_r\) such that \[ (A-\lambda I)v_1=0,\qquad (A-\lambda I)v_{i}=v_{i-1}\ \text{for } i=2,\dots,r. \] The first vector \(v_1\) is an actual eigenvector, while the later vectors are generalized eigenvectors whose repeated application under \(A-\lambda I\) walks backward through the chain. Each chain corresponds to one Jordan block of size \(r\).

4.3 Dimension counts for generalized eigenspaces

For an eigenvalue \(\lambda\), the generalized eigenspace \(G_\lambda=\ker(A-\lambda I)^{r}\) (with \(r\) at least as large as the maximum Jordan block size for \(\lambda\)) has dimension equal to the total size of the Jordan blocks for \(\lambda\). Within \(G_\lambda\), the true eigenspace is \(\ker(A-\lambda I)\), and its smaller dimension explains why diagonalization is impossible.

4.4 Relationship to the minimal polynomial

The minimal polynomial reflects the longest Jordan chains. Specifically, the exponent of \((t-\lambda)\) in \(m_A(t)\) equals the maximum size of a Jordan block for eigenvalue \(\lambda\). Therefore, when a Jordan block exceeds size \(1\), the minimal polynomial includes a repeated factor, and non-diagonalizability follows.

5 Polynomial and Similarity Viewpoints

5.1 Using polynomials in matrices (functional calculus basics)

Given a polynomial \(p(t)=\sum_{j=0}^d c_j t^j\), one defines \[ p(A)=\sum_{j=0}^d c_j A^j. \] This framework underlies statements about the minimal polynomial (the smallest polynomial that kills \(A\)) and about generalized eigenspaces (kernels of \((A-\lambda I)^k\)). It also explains why Jordan blocks govern how \(A^k\) grows: nilpotent parts contribute polynomial-in-\(k\) factors.

5.2 Similar matrices and invariants

If \(A\) and \(B\) are similar (\(A=PBP^{-1}\)), they share key invariants: characteristic polynomial, minimal polynomial, and Jordan block structure over an algebraic closure. Since diagonalizability is equivalent to having all Jordan blocks of size \(1\), it is similarity-invariant: either both matrices are diagonalizable or neither is.

5.3 Commuting matrices and shared eigen-structure

If a matrix \(B\) commutes with \(A\) (\(AB=BA\)), restrictions arise on how \(B\) acts on the generalized eigenspaces of \(A\). In many settings, commuting operators preserve the Jordan chain structure, leading to shared eigenvectors or at least compatible decomposition into invariant subspaces. This perspective is often used to analyze families of operators simultaneously.

5.4 Stabilizing subspaces under matrix powers

Non-diagonalizable behavior frequently appears in the long-term action of \(A^k\). Jordan decomposition shows that powers involve terms like \((\lambda I+N)^k\), where \(N\) is nilpotent and contributes polynomial factors in \(k\) up to the nilpotency index. These polynomial factors influence which subspaces effectively dominate the dynamics and how rapidly vectors align with leading generalized eigenspaces.

6 Computing with Non-diagonalizable Matrices

6.1 Practical workflow for determining diagonalizability

A common computational strategy:

  1. Compute the characteristic polynomial and factor it to identify eigenvalues and algebraic multiplicities.
  2. For each eigenvalue \(\lambda\), compute \(\dim\ker(A-\lambda I)\) to obtain geometric multiplicity.
  3. Compare \(m_g(\lambda)\) with \(m_a(\lambda)\). If all match, \(A\) is diagonalizable; otherwise it is non-diagonalizable.

As an alternative, one may compute or infer the minimal polynomial and check whether it has repeated linear factors.

6.2 Finding generalized eigenvectors in examples

Once an eigenvalue is identified with \(m_g(\lambda) < m_a(\lambda)\), generalized eigenvectors are found by solving \[ (A-\lambda I)^k v=0 \] for the smallest \(k\) that enlarges the kernel beyond the eigenspace. One typically:

  • Solve \((A-\lambda I)v=0\) for eigenvectors.
  • Then solve \((A-\lambda I)^2 v=0\) and extend to obtain vectors not in the eigenspace but annihilated at the next power.

Iterating yields a basis compatible with Jordan chains.

6.3 Building a Jordan basis

To build a Jordan basis, one constructs Jordan chains:

  • Start from eigenvectors (vectors in \(\ker(A-\lambda I)\)).
  • Extend backward by finding vectors \(w\) such that \((A-\lambda I)w\) equals a previously selected vector in the chain.

The procedure continues until no extension is possible, producing chain lengths that determine Jordan block sizes.

6.4 Computing matrix powers and exponentials via Jordan form

Jordan form enables closed-form expressions for \(A^k\) and \(\exp(At)\). For each Jordan block \(J_r(\lambda)=\lambda I+N\) with nilpotent \(N\) satisfying \(N^r=0\), one uses the binomial expansion: \[ (\lambda I+N)^k=\sum_{j=0}^{r-1} \binom{k}{j}\lambda^{k-j}N^j. \] Similarly, \[ e^{(\lambda I+N)t}=e^{\lambda t}\sum_{j=0}^{r-1}\frac{t^j}{j!}N^j. \] As a result, non-diagonalizable matrices produce exponential growth/decay terms multiplied by polynomials in \(k\) or \(t\), reflecting the nilpotent coupling.

7 Applications and Connections

7.1 Differential equations and repeated-root systems

In linear systems of differential equations, repeated roots in the characteristic polynomial of the associated operator lead to solution forms that include polynomial factors times exponentials. This phenomenon corresponds exactly to non-diagonalizable structures in the matrix governing the system. Jordan form provides a systematic way to derive the correct time-domain behavior.

7.2 Markov chain dynamics (non-diagonalizable transition matrices)

For discrete-time Markov chains, the transition matrix is often analyzed through eigenvalues and their multiplicities. When the transition matrix is non-diagonalizable, convergence toward the stationary distribution can include additional polynomial terms, not just pure exponential decay. Jordan blocks determine how quickly powers of the transition operator smooth out initial discrepancies.

7.3 Vibrations and linear systems with defects

In mechanical or network models, damping-free idealizations can yield repeated eigenvalues, and structural imperfections or coupling can produce defective modal descriptions. Non-diagonalizable dynamics can manifest as responses involving terms proportional to \(t\) (or higher-degree polynomials) in addition to oscillations, reflecting repeated natural frequencies and incomplete modal bases.

7.4 Control systems and state transition behavior

In control theory, state transition matrices describe how system states evolve under inputs or in open-loop settings. Non-diagonalizable system matrices lead to transient responses that may grow polynomially before eventually aligning with the eigenvalue-determined asymptotics. Understanding Jordan structure helps predict overshoot behavior, transient amplification, and the design of observers or controllers.

8 Illustrative Examples

8.1 A 2×2 defective matrix example

Consider \[ A=\begin{pmatrix}1&1\\0&1\end{pmatrix}. \] Its characteristic polynomial is \((t-1)^2\), so the only eigenvalue is \(\lambda=1\) with algebraic multiplicity \(2\). Compute \[ A-I=\begin{pmatrix}0&1\\0&0\end{pmatrix}, \] whose kernel is all vectors \(\begin{pmatrix}x\\0\end{pmatrix}\), hence \(m_g(1)=1\). Since \(m_g(1)\ne m_a(1)\), \(A\) is non-diagonalizable. In Jordan form, \(A\) already equals the Jordan block \(J_2(1)\).

8.2 A 3×3 Jordan block example

Let \[ A=\begin{pmatrix} \lambda&1&0\\ 0&\lambda&1\\ 0&0&\lambda \end{pmatrix}=J_3(\lambda). \] The eigenvalue is \(\lambda\) with algebraic multiplicity \(3\). The eigenspace \(\ker(A-\lambda I)\) consists of vectors with only the first component possibly nonzero, so its dimension is \(1\). Therefore \(m_g(\lambda)=1<m_a(\lambda)=3\), confirming non-diagonalizability. Here the Jordan block size is \(3\), so the minimal polynomial contains \((t-\lambda)^3\).

8.3 Mixed case: diagonalizable part plus Jordan part

A matrix may decompose into a direct sum of a diagonalizable block and a defective Jordan block. For instance, over \(\mathbb{C}\), a matrix similar to \[ \begin{pmatrix} \mu &0&0\\ 0&\lambda&1\\ 0&0&\lambda \end{pmatrix} \] has two eigenvalues: \(\mu\) contributes a \(1\times 1\) block (fully diagonal), while \(\lambda\) contributes a \(2\times 2\) Jordan block. Since not all blocks are \(1\times 1\), the whole matrix is non-diagonalizable even though part of it behaves diagonally.

8.4 Checking diagonalizability for a given matrix

For a concrete matrix, one can implement a checklist:

  • Factor the characteristic polynomial to find eigenvalues and their multiplicities.
  • For each eigenvalue, compute \(\ker(A-\lambda I)\) and determine its dimension.
  • If any eigenvalue has geometric multiplicity strictly less than algebraic multiplicity, the matrix is non-diagonalizable.

This method avoids constructing Jordan form explicitly, though Jordan form can be derived afterward if needed.

9 Summary and Reference Checklist

9.1 Quick tests for non-diagonalizability

  • Multiplicity check: For some eigenvalue \(\lambda\), compute \(m_g(\lambda)\) and \(m_a(\lambda)\). If \(m_g(\lambda)<m_a(\lambda)\), then \(A\) is non-diagonalizable.
  • Minimal polynomial check: If \(m_A(t)\) has any repeated linear factor, \(A\) is non-diagonalizable.
  • Jordan viewpoint: If any Jordan block has size greater than \(1\), the matrix cannot be diagonalized.

9.2 Common pitfalls and misconceptions

  • Repeated roots alone are not enough: A repeated eigenvalue may still be diagonalizable if the eigenspace dimension matches the algebraic multiplicity.
  • Confusing eigenvalues with eigenvectors: Eigenvalues determine only part of the story; the size of eigenspaces is what controls diagonalizability.
  • Overlooking field issues: Diagonalization over \(\mathbb{R}\) may differ from diagonalization over \(\mathbb{C}\), since eigenvalues might not split over the real numbers.

9.3 Further reading and standard theorems (high level)

For deeper study, standard references cover: the spectral theorem for special classes of matrices, the Jordan canonical form theorem, the relationship between minimal polynomials and Jordan block sizes, and the use of invariant subspaces and generalized eigenspaces in linear operator theory. These results form the backbone for analyzing non-diagonalizable transformations in both pure and applied settings.