Find a basis for the eigenspace corresponding to the eigenvalue of A given below. 0 2 A=3 0 4 1 = 2 2 -1 5 A basis for the eigenspace corresponding to A = 2 is | (Use a comma to separate answers as needed.)
Find a basis for the eigenspace corresponding to the eigenvalue of A given below. 0 2 A=3 0 4 1 = 2 2 -1 5 A basis for the eigenspace corresponding to A = 2 is | (Use a comma to separate answers as needed.)
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%
![**Finding a Basis for the Eigenspace**
In this lesson, we will learn how to find a basis for the eigenspace corresponding to a given eigenvalue of a matrix. Consider the problem below, where we are given a matrix \(A\) and an eigenvalue \(\lambda\):
Given matrix \(A\) and eigenvalue \(\lambda\):
\[A = \begin{bmatrix}
3 & 0 & 2 \\
3 & 0 & 4 \\
2 & -1 & 5
\end{bmatrix}, \quad \lambda = 2\]
We are required to find a basis for the eigenspace corresponding to the eigenvalue \(\lambda = 2\).
To proceed, follow these steps:
1. Subtract \(\lambda I\) (where \(I\) is the identity matrix of the same dimension as \(A\)) from the matrix \(A\).
2. Solve the resulting system of linear equations to find the null space of the matrix \(A - 2I\).
3. The basis for the null space found in step 2 will form the basis for the eigenspace corresponding to the given eigenvalue.
We then get the augmented matrix equation and solve it step-by-step to find the eigenvectors. Note that the specific computations are subject to detailed mathematical operations involving linear algebra techniques.
### Example Solution Format
\[ \text{A basis for the eigenspace corresponding to} \ \lambda = 2 \ \text{is} \quad [\text{Vectors here}].\]
Remember to use a comma to separate multiple vectors in the basis if needed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9225fa68-a539-4743-ab4b-10a3968e5a82%2F0d3a83aa-d615-4727-bba2-2458ca54ce6d%2Fh1yr97u_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding a Basis for the Eigenspace**
In this lesson, we will learn how to find a basis for the eigenspace corresponding to a given eigenvalue of a matrix. Consider the problem below, where we are given a matrix \(A\) and an eigenvalue \(\lambda\):
Given matrix \(A\) and eigenvalue \(\lambda\):
\[A = \begin{bmatrix}
3 & 0 & 2 \\
3 & 0 & 4 \\
2 & -1 & 5
\end{bmatrix}, \quad \lambda = 2\]
We are required to find a basis for the eigenspace corresponding to the eigenvalue \(\lambda = 2\).
To proceed, follow these steps:
1. Subtract \(\lambda I\) (where \(I\) is the identity matrix of the same dimension as \(A\)) from the matrix \(A\).
2. Solve the resulting system of linear equations to find the null space of the matrix \(A - 2I\).
3. The basis for the null space found in step 2 will form the basis for the eigenspace corresponding to the given eigenvalue.
We then get the augmented matrix equation and solve it step-by-step to find the eigenvectors. Note that the specific computations are subject to detailed mathematical operations involving linear algebra techniques.
### Example Solution Format
\[ \text{A basis for the eigenspace corresponding to} \ \lambda = 2 \ \text{is} \quad [\text{Vectors here}].\]
Remember to use a comma to separate multiple vectors in the basis if needed.
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
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.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,
