A matrix A is said to be nilpotent if there exists some
Chapter 8, Problem 26E(choose chapter or problem)
A matrix A is said to be nilpotent if there exists some positive integer m such that \(\mathbf{A}^{m}=\mathbf{0}\). Verify that
\(\mathbf{A}=\left(\begin{array}{lll}
-1 & 1 & 1 \\
-1 & 0 & 1 \\
-1 & 1 & 1
\end{array}\right)
\)
is nilpotent. Discuss why it is relatively easy to compute \(e^{\mathbf{A} t}\) when A is nilpotent. Compute \(e^{\mathbf{A} t}\) and then use (1) to solve the system \(\mathbf{X}^{\prime}=\mathbf{A X}\).
Text Transcription:
mathbf A^m = mathbf 0
mathbf A = ({array} lll -1 & 1 & 1 -1 & 0 & 1 -1 & 1 & 1 {array})
e^mathbf A t
mathbf X^prime = mathbf A X
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