Find 2 linearly independent eigenvectors and a eigenvalue

In summary, to find 2 linearly independent eigenvectors and an eigenvalue, you first need to find the eigenvalues of the matrix and then plug them into the original matrix to find the corresponding eigenvectors. It is important to find linearly independent eigenvectors because they can simplify calculations and provide insight into the behavior of the matrix. A matrix can have at most n linearly independent eigenvectors, where n is the dimension of the matrix. Eigenvectors and eigenvalues are used in various real-world applications, including image and signal processing, data compression, and machine learning. A matrix can have complex eigenvectors and eigenvalues, which can still provide valuable information about the behavior of the matrix.
  • #1
bodensee9
178
0
Hi

I am supposed to, without calculation, find 2 linearly independent eigenvectors and a eigenvalue of the following matrix A

5 5 5
5 5 5
5 5 5

The eigenvalue is easy -- it is 15. And I can find one eigenvector, [1 1 1] (written vertically), but another without calculation? Is there a trick that I should see? Of course, zero is also a eigenvalue, but how am I supposed to find the other eigenvector without calculation? Is there something that I should see?

Thanks.
 
Physics news on Phys.org
  • #2


Pick the vector (x,y,z) so that 5x+5y+5z=0. Say, (1,1,-2)?
 
  • #3


oh thanks!
 

Related to Find 2 linearly independent eigenvectors and a eigenvalue

1. How do you find 2 linearly independent eigenvectors and an eigenvalue?

To find 2 linearly independent eigenvectors and an eigenvalue, you will need to first find the eigenvalues of the matrix. Once you have the eigenvalues, you can plug them into the original matrix to find the corresponding eigenvectors. To ensure that the eigenvectors are linearly independent, you can check if their determinants are non-zero.

2. Why is it important to find linearly independent eigenvectors?

Finding linearly independent eigenvectors is important because they can be used as a basis for the eigenspace of a matrix. This allows for simplification of calculations and can provide insight into the behavior of the matrix.

3. Can a matrix have more than 2 linearly independent eigenvectors?

No, a matrix can have at most n linearly independent eigenvectors, where n is the dimension of the matrix. This is because the number of linearly independent eigenvectors is equal to the number of distinct eigenvalues of the matrix.

4. How are eigenvectors and eigenvalues used in real-world applications?

Eigenvectors and eigenvalues are used in a variety of real-world applications, including image and signal processing, data compression, and machine learning. They can also be used to model physical systems and analyze their behaviors.

5. Can a matrix have complex eigenvectors and eigenvalues?

Yes, a matrix can have complex eigenvectors and eigenvalues. This is especially common in matrices with complex entries or in applications such as quantum mechanics. Complex eigenvectors and eigenvalues can still provide valuable information about the behavior of a matrix, even though they may not be easily visualized.

Similar threads

Prove 9 is eigenvalue of ##T^2\iff## 3 or -3 eigenvalue of ##T##.
    • Calculus and Beyond Homework Help
Replies
5
Views
591
Operator T, ##T^2=I##, -1 not an eigenvalue of T, prove ##T=I##.
    • Calculus and Beyond Homework Help
Replies
2
Views
369
Question regarding eigenvectors
    • Calculus and Beyond Homework Help
Replies
7
Views
1K
Linear homogenous system with repeated eigenvalues
    • Calculus and Beyond Homework Help
Replies
2
Views
577
Solving a system of differential equations by fundamental matrix
    • Calculus and Beyond Homework Help
Replies
8
Views
2K
Find Eigenvalues & Eigenvectors for Exercise 3 (2), Explained!
    • Calculus and Beyond Homework Help
Replies
8
Views
974
Find eigenvalues & eigenvectors
    • Calculus and Beyond Homework Help
Replies
3
Views
2K
Setting Free variables when finding eigenvectors
    • Calculus and Beyond Homework Help
Replies
1
Views
2K
Are Similar Matrices' Eigenvalues the Same? Solving for Symmetric Matrices
    • Calculus and Beyond Homework Help
Replies
9
Views
1K
Diagonalizing a matrix given the eigenvalues
    • Calculus and Beyond Homework Help
Replies
8
Views
1K

Hot Threads

Recent Insights

Back
Top