The matrix -11 A= - 0 24 0 9 has two real eigenvalues, A₁ - 1 of multiplicity 2, and A₂-3 of multiplicity 1. Find an orthonormal basis for the eigenspace corresponding to A₁
The matrix -11 A= - 0 24 0 9 has two real eigenvalues, A₁ - 1 of multiplicity 2, and A₂-3 of multiplicity 1. Find an orthonormal basis for the eigenspace corresponding to A₁
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
![### Orthogonal Basis for Eigenspace - Example Problem
**Problem:**
Given the matrix
\[ A = \begin{pmatrix}
-11 & 0 & 24 \\
0 & 1 & 0 \\
-4 & 0 & 9
\end{pmatrix}, \]
which has two real eigenvalues λ₁ = 1 of multiplicity 2 and λ₂ = -3 of multiplicity 1. Find an orthonormal basis for the eigenspace corresponding to λ₁.
**Solution:**
1. **Find the Eigenvectors:**
- For λ₁ = 1, solve the equation \( (A - I)v = 0 \), where \( I \) is the identity matrix.
2. **Construct the Eigenvectors:**
- Find the general solution to \( (A - I)v = 0 \) to get the basis vectors for the eigenspace.
3. **Use the Gram-Schmidt Process:**
- Apply the Gram-Schmidt process to turn the basis vectors into an orthonormal basis.
### Matrix and Eigenvalues
- **Eigenvalue λ₁ = 1:**
- Multiplicity: 2
- Procedure: Solve \( (A - I)v = 0 \)
- **Eigenvalue λ₂ = -3:**
- Multiplicity: 1 (not needed for this problem)
### Box Representation for Basis Vectors
To construct the orthonormal basis for the eigenspace corresponding to λ₁:
\[ \left\{
\begin{pmatrix}
a \\
b \\
c
\end{pmatrix},
\begin{pmatrix}
d \\
e \\
f
\end{pmatrix}
\right\} \]
where \( \begin{pmatrix}
a \\
b \\
c
\end{pmatrix} \) and \( \begin{pmatrix}
d \\
e \\
f
\end{pmatrix} \) are the normalized vectors after applying the Gram-Schmidt process.
### Diagrams in the Solution:
The rectangular areas represent the slots for the components of the orthonormal basis vectors. The entire set forms the orthonormal basis needed for the eigenspace corresponding to λ₁.
Ensure to follow the steps precisely to arrive at the final orthonormal basis for academic](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa59c8a62-6d27-4c14-9dc5-c4241c4a7fe0%2Fc04ec0e8-149e-452e-8f5d-2d8003117ba8%2Fvnq2b2i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Orthogonal Basis for Eigenspace - Example Problem
**Problem:**
Given the matrix
\[ A = \begin{pmatrix}
-11 & 0 & 24 \\
0 & 1 & 0 \\
-4 & 0 & 9
\end{pmatrix}, \]
which has two real eigenvalues λ₁ = 1 of multiplicity 2 and λ₂ = -3 of multiplicity 1. Find an orthonormal basis for the eigenspace corresponding to λ₁.
**Solution:**
1. **Find the Eigenvectors:**
- For λ₁ = 1, solve the equation \( (A - I)v = 0 \), where \( I \) is the identity matrix.
2. **Construct the Eigenvectors:**
- Find the general solution to \( (A - I)v = 0 \) to get the basis vectors for the eigenspace.
3. **Use the Gram-Schmidt Process:**
- Apply the Gram-Schmidt process to turn the basis vectors into an orthonormal basis.
### Matrix and Eigenvalues
- **Eigenvalue λ₁ = 1:**
- Multiplicity: 2
- Procedure: Solve \( (A - I)v = 0 \)
- **Eigenvalue λ₂ = -3:**
- Multiplicity: 1 (not needed for this problem)
### Box Representation for Basis Vectors
To construct the orthonormal basis for the eigenspace corresponding to λ₁:
\[ \left\{
\begin{pmatrix}
a \\
b \\
c
\end{pmatrix},
\begin{pmatrix}
d \\
e \\
f
\end{pmatrix}
\right\} \]
where \( \begin{pmatrix}
a \\
b \\
c
\end{pmatrix} \) and \( \begin{pmatrix}
d \\
e \\
f
\end{pmatrix} \) are the normalized vectors after applying the Gram-Schmidt process.
### Diagrams in the Solution:
The rectangular areas represent the slots for the components of the orthonormal basis vectors. The entire set forms the orthonormal basis needed for the eigenspace corresponding to λ₁.
Ensure to follow the steps precisely to arrive at the final orthonormal basis for academic
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 3 steps with 3 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,
