Jordan Canonical Form and Defective Matrices

Defective matrices are no longer a textbook curiosity. The liveliest arena is non-Hermitian physics: an open quantum system, a lossy cavity, or a $\mathcal{PT}$-symmetric photonic lattice is governed by an effective non-Hermitian Hamiltonian whose eigenvalues and eigenvectors can coalesce at an exceptional point — where the Hamiltonian is a genuine Jordan block, not a diagonalizable near-degeneracy. The coalescence produces effects with no Hermitian analogue: a resonance-splitting scaling as the $k$-th root of a perturbation near an order-$k$ point (exploited for enhanced sensing in microcavities and gyroscopes), and asymmetric mode conversion when a control loop encircles the point. A second arena is non-normal stability theory: the shear-flow operator is far from normal, so even when every eigenvalue predicts decay the near-defective structure permits enormous transient amplification — the pseudospectral mechanism now believed to seed the transition to turbulence. The same secular $t\,e^{\lambda t}$ growth appears as secular terms in celestial mechanics and perturbative resonance. And numerically, because block sizes are infinitely sensitive to perturbation, robust software works with the Schur form or pseudospectra instead — defectiveness is a knife-edge, and physics tends to live right on it.

Picture an eigenvector as a direction the operator merely rescales: feed in $v$, get back $\lambda v$, no leakage. A defective eigenvalue promises more of these directions than it can deliver. Where a second eigenvector should sit, the operator offers a partner $v_2$ obeying $Av_2=\lambda v_2+v_1$ — almost an eigenvector, but with a thin spill into the true eigendirection $v_1$. That spill is the lone $1$ on the superdiagonal — no diagonal matrix owns it. Stack the partners into a tower and you get a Jordan chain: $(A-\lambda I)$ walks you down it one rung at a time to the eigenvector at the bottom and then zero. The tower's height is the block size.

The purest incarnation is the nilpotent shift. The matrix with $1$s on the superdiagonal and zeros elsewhere sends $e_2\mapsto e_1$, $e_3\mapsto e_2$, $e_1\mapsto 0$: single eigenvalue $0$, yet only one eigenvector. A Jordan block is exactly this shift $N$ on a pure scaling, $J_k(\lambda)=\lambda I+N$; since $N^k=0$, every defective structure is a nilpotent embellishment of an eigenvalue, and that nilpotency makes the physics computable. The series $e^{Nt}=I+Nt+\cdots$ terminates after $k$ terms, so a defective mode's propagator is not a pure exponential but an exponential times a polynomial in $t$ of degree $k-1$. The second figure shows that secular factor in the flesh — the $t\,e^{\lambda t}$ mode a diagonalizable system can never produce.

Similarity can always reach block-diagonal form, even when it cannot reach diagonal form. This is the master identity: a single invertible $P$, built from generalized eigenvectors, conjugates $A$ into a direct sum of Jordan blocks.

where: $A$ — an $n\times n$ matrix over $\mathbb{C}$; $P$ — invertible, its columns the concatenated Jordan chains; $J_{k_i}(\lambda_i)$ — the Jordan block of size $k_i$ carrying eigenvalue $\lambda_i$; $\bigoplus$ — direct sum, i.e. block-diagonal placement. The form is unique up to the order of blocks, so it is a genuine invariant: two matrices are similar iff they share the same list of blocks. A single Jordan block is the atom of the construction: upper bidiagonal, the eigenvalue repeated down the diagonal and $1$s on the superdiagonal, with everything about defectiveness packed into those off-diagonal ones.

where: $\lambda$ — the block's single eigenvalue; $k$ — the block size and chain length; $N$ — the nilpotent shift with $N_{i,i+1}=1$, obeying $N^k=0$ but $N^{k-1}\neq0$. A $1\times1$ block is an honest eigenvector; every larger block is defective. The chain relation forces that superdiagonal. Where an eigenvector satisfies $(A-\lambda I)v_1=0$, a generalized eigenvector one rung up is sent by $(A-\lambda I)$ not to zero but to the vector below it.

where: $v_1$ — an ordinary eigenvector at the bottom of the chain; $v_j$ — the $j$-th generalized eigenvector, killed by $(A-\lambda I)^j$ but not $(A-\lambda I)^{j-1}$. Reading $Av_j=\lambda v_j+v_{j-1}$ column by column reproduces block (2) exactly. Two multiplicities govern the coarse structure. The algebraic multiplicity is the total size of the blocks for $\lambda$; the geometric multiplicity — the number of independent eigenvectors — counts the blocks themselves, one per chain.

where: $m_a(\lambda)$ — algebraic multiplicity (root order in the characteristic polynomial); $m_g(\lambda)$ — geometric multiplicity. The gap $m_a-m_g$ measures defectiveness: it is zero exactly when every block is $1\times1$. The two multiplicities do not pin down the individual block sizes. For that one needs the full rank sequence of powers of $(A-\lambda I)$; the count of blocks of each size is a second difference of those ranks.

where: $b_j(\lambda)$ — the number of Jordan blocks of size exactly $j$; $r_j$ — the rank of the $j$-th power, with $r_0=n$. In terms of nullities $n_j=n-r_j$, the number of blocks of size $\ge j$ is $n_j-n_{j-1}$. The additive form of the same content is the Jordan–Chevalley decomposition: split $A$ into a diagonalizable piece and a commuting nilpotent piece. This is the version numerical work uses, because $D$ and $N$ are polynomials in $A$.

where: $D$ — the semisimple (diagonalizable) part; $N$ — the nilpotent part with $N^n=0$. The decomposition is unique, and both parts are polynomials in $A$. Physicists care because any analytic function of $A$ reduces to a finite sum on each block, with derivatives of $f$ climbing the superdiagonal — nilpotency truncates the Taylor series after $k$ terms.

where: $f$ — a function analytic at $\lambda$; $f^{(j)}(\lambda)$ — its $j$-th derivative at the eigenvalue; $N^{j}$ — the shift raised to the $j$-th power, which populates the $j$-th superdiagonal. The formula never needs a limit or a contour: it is a finite, exact sum. The single most important instance is $f(x)=e^{xt}$, the propagator of $\dot x=Ax$. On a defective block it is an exponential dressed by a polynomial whose degree is one less than the block size — the secular growth of the second figure.

where: $t$ — the evolution parameter (time); $e^{\lambda t}$ — the scalar envelope; the highest term $t^{k-1}/(k-1)!$ is the fastest-growing polynomial factor, present only when $k\ge2$. In a diagonalizable system all blocks are $1\times1$ and the sum collapses to $e^{\lambda t}$ alone.

Jordan form interpolates between the diagonalizable extreme (every block $1\times1$) and the maximally defective extreme (one chain filling the space).

1 · A defective 3×3: Jordan form, the chains, and the propagator (fully worked)

Let $A=\begin{pmatrix}1&1&0\\-1&3&0\\0&0&2\end{pmatrix}$. Find its eigenvalues, its Jordan form $J$ and a transforming matrix $P$, and then compute $e^{At}$.

Step 1. The block-triangular structure gives the characteristic polynomial from the top-left $2\times2$ (trace $4$, determinant $1\cdot3-1\cdot(-1)=4$) times $(2-x)$: $\chi(x)=(x^2-4x+4)(2-x)=-(x-2)^3$. So $\lambda=2$ is the only eigenvalue, with algebraic multiplicity 3.

Step 2. Form $N=A-2I=\begin{pmatrix}-1&1&0\\-1&1&0\\0&0&0\end{pmatrix}$. Its rows span a single direction, so $\operatorname{rank}N=1$ and the geometric multiplicity is $3-1=2$. A direct check gives $N^2=0$, so $r_1=1,\ r_2=0$. The block-counting formula (5) yields $b_1=r_0-2r_1+r_2=3-2+0=\mathbf{1}$ block of size $1$ and $b_2=r_1-2r_2+r_3=1-0+0=\mathbf{1}$ block of size $2$. Hence $J=\begin{pmatrix}2&1&0\\0&2&0\\0&0&2\end{pmatrix}$ — one $2\times2$ block and one $1\times1$ block, sizes $(2,1)$.

Step 3. Build the chains. Solving $Nv=0$ gives the eigenspace $\operatorname{span}\{(1,1,0),(0,0,1)\}$. The image of $N$ is $\operatorname{span}\{(1,1,0)\}$, so the eigenvector $v_1=(1,1,0)$ is the one that admits a partner. Solve $Nv_2=v_1$: the equation $-v_{2,1}+v_{2,2}=1$ is met by $v_2=(0,1,0)$. The isolated eigenvector for the size-$1$ block is $v_3=(0,0,1)$. Assemble $P=\begin{pmatrix}1&0&0\\1&1&0\\0&0&1\end{pmatrix}$ with columns $v_1,v_2,v_3$.

Step 4. Verify $AP=PJ$ column by column: $Av_1=(2,2,0)=2v_1$; $Av_2=(1,3,0)=2v_2+v_1$; $Av_3=(0,0,2)=2v_3$. All three hold exactly, so $J=P^{-1}AP$ with $P^{-1}=\begin{pmatrix}1&0&0\\-1&1&0\\0&0&1\end{pmatrix}$.

Step 5. Now the propagator. On the blocks, $e^{Jt}=e^{2t}\begin{pmatrix}1&t&0\\0&1&0\\0&0&1\end{pmatrix}$, and $e^{At}=P\,e^{Jt}\,P^{-1}$. Multiplying out,

Two sanity checks confirm it: at $t=0$ it is the identity, and $\dfrac{d}{dt}e^{At}\big|_{t=0}=2I+N=A$. The stray factor $t$ in the $2\times2$ corner is the secular signature of the single defective block.

2 · Distinguishing $(3,1)$ from $(2,2)$ by rank (the trap made concrete)

A $4\times4$ matrix has one eigenvalue $\lambda$ with algebraic multiplicity $4$ and geometric multiplicity $2$. What Jordan forms are possible, and what single measurement decides between them?

Step 1. Two blocks ($m_g=2$) summing to size $4$ ($m_a=4$) leave exactly two partitions: sizes $(3,1)$ or sizes $(2,2)$. The multiplicities alone cannot choose, because both give $r_1=\operatorname{rank}(A-\lambda I)=n-m_g=\mathbf{2}$.

Step 2. Go one power up. For sizes $(3,1)$ the size-$3$ block contributes rank $2$ to $(A-\lambda I)^2$ and the size-$1$ block contributes $0$, so $r_2=\mathbf{1}$. For sizes $(2,2)$ each size-$2$ block dies at the second power, so $r_2=\mathbf{0}$.

Step 3. Apply formula (5). With $(r_0,\dots,r_4)=(4,2,1,0,0)$: $b_1=4-4+1=1$ and $b_3=1-0+0=1$ — one block of size $3$, one of size $1$. With $(4,2,0,0,0)$: $b_2=2-0+0=2$ — two blocks of size $2$. The single number $r_2=\operatorname{rank}(A-\lambda I)^2$ decides: $r_2=1\Rightarrow(3,1)$, $r_2=0\Rightarrow(2,2)$.

3 · The critically damped oscillator (defectiveness in a physics you already know)

A damped oscillator $\ddot q+2\gamma\dot q+\omega_0^2q=0$ is written as $\dot{\mathbf x}=A\mathbf x$ with $\mathbf x=(q,\dot q)$. Take $\omega_0=2$ rad/s and tune to critical damping $\gamma=\omega_0=2$. Solve for $q(t)$ with $q(0)=1,\ \dot q(0)=0$.

Step 1. The companion matrix $A=\begin{pmatrix}0&1\\-4&-4\end{pmatrix}$ has characteristic equation $(x+2)^2=0$. The doubled root $\lambda=-2$ has geometric multiplicity $1$ (a companion matrix always has a single eigenvector), so $A$ is a single defective $2\times2$ Jordan block — critical damping is exactly where the system goes defective.

Step 2. By equation (8) the solution carries the secular factor: $q(t)=(a+bt)\,e^{-2t}$. Impose $q(0)=1\Rightarrow a=1$, and $\dot q=(b-2a-2bt)e^{-2t}$ with $\dot q(0)=b-2=0\Rightarrow b=2$. Hence $\mathbf{q(t)=(1+2t)\,e^{-2t}}$.

Step 3. Verify against the ODE: $\dot q=-4t\,e^{-2t}$ (so $\dot q(0)=0$ ✓) and $\ddot q=(-4+8t)e^{-2t}$, giving $\ddot q+4\dot q+4q=e^{-2t}[(-4+8t)+4(-4t)+4(1+2t)]=0$ ✓. The $t\,e^{-2t}$ term is the second solution the missing eigenvector cannot provide — and what makes critical damping return to equilibrium fastest without overshoot.

Jordan form is a rare case where pure algebra — worked out for substitution groups — became the exact language physics needed for resonance, damping, and the non-Hermitian frontier.

Texts. Stone & Goldbart, Mathematics for Physics, ch. on linear operators and the Jordan form · Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed., ch. on matrices and eigensystems · Hassani, Mathematical Physics, ch. on spectral decomposition and canonical forms · Byron & Fuller, Mathematics of Classical and Quantum Physics, ch. 4–5 · Nakahara, Geometry, Topology and Physics, §on linear operators · Horn & Johnson, Matrix Analysis, ch. 3 (the definitive reference on Jordan canonical form).

Provenance. The block-counting identity $b_j=r_{j-1}-2r_j+r_{j+1}$ and the multiplicity relations (4) are verifiable against Horn & Johnson ch. 3 and Stone & Goldbart; the block-exponential (7)–(8) and the critically damped $t\,e^{-\gamma t}$ result are standard (Arfken–Weber–Harris; Riley–Hobson–Bence). Every number here was re-verified numerically: for the worked $3\times3$ the eigenvalues are all $2$, $\operatorname{rank}N=1$, $\operatorname{rank}N^2=0$, $AP=PJ$ to machine precision, and $e^{At}=e^{2t}\begin{pmatrix}1-t&t&0\\-t&1+t&0\\0&0&1\end{pmatrix}$; the $(3,1)$-vs-$(2,2)$ discriminator $r_2\in\{1,0\}$ and the secular curve $t\,e^{-2t}$ (peak $0.184$ at $t=0.5$, crossing $e^{-2t}$ at $t=1$) check out. A verifier should confirm (i) $AP=PJ$ column by column, (ii) $N^k=0,\ N^{k-1}\neq0$ for a size-$k$ block, and (iii) the truncation of the exponential series.

Prerequisites. Eigenvalues, Eigenvectors, and the Characteristic Equation · Diagonalization and the Spectral Decomposition · Change of Basis and Similarity Transformations · Rank, Nullity, and the Fundamental Theorem of Linear Maps · Linear Operators and Matrix Representations. Leads to. Diagonalization and the Spectral Decomposition · Adjoint and Hermitian Operators · Unitary and Orthogonal Operators.

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