Diagonalization, Jordan Form, and Linear Systems

date2026-07-16document_iddoc_1391a6b57f52591851f8c1b3116570c4descriptionJordan 標準形と一般化固有ベクトルの鎖を、定数係数線型系の指数関数×多項式の時間発展へ翻訳する。prerequisites一階連立系と行列指数関数 / ジョルダン標準形の入口type講義content_typelecturestatusactive

mathdifferential-equationslinear-systemslecture

1Purpose of the Lecture

For the constant-coefficient system

x^{'} = A x, x (t_{0}) = x_{0},

Jordan form translates generalized eigenvector chains into solutions consisting of an exponential multiplied by a finite-degree polynomial. We work over $C$ . A real matrix may be complexified and placed in complex Jordan form; conjugate complex solutions then yield real solutions through their real and imaginary parts. Real Jordan form is outside the scope of this lecture.

data/lecture/math/linear-algebra/introduction-to-jordan-canonical-form.lecture.n.md

2Exponential of a Jordan Block

For a block of size $k$ , write

J_{k} (λ) = λ I + N, N = (\begin{matrix} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 1 \\ 0 & 0 & 0 & \dots & 0 \end{matrix}) .

The matrices $λ I$ and $N$ commute. Hence, with $τ = t - t_{0}$ ,

e^{J_{k} (λ) τ} = e^{λ τ} \sum_{r = 0}^{k - 1} \frac{τ^{r}}{r!} N^{r} .

Here $N^{k} = 0$ and, for $k > 1$ , $N^{k - 1} \neq 0$ . Nilpotency terminates the series and produces polynomial factors through degree $k - 1$ . This factorization relies on commutativity and is not valid for arbitrary matrices $X, Y$ .

3Return to the Original Coordinates

P^{- 1} A P = J,

then

e^{A τ} = P e^{J τ} P^{- 1}, x (t) = P e^{J (t - t_{0})} P^{- 1} x_{0.}

Each Jordan block may be exponentiated independently.

4Generalized Eigenvector Chains

Let

(A - λ I) v_{1} = 0, (A - λ I) v_{j} = v_{j - 1} (2 [PARSE ERROR: Undefined("Command(\"le\")")] j [PARSE ERROR: Undefined("Command(\"le\")")] k)

be a Jordan chain. The solution with initial value $v_{j}$ is

x_{j} (t) = e^{λ τ} \sum_{ℓ = 0}^{j - 1} \frac{τ^{ℓ}}{ℓ!} v_{j - ℓ} .

In particular,

x_{1} = e^{λ τ} v_{1}, x_{2} = e^{λ τ} (v_{2} + τ v_{1}), x_{3} = e^{λ τ} (v_{3} + τ v_{2} + \frac{τ^{2}}{2} v_{1}) .

Differentiation and the chain relations verify the system directly. The exponential derivative contributes the term with $λ$ , while differentiating the polynomial lowers the chain index by one; $(A - λ I) v_{j} = v_{j - 1}$ then gives $x_{j^{'}} = A x_{j}$ .

5Example in Non-Jordan Coordinates

For

A = (\begin{matrix} 2 & 1 \\ - 1 & 0 \end{matrix}), v_{1} = (\begin{matrix} 1 \\ - 1 \end{matrix}), v_{2} = (\begin{matrix} 1 \\ 0 \end{matrix}),

we have $(A - I) v_{1} = 0$ and $(A - I) v_{2} = v_{1}$ . Moreover,

\det (μ I - A) = (μ - 1)^{2}, P = (\begin{matrix} 1 & 1 \\ - 1 & 0 \end{matrix}), J = P^{- 1} A P = (\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}) .

Consequently,

e^{A τ} = e^{τ} (\begin{matrix} 1 + τ & τ \\ - τ & 1 - τ \end{matrix}) .

6Asymptotic Behavior

For $τ [PARSE ERROR: Undefined("Command(\"ge\")")] 0$ , a Jordan block satisfies

∥ e^{J_{k} (λ) τ} ∥ [PARSE ERROR: Undefined("Command(\"le\")")] C (1 + τ^{k - 1}) e^{Re λ τ} .

Thus, negative real part implies decay despite polynomial factors, while positive real part produces an exponentially growing direction. When the real part is zero, blocks of size one are bounded, whereas larger blocks admit polynomially unbounded solutions. Both eigenvalue real parts and Jordan block sizes are therefore required at the spectral boundary.

7Numerical Boundary

Jordan structure is sensitive to perturbations and is primarily a theoretical classification. For example,

(\begin{matrix} λ & 1 \\ ε & λ \end{matrix})

is one Jordan block when $ε = 0$ , whereas its eigenvalues split into $λ \pm \sqrt{ε}$ when $ε \neq 0$ . Numerical computation of $e^{A t}$ should not generally assume that an exact Jordan block structure can be identified reliably in floating-point arithmetic.

8Previous Lecture

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9Next Lecture

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10Exercises

data/exercise/math/differential-equations/jordan-form-and-linear-systems.exercise.n.md