write the Matrix exponential For to show why ett - fet o #* = [6* > ] O A = [1 2] 3
write the Matrix exponential For to show why ett - fet o #* = [6* > ] O A = [1 2] 3
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
to show why (e^(At))=the shown matrix, write the matrix exponential (as an infinite sum) for the system of
![**Matrix Exponential Example**
In this task, we explore the matrix exponential to demonstrate why \( e^{At} \) results in a specific form.
Given a matrix \( A \):
\[ A = \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix} \]
We aim to show that:
\[ e^{At} = \begin{bmatrix} e^t & 0 \\ 0 & e^{3t} \end{bmatrix} \]
### Explanation
The exponential of a matrix \( A \), denoted as \( e^{At} \), is defined as the power series:
\[ e^{At} = I + At + \frac{(At)^2}{2!} + \frac{(At)^3}{3!} + \cdots \]
For a diagonal matrix \( A \), such as the one given, the matrix exponential can be computed by exponentiating each of the diagonal elements separately.
**Matrix Exponential Calculation:**
1. **Diagonal Elements**:
- The element in the first row and column of matrix \( A \) is 1, leading to \( e^{1 \cdot t} = e^t \).
- The element in the second row and column is 3, leading to \( e^{3 \cdot t} = e^{3t} \).
2. **Off-Diagonal Elements**:
- The off-diagonal elements remain 0 since matrix \( A \) has 0's in those positions.
Thus, by exponentiating the diagonal elements independently, the resulting matrix exponential \( e^{At} \) is:
\[ e^{At} = \begin{bmatrix} e^t & 0 \\ 0 & e^{3t} \end{bmatrix} \]
This concise representation demonstrates the power of computing the matrix exponential for diagonal matrices.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6d0ee2cb-cfe2-4eb9-a097-d38324436758%2F6e46d21f-d881-4c04-b757-391763c09683%2Fzcnsiv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Matrix Exponential Example**
In this task, we explore the matrix exponential to demonstrate why \( e^{At} \) results in a specific form.
Given a matrix \( A \):
\[ A = \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix} \]
We aim to show that:
\[ e^{At} = \begin{bmatrix} e^t & 0 \\ 0 & e^{3t} \end{bmatrix} \]
### Explanation
The exponential of a matrix \( A \), denoted as \( e^{At} \), is defined as the power series:
\[ e^{At} = I + At + \frac{(At)^2}{2!} + \frac{(At)^3}{3!} + \cdots \]
For a diagonal matrix \( A \), such as the one given, the matrix exponential can be computed by exponentiating each of the diagonal elements separately.
**Matrix Exponential Calculation:**
1. **Diagonal Elements**:
- The element in the first row and column of matrix \( A \) is 1, leading to \( e^{1 \cdot t} = e^t \).
- The element in the second row and column is 3, leading to \( e^{3 \cdot t} = e^{3t} \).
2. **Off-Diagonal Elements**:
- The off-diagonal elements remain 0 since matrix \( A \) has 0's in those positions.
Thus, by exponentiating the diagonal elements independently, the resulting matrix exponential \( e^{At} \) is:
\[ e^{At} = \begin{bmatrix} e^t & 0 \\ 0 & e^{3t} \end{bmatrix} \]
This concise representation demonstrates the power of computing the matrix exponential for diagonal matrices.
Expert Solution

This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 4 images

Recommended textbooks for you

Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated

Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY

Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated

Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY

Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,

