4. Find the eigenvalues and a basis for each eigenspace in C². Show all of the work (no calculator). 3 -2 3
4. Find the eigenvalues and a basis for each eigenspace in C². Show all of the work (no calculator). 3 -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
100%
![### Problem 4: Finding Eigenvalues and Eigenvectors
**Problem Statement:**
Given the matrix
\[
\begin{pmatrix}
3 & -2 \\
2 & 3
\end{pmatrix}
\]
find the eigenvalues and a basis for each eigenspace in \(\mathbb{C}^2\). Show all of the work (no calculator).
**Step-by-Step Solution:**
**Step 1: Find the Eigenvalues**
The eigenvalues \(\lambda\) of a matrix can be found by solving the characteristic equation:
\[ \text{det}(A - \lambda I) = 0 \]
where \(A\) is the matrix and \(I\) is the identity matrix.
For our matrix:
\[
A = \begin{pmatrix}
3 & -2 \\
2 & 3
\end{pmatrix}
\]
The identity matrix \(I\) in \(\mathbb{C}^2\) is:
\[
I = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
\]
The matrix \(A - \lambda I\) is:
\[
A - \lambda I = \begin{pmatrix}
3 - \lambda & -2 \\
2 & 3 - \lambda
\end{pmatrix}
\]
The characteristic polynomial is given by the determinant:
\[
\text{det}(A - \lambda I) = \begin{vmatrix}
3 - \lambda & -2 \\
2 & 3 - \lambda
\end{vmatrix}
\]
\[
= (3 - \lambda)(3 - \lambda) - (-2)(2)
\]
\[
= (3 - \lambda)^2 - 4
\]
\[
= 9 - 6\lambda + \lambda^2 - 4
\]
\[
= \lambda^2 - 6\lambda + 5
\]
Set the characteristic polynomial to zero and solve for \(\lambda\):
\[
\lambda^2 - 6 \lambda + 5 = 0
\]
Factoring the quadratic equation, we get:
\[
(\lambda - 1)(\lambda - 5) = 0
\]
Therefore, the eigenvalues are:
\[](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d07e113-70c7-44ea-a54a-4dad627b44c8%2Fe87798ff-79f6-4f45-aa92-c36b532443c0%2Fccm7hcq_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 4: Finding Eigenvalues and Eigenvectors
**Problem Statement:**
Given the matrix
\[
\begin{pmatrix}
3 & -2 \\
2 & 3
\end{pmatrix}
\]
find the eigenvalues and a basis for each eigenspace in \(\mathbb{C}^2\). Show all of the work (no calculator).
**Step-by-Step Solution:**
**Step 1: Find the Eigenvalues**
The eigenvalues \(\lambda\) of a matrix can be found by solving the characteristic equation:
\[ \text{det}(A - \lambda I) = 0 \]
where \(A\) is the matrix and \(I\) is the identity matrix.
For our matrix:
\[
A = \begin{pmatrix}
3 & -2 \\
2 & 3
\end{pmatrix}
\]
The identity matrix \(I\) in \(\mathbb{C}^2\) is:
\[
I = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
\]
The matrix \(A - \lambda I\) is:
\[
A - \lambda I = \begin{pmatrix}
3 - \lambda & -2 \\
2 & 3 - \lambda
\end{pmatrix}
\]
The characteristic polynomial is given by the determinant:
\[
\text{det}(A - \lambda I) = \begin{vmatrix}
3 - \lambda & -2 \\
2 & 3 - \lambda
\end{vmatrix}
\]
\[
= (3 - \lambda)(3 - \lambda) - (-2)(2)
\]
\[
= (3 - \lambda)^2 - 4
\]
\[
= 9 - 6\lambda + \lambda^2 - 4
\]
\[
= \lambda^2 - 6\lambda + 5
\]
Set the characteristic polynomial to zero and solve for \(\lambda\):
\[
\lambda^2 - 6 \lambda + 5 = 0
\]
Factoring the quadratic equation, we get:
\[
(\lambda - 1)(\lambda - 5) = 0
\]
Therefore, the eigenvalues are:
\[
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 2 steps with 2 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,
