(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.

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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.

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