A real 3 x 3 matrix H has the eigenpairs (• ()) (-() (- C) 1– i2 1+ i2 2 – i2 2+ i 2+ i2 i6 , -i6, 0, 1 2 – i
A real 3 x 3 matrix H has the eigenpairs (• ()) (-() (- C) 1– i2 1+ i2 2 – i2 2+ i 2+ i2 i6 , -i6, 0, 1 2 – i
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
(a) Give an invertible matrix V and a diagonal matrix D such that H =VDV-1.
(You do not have to compute either V-1 or H!)
(b) Give a real fundamental matrix for the system x'= Hx.
![A real \(3 \times 3\) matrix \(\mathbf{H}\) has the eigenpairs
\[
\left( 0, \begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix} \right), \quad \left( i6, \begin{pmatrix} 1 - i2 \\ 2 + i2 \\ 2 - i \end{pmatrix} \right), \quad \left( -i6, \begin{pmatrix} 1 + i2 \\ 2 - i2 \\ 2 + i \end{pmatrix} \right).
\]
### Explanation:
- **Eigenpairs**: Each eigenpair consists of an eigenvalue and its corresponding eigenvector.
1. **First Eigenpair:**
- Eigenvalue: \(0\)
- Eigenvector: \(\begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}\)
2. **Second Eigenpair:**
- Eigenvalue: \(i6\) (complex number)
- Eigenvector: \(\begin{pmatrix} 1 - i2 \\ 2 + i2 \\ 2 - i \end{pmatrix}\)
3. **Third Eigenpair:**
- Eigenvalue: \(-i6\) (complex number)
- Eigenvector: \(\begin{pmatrix} 1 + i2 \\ 2 - i2 \\ 2 + i \end{pmatrix}\)
These eigenpairs can help in understanding the behavior of the matrix \(\mathbf{H}\), such as diagonalization or transformations it represents.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F107751dd-22a7-4c13-a90b-ff8221b081c3%2F9d058d69-e17a-4cc8-96b3-e99a5b397921%2Fphgkhes_processed.png&w=3840&q=75)
Transcribed Image Text:A real \(3 \times 3\) matrix \(\mathbf{H}\) has the eigenpairs
\[
\left( 0, \begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix} \right), \quad \left( i6, \begin{pmatrix} 1 - i2 \\ 2 + i2 \\ 2 - i \end{pmatrix} \right), \quad \left( -i6, \begin{pmatrix} 1 + i2 \\ 2 - i2 \\ 2 + i \end{pmatrix} \right).
\]
### Explanation:
- **Eigenpairs**: Each eigenpair consists of an eigenvalue and its corresponding eigenvector.
1. **First Eigenpair:**
- Eigenvalue: \(0\)
- Eigenvector: \(\begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}\)
2. **Second Eigenpair:**
- Eigenvalue: \(i6\) (complex number)
- Eigenvector: \(\begin{pmatrix} 1 - i2 \\ 2 + i2 \\ 2 - i \end{pmatrix}\)
3. **Third Eigenpair:**
- Eigenvalue: \(-i6\) (complex number)
- Eigenvector: \(\begin{pmatrix} 1 + i2 \\ 2 - i2 \\ 2 + i \end{pmatrix}\)
These eigenpairs can help in understanding the behavior of the matrix \(\mathbf{H}\), such as diagonalization or transformations it represents.
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