Finding Eigenvectors & Eigenvalues of A Matrix

In summary, the conversation is about using direct multiplication to show that certain vectors are eigenvectors of a given matrix and finding the corresponding eigenvalues. The person asking for help is unsure of how to approach the problem and is asking for clarification.
  • #1
innightmare
35
0

Homework Statement




Use direct multiplication to show that for each of the following matrices A, the given vectors v1, v2, and v3 are eigenvectors of A and to find the eigen values lama1, lama2, and lama3 of A:

A=top row: (2 -1 3) second row: (-1 6 -1) third row: (3 -5 2) v1=(1,0,-1) v2=(2,2,2), v3 =(7, -9,11)

Homework Equations



Plug the v's back into the A matrice

The Attempt at a Solution



Have no idea what they mean nor how to go about the direct multiplication here. Thanks What exactly do they want me to do?
 
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  • #2
matrix * vector = eigenvalue * (same)matrix
 
  • #3
malawi_glenn said:
matrix * vector = eigenvalue * (same)vector

There's a minor (typo) error in there, see the correction in bold :smile:
 
  • #4
yeah LOL :)

otherwise it would have been called eigenmatrix ;)
 

Related to Finding Eigenvectors & Eigenvalues of A Matrix

1. What are eigenvectors and eigenvalues?

Eigenvectors and eigenvalues are concepts used in linear algebra to describe special vectors and scalars associated with a square matrix. Eigenvectors are vectors that, when multiplied by a matrix, result in a scalar multiple of itself. Eigenvalues are the corresponding scalars that represent the amount by which the eigenvector is scaled.

2. Why are eigenvectors and eigenvalues important?

Eigenvectors and eigenvalues are important because they provide a way to simplify complex matrix operations, such as finding the inverse or determinant of a matrix. They also have applications in physics, engineering, and other fields for solving problems involving systems of linear equations.

3. How do you find the eigenvectors and eigenvalues of a matrix?

To find the eigenvectors and eigenvalues of a matrix, you first need to find the characteristic polynomial of the matrix. You can then use this polynomial to find the eigenvalues, which are the roots of the polynomial. To find the corresponding eigenvectors, you can use the eigenvalues to solve a system of equations involving the matrix and the eigenvector variables.

4. Can a matrix have more than one eigenvector and eigenvalue?

Yes, a matrix can have multiple eigenvectors and eigenvalues. In fact, the number of eigenvectors and eigenvalues of a matrix is equal to its dimension. So, a 3x3 matrix will have three eigenvectors and three eigenvalues.

5. How are eigenvectors and eigenvalues used in data analysis?

In data analysis, eigenvectors and eigenvalues can be used to reduce the dimensionality of a dataset. This means that you can represent a large set of data in a smaller number of dimensions without losing important information. This can be useful for visualizing data or for making predictions based on the data.

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