A = -27 32 -18 8 -8 5 -24 23 -18 Recall that for x € C, e^x= Σæk/k! = 1+x+x²/2! + x³/3! + ... k=1 Define e^A = ΣAk/k! k=1 What is e^A. Explain.
A = -27 32 -18 8 -8 5 -24 23 -18 Recall that for x € C, e^x= Σæk/k! = 1+x+x²/2! + x³/3! + ... k=1 Define e^A = ΣAk/k! k=1 What is e^A. Explain.
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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Question
![**Matrix Exponential**
Consider the matrix \( A \):
\[
A =
\begin{bmatrix}
-27 & 32 & -24 \\
-18 & 23 & -18 \\
8 & -8 & 5
\end{bmatrix}
\]
### Exponential Function for Complex Numbers
Recall that for \( x \in \mathbb{C} \), the exponential function \( e^x \) is defined by the series:
\[
e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots
\]
### Matrix Exponential
In a similar way, the exponential of a matrix \( A \), denoted by \( e^A \), is defined by:
\[
e^A = \sum_{k=0}^{\infty} \frac{A^k}{k!}
\]
This series converges for any square matrix \( A \).
### Explanation
To find \( e^A \), calculate the series by computing successive powers of \( A \) and dividing by factorials, continuing this process until the additional terms become negligible.
The matrix exponential is widely used in solving systems of linear differential equations, among other applications. It is a crucial concept in the study of linear algebra and functional analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70271567-bd34-49ee-a845-ddba2468b76c%2Fa22dac3f-6517-48a6-9930-57432f565614%2Fa9cvpm_processed.png&w=3840&q=75)
Transcribed Image Text:**Matrix Exponential**
Consider the matrix \( A \):
\[
A =
\begin{bmatrix}
-27 & 32 & -24 \\
-18 & 23 & -18 \\
8 & -8 & 5
\end{bmatrix}
\]
### Exponential Function for Complex Numbers
Recall that for \( x \in \mathbb{C} \), the exponential function \( e^x \) is defined by the series:
\[
e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots
\]
### Matrix Exponential
In a similar way, the exponential of a matrix \( A \), denoted by \( e^A \), is defined by:
\[
e^A = \sum_{k=0}^{\infty} \frac{A^k}{k!}
\]
This series converges for any square matrix \( A \).
### Explanation
To find \( e^A \), calculate the series by computing successive powers of \( A \) and dividing by factorials, continuing this process until the additional terms become negligible.
The matrix exponential is widely used in solving systems of linear differential equations, among other applications. It is a crucial concept in the study of linear algebra and functional analysis.
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