Exercise 1495Calculus › Differential equations › Linear systemsL3Added 2 Nov 2026

Matrix exponential with a nilpotent part

#matrix exponential#nilpotent#Jordan form

Statement

Compute $\exp(tA)$ for
$A = \begin{pmatrix} \lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda \end{pmatrix}.$

Hint

Write $A = \lambda I + N$ where $N$ is nilpotent ( $N^3=0$ ). Since $\lambda I$ and $N$ commute, $\exp(tA) = e^{\lambda t}\exp(tN)$ .

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Solution

Decomposition. $A = \lambda I + N$ with
$N = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}, \qquad N^2 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \qquad N^3 = 0.$

Exponential of $N$ .
$\exp(tN) = I + tN + \frac{t^2}{2}N^2 = \begin{pmatrix} 1 & t & t^2/2 \\ 0 & 1 & t \\ 0 & 0 & 1 \end{pmatrix}.$

Exponential of $A$ .
$\exp(tA) = e^{\lambda t}\begin{pmatrix} 1 & t & t^2/2 \\ 0 & 1 & t \\ 0 & 0 & 1 \end{pmatrix}.$

Matrix exponential with a nilpotent part

Statement

Solution

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