Let A= (11). It can be shown that etA - et Use this fact to write the solution to the initial value problem = Ay, y (0) = (9) Be sure to verify that your solution satisfies the initial value problem. -
Let A= (11). It can be shown that etA - et Use this fact to write the solution to the initial value problem = Ay, y (0) = (9) Be sure to verify that your solution satisfies the initial value problem. -
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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q2. Linear algerba
![### Linear Algebra and Matrix Theory Problems
1. **Eigenvalues of Matrices**
(a) Are the eigenvalues of \( AB \) necessarily all real? Justify your answer.
(b) Are the eigenvalues of \( AB \) necessarily all real? Justify your answer.
\[ \text{[Hint: Consider the matrices } A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \text{ and } B = \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}. \text{]} \]
2. **Solving Initial Value Problems Using Exponentiation of Matrices**
Let \( A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \). It can be shown that:
\[ e^{tA} = \begin{pmatrix} e^t & te^t \\ 0 & e^t \end{pmatrix}. \]
Use this fact to write the solution to the initial value problem:
\[ \begin{cases}
y' = Ay, \\
y(0) = \begin{pmatrix} 0 \\ 1 \end{pmatrix}.
\end{cases} \]
Be sure to verify that your solution satisfies the initial value problem.
3. **Properties of Skew Hermitian Matrices**
Recall that a skew Hermitian matrix is a matrix \( B \) with the property \( B^H = -B \). Let \( B \) be an \( n \times n \) skew Hermitian matrix and let \( \mathbf{x} \) be a vector in \( \mathbb{C}^n \).
Show that \( \alpha = \mathbf{x}^H A \mathbf{x} \) is a purely imaginary number; that is, \( \alpha = bi \) with \( b \) a real number.
\[ \text{[Hint: Recall that for a complex number } z, \, z^H = \bar{z}. \]
### Explanation of Graphs and Diagrams
There are several matrices provided in the text:
- \( A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \)
- \( B = \begin{pmatrix} 0 & i \\ -i](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67a79aa2-f715-406c-8bee-8178252bb86d%2Ffb177e8d-f63d-4650-9053-5d2d296fb8b9%2Fx4cwjq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Algebra and Matrix Theory Problems
1. **Eigenvalues of Matrices**
(a) Are the eigenvalues of \( AB \) necessarily all real? Justify your answer.
(b) Are the eigenvalues of \( AB \) necessarily all real? Justify your answer.
\[ \text{[Hint: Consider the matrices } A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \text{ and } B = \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}. \text{]} \]
2. **Solving Initial Value Problems Using Exponentiation of Matrices**
Let \( A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \). It can be shown that:
\[ e^{tA} = \begin{pmatrix} e^t & te^t \\ 0 & e^t \end{pmatrix}. \]
Use this fact to write the solution to the initial value problem:
\[ \begin{cases}
y' = Ay, \\
y(0) = \begin{pmatrix} 0 \\ 1 \end{pmatrix}.
\end{cases} \]
Be sure to verify that your solution satisfies the initial value problem.
3. **Properties of Skew Hermitian Matrices**
Recall that a skew Hermitian matrix is a matrix \( B \) with the property \( B^H = -B \). Let \( B \) be an \( n \times n \) skew Hermitian matrix and let \( \mathbf{x} \) be a vector in \( \mathbb{C}^n \).
Show that \( \alpha = \mathbf{x}^H A \mathbf{x} \) is a purely imaginary number; that is, \( \alpha = bi \) with \( b \) a real number.
\[ \text{[Hint: Recall that for a complex number } z, \, z^H = \bar{z}. \]
### Explanation of Graphs and Diagrams
There are several matrices provided in the text:
- \( A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \)
- \( B = \begin{pmatrix} 0 & i \\ -i
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