Understanding Eigenvalues and Eigenvectors in Reflection Matrices

In summary, the conversation discusses the properties of a matrix A representing a reflection in two dimensions across a line generated by a vector v. The conversation then lists six statements and asks the reader to check which ones are true. These statements pertain to the eigenvalues and eigenvectors of A. The correct statements are B, C, and E, while the incorrect statements are A, D, and F. The conversation ends with a request for further explanation and solution for the problem.
  • #1
Whiz
20
0

Homework Statement



Let A be a matrix corresponding to reflection in 2 dimensions across the line generated by a vector v . Check all true statements:

A. lambda =1 is an eigenvalue for A
B. Any vector w perpendicular to v is an eigenvector for A corresponding to the eigenvalue lambda =1.
C. The vector v is an eigenvector for A corresponding to the eigenvalue lambda =1.
D. Any vector w perpendicular to v is an eigenvector for A corresponding to the eigenvalue lambda =−1.
E. lambda =−1 is an eigenvalue for A
F. None of the above

The Attempt at a Solution



I honestly have no clue how to do this question. Can somebody explain to me what the question is asking and how to solve it?
 
Physics news on Phys.org
  • #2


Do you know what the definition of an eigenvalue/eigenvector is? Think about the case where A just reflects over the x-axis first in R2 in order to get a handle on what the question is asking
 

1. What are eigenvectors and why are they important in math?

Eigenvectors are special vectors that, when multiplied by a matrix, produce a scalar multiple of the original vector. They are important in math because they help us understand how a matrix transforms a vector and can reveal important patterns and structures in data.

2. How are eigenvectors and eigenvalues related?

Eigenvectors and eigenvalues are related because an eigenvector is associated with a specific eigenvalue of a matrix. The eigenvalue represents the scalar multiple that the eigenvector is multiplied by when transformed by the matrix.

3. How do you calculate eigenvectors and eigenvalues?

To calculate eigenvectors and eigenvalues, you must first find the characteristic polynomial of the matrix. Then, solve for the roots of the polynomial to find the eigenvalues. Next, plug each eigenvalue into the equation (A - λI)x = 0 and solve for x to find the corresponding eigenvector.

4. What are some real-world applications of eigenvectors?

Eigenvectors have many real-world applications, such as in data analysis, image processing, and machine learning. They can be used to reduce the dimensionality of data, identify important features, and classify data points.

5. Can there be multiple eigenvectors for one eigenvalue?

Yes, there can be multiple eigenvectors for one eigenvalue. This is because eigenvectors are not unique and can be scaled by any non-zero constant and still be considered an eigenvector. However, the eigenvectors for a particular eigenvalue must be linearly independent.

Similar threads

Prove 9 is eigenvalue of ##T^2\iff## 3 or -3 eigenvalue of ##T##.
    • Calculus and Beyond Homework Help
Replies
5
Views
527
Prove that ##\lambda## or ##-\lambda## is an eigenvalue for ##T##.
    • Calculus and Beyond Homework Help
Replies
11
Views
1K
Linear homogenous system with repeated eigenvalues
    • Calculus and Beyond Homework Help
Replies
2
Views
525
Understanding Eigenvectors: Solving for Eigenvalues and Corresponding Vectors
    • Calculus and Beyond Homework Help
Replies
9
Views
1K
Does there exist a 2x2 non-singular matrix with only one 1d eigenspace?
    • Calculus and Beyond Homework Help
Replies
2
Views
391
How to diagonalize a matrix with complex eigenvalues?
    • Calculus and Beyond Homework Help
Replies
5
Views
2K
Help me prove something on eigenvectors?
    • Calculus and Beyond Homework Help
Replies
7
Views
1K
Solving a system of differential equations by fundamental matrix
    • Calculus and Beyond Homework Help
Replies
8
Views
2K
Find the eigenvalues and eigenvectors
    • Calculus and Beyond Homework Help
Replies
19
Views
3K
If A is nxn nilpotent matrix, this char(A) = x^n
    • Calculus and Beyond Homework Help
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top