If J, is a 4x4 Jordan block find eit
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement:**
If \( J \) is a \( 4 \times 4 \) Jordan block, find \( e^{Jt} \).
**Explanation:**
This problem involves finding the matrix exponential of a Jordan block, which is a specific type of matrix used in linear algebra. A Jordan block of size \( n \times n \) associated with an eigenvalue \(\lambda\) has the form:
\[
\begin{bmatrix}
\lambda & 1 & 0 & \cdots & 0 \\
0 & \lambda & 1 & \ddots & \vdots \\
\vdots & \ddots & \ddots & \ddots & 0 \\
\vdots & & \ddots & \ddots & 1 \\
0 & \cdots & \cdots & 0 & \lambda
\end{bmatrix}
\]
For a \( 4 \times 4 \) Jordan block with eigenvalue \(\lambda\), the matrix \( J \) would be:
\[
\begin{bmatrix}
\lambda & 1 & 0 & 0 \\
0 & \lambda & 1 & 0 \\
0 & 0 & \lambda & 1 \\
0 & 0 & 0 & \lambda
\end{bmatrix}
\]
To find \( e^{Jt} \), you use the series expansion for the matrix exponential and properties of Jordan blocks. The detailed process often involves considering each submatrix and how it contributes to the exponential form.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F71f6a4b8-527f-4030-b85a-828002701c6f%2F0bdf5fb6-6c9a-4b2d-9ee5-7137f930991f%2Fggbpb3e_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
If \( J \) is a \( 4 \times 4 \) Jordan block, find \( e^{Jt} \).
**Explanation:**
This problem involves finding the matrix exponential of a Jordan block, which is a specific type of matrix used in linear algebra. A Jordan block of size \( n \times n \) associated with an eigenvalue \(\lambda\) has the form:
\[
\begin{bmatrix}
\lambda & 1 & 0 & \cdots & 0 \\
0 & \lambda & 1 & \ddots & \vdots \\
\vdots & \ddots & \ddots & \ddots & 0 \\
\vdots & & \ddots & \ddots & 1 \\
0 & \cdots & \cdots & 0 & \lambda
\end{bmatrix}
\]
For a \( 4 \times 4 \) Jordan block with eigenvalue \(\lambda\), the matrix \( J \) would be:
\[
\begin{bmatrix}
\lambda & 1 & 0 & 0 \\
0 & \lambda & 1 & 0 \\
0 & 0 & \lambda & 1 \\
0 & 0 & 0 & \lambda
\end{bmatrix}
\]
To find \( e^{Jt} \), you use the series expansion for the matrix exponential and properties of Jordan blocks. The detailed process often involves considering each submatrix and how it contributes to the exponential form.
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