Chapter 36 — Key Takeaways

The one idea

A defective matrix is not broken — it is shearing, and the Jordan form is the honest record of that shear. When a matrix runs out of eigenvectors (geometric multiplicity below algebraic, the gap Chapter 24 first measured), diagonalization fails — but the matrix still has a structure: alongside each eigenvalue's genuine invariant direction sits a "leftover" sliding motion that no eigenvector can hold. Generalized eigenvectors name that leftover, Jordan chains organize it, and the resulting Jordan block — an eigenvalue on the diagonal with $1$'s on the superdiagonal — is as close to diagonal as the matrix will allow. Every complex matrix has a Jordan canonical form $A = PJP^{-1}$; diagonalizable matrices are simply the lucky case where every block has size $1$. This is recurring theme #6 (eigenvalues reveal what a matrix really does) carried to its honest conclusion: when the eigenvectors run short, the Jordan form measures exactly how much of the transformation they could not capture.

The big ideas, in order

  1. Defective = too few eigenvectors. For some eigenvalue, geometric multiplicity ($\dim N(A-\lambda I)$, the number of independent eigenvectors) is strictly less than algebraic multiplicity (its multiplicity as a root). The defect $d_\lambda = \text{algebraic} - \text{geometric}$ counts the missing basis vectors. Only repeated eigenvalues can be defective; distinct eigenvalues always diagonalize, and symmetric matrices (Chapter 27) are never defective.

  2. Generalized eigenvectors complete the basis. A generalized eigenvector solves the inhomogeneous system $(A-\lambda I)\mathbf{w} = \mathbf{v}$, where $\mathbf{v}$ is an eigenvector — one rung above an eigenvector instead of in the null space. It is solvable only when $\mathbf{v}$ lies in the column space of $A - \lambda I$, which is why longer chains must start from the right eigenvector.

  3. A Jordan chain relays down to zero. A length-$k$ chain $\mathbf{v}_1, \dots, \mathbf{v}_k$ satisfies $(A-\lambda I)\mathbf{v}_j = \mathbf{v}_{j-1}$ (and $(A-\lambda I)\mathbf{v}_1 = \mathbf{0}$); its vectors are provably independent and become one $k\times k$ block. The number of blocks for $\lambda$ equals the geometric multiplicity; their total size equals the algebraic multiplicity; individual sizes need the ranks of $(A-\lambda I)^j$.

  4. A Jordan block is a shear in the eigen-direction. $J_k(\lambda) = \lambda I + N$ splits into a scaling part ($\lambda I$) and a nilpotent part ($N$, the superdiagonal $1$'s, with $N^k = 0$). Geometrically it scales along the eigen-direction and shears the rest toward it; the off-diagonal $1$ is the leftover that prevents diagonalization, made visible as a slide.

  5. The Jordan canonical form exists over $\mathbb{C}$, uniquely up to block order. Every square complex matrix is $A = PJP^{-1}$ with $J$ block-diagonal. Over $\mathbb{R}$ it can fail (rotations have complex eigenvalues). Two matrices are similar iff they share a Jordan form up to block order — so the block sizes are a complete similarity invariant, carrying information beyond the eigenvalues and multiplicities alone.

  6. Diagonalization is the all-blocks-size-$1$ special case. Chapter 25 is not replaced but completed: "is $A$ diagonalizable?" becomes "are all Jordan blocks $1\times 1$?", true exactly when geometric equals algebraic multiplicity for every eigenvalue.

  7. Powers and the matrix exponential of a defective matrix carry polynomial-times-geometric terms. $A^m = PJ^mP^{-1}$ with $J^m$ computed by the terminating binomial $(\lambda I + N)^m$; a size-$k$ block produces entries that are degree-$(k-1)$ polynomials in $m$ times $\lambda^m$. For $2\times 2$, $J^m = \begin{bmatrix}\lambda^m & m\lambda^{m-1}\\0&\lambda^m\end{bmatrix}$ and $e^{Jt} = e^{\lambda t}\begin{bmatrix}1&t\\0&1\end{bmatrix}$. The $m\lambda^{m-1}$ and $t\,e^{\lambda t}$ terms are the signatures a diagonalizable matrix never shows.

  8. The Jordan form is numerically fragile — theory, not computation. Defectiveness is destroyed by any perturbation; a perturbation of size $\epsilon$ splits a repeated eigenvalue by $\sim\sqrt{\epsilon}$ and makes the eigenvector matrix ill-conditioned. No general numerical library computes it (numpy does not offer it); the stable Schur decomposition and SVD stand in. sympy computes it exactly because it never rounds. Use it to understand, not to compute on a finite machine — Chapter 38 makes this precise.

Skills you gained

  • Diagnose defectiveness by comparing $\dim N(A-\lambda I)$ to the algebraic multiplicity — never assuming a repeated root supplies enough eigenvectors.
  • Find a generalized eigenvector by solving $(A-\lambda I)\mathbf{w} = \mathbf{v}$, and recognize when $\mathbf{v}$ must lie in the column space of $A - \lambda I$.
  • Assemble $P$ from Jordan chains (eigenvector first, then generalized vectors) and verify $P^{-1}AP = J$ via $AP = PJ$, by hand and against numpy/sympy.
  • Read a Jordan block geometrically as a scaling-plus-shear, and count blocks (geometric multiplicity) and total size (algebraic multiplicity).
  • Compute powers and exponentials of a defective matrix via the nilpotent split, predicting the $m\lambda^{m-1}$ / $t\,e^{\lambda t}$ terms.
  • Judge when the Jordan form is appropriate — symbolic understanding yes, floating-point computation no.

Terms to know

defective matrix · algebraic multiplicity · geometric multiplicity · defect · generalized eigenvector · Jordan chain · index of an eigenvalue · Jordan block · superdiagonal · nilpotent part · Jordan normal form / Jordan canonical form · generalized eigenspace · block-diagonal · similarity · Schur decomposition

Connections — backward and forward

  • Back to Chapter 24: the algebraic-vs-geometric multiplicity gap, and the multiplicity inequality, are exactly the defective case this chapter resolves; Chapter 24 explicitly flagged it.
  • Back to Chapter 25: diagonalization $A = PDP^{-1}$ is the special case where $J = D$ (all blocks size $1$); the Jordan form is its honest generalization.
  • Back to Chapter 16: $A = PJP^{-1}$ is a similarity — the same transformation in coordinates adapted to the chains; the Jordan form is the canonical representative of a similarity class.
  • Back to Chapter 26: the "over $\mathbb{C}$" condition is the same move to complex scalars that handled rotations; a real matrix has a real Jordan form only if all eigenvalues are real.
  • Forward to Chapter 37 (the immediate payoff): the matrix exponential $e^{At} = Pe^{Jt}P^{-1}$ solves every linear system of ODEs as $\mathbf{x}(t) = e^{At}\mathbf{x}(0)$, and the $t\,e^{\lambda t}$ term this chapter derived is exactly what a defective system contributes — the critically-damped "creep" of Case Study 1, drawn as a degenerate node in a phase portrait. Chapter 37 is where the Jordan form earns its keep.
  • Forward to Chapter 38: the numerical fragility of the Jordan form is the headline example of conditioning and stability — why exact theory and finite-precision computation are different disciplines, and why the Schur form is the stable substitute.
  • Cross-book: the matrix exponential connects directly to differential equations, where $e^{At}$ is the solution operator; the fragility connects to the numerical methods that production software runs in place of the Jordan form.

Remember this: when a matrix runs out of eigenvectors, it is hiding a shear. Generalized eigenvectors find the shear, Jordan blocks record it, and the Jordan form $A = PJP^{-1}$ — diagonal except for a few $1$'s above the diagonal — is the most a defective matrix will let you simplify it to. The next chapter turns that almost-diagonal form into the solution of a differential equation.