Calculate the eigenvectors of a specific matrix

• Dassinia
In summary, the conversation discusses a problem with calculating eigenvectors for a specific matrix and how to use these eigenvectors for differential equations. The matrix A is given along with its eigenvalues and eigenvectors. The individual is struggling with finding the eigenvectors for eigenvalue 2 and is unsure of what to do when the first eigenvector is equal to 0.
Dassinia
Hello,
I'm really having a problem to calculate the eigenvectors of a specific matrix, I'm used to do this but i don't know why I'm stuck at this one

Homework Statement

A=
2 1 0 1
0 3 -1 0
0 1 1 0
0 -1 0 3

λ1=2 multiplicity 3
λ2=3 multiplicity 1

A-2I=
0 1 0 1
0 1 -1 0
0 1 -1 0
0 -1 0 1

The eigenvectors are given
[-1 -1 -1 -1]
[0 1 2 0]
[1 0 0 0]

If I solve
(A-2I)xi=0
I have
x1=0
x2+x4=0
x2-x3=0
-x2+x4=0

I have to use this for differential equations, and my linear algebra course is far behind, I don't remember what I'm supposed to do when we get the first eigenvector =0, because when the multiplicity of the eigenvalue is > 0 we use the first one to find the following eigenvectors

Thanks!

Dassinia said:
Hello,
I'm really having a problem to calculate the eigenvectors of a specific matrix, I'm used to do this but i don't know why I'm stuck at this one

Homework Statement

A=
2 1 0 1
0 3 -1 0
0 1 1 0
0 -1 0 3

λ1=2 multiplicity 3
λ2=3 multiplicity 1

A-2I=
0 1 0 1
0 1 -1 0
0 1 -1 0
0 -1 0 1

The eigenvectors are given
[-1 -1 -1 -1]
[0 1 2 0]
[1 0 0 0]

If I solve
(A-2I)xi=0
I have
x1=0
x2+x4=0
x2-x3=0
-x2+x4=0

I have to use this for differential equations, and my linear algebra course is far behind, I don't remember what I'm supposed to do when we get the first eigenvector =0, because when the multiplicity of the eigenvalue is > 0 we use the first one to find the following eigenvectors

Thanks!

There is a nonzero eigenvector corresponding to eigenvalue 2. Rethink your conclusion that solving that matrix gives you x1=0.

1. What is the purpose of calculating eigenvectors for a matrix?

The eigenvectors of a matrix represent the directions in which the matrix acts as a simple scaling transformation. They are useful for understanding the behavior and properties of a matrix, and are often used in applications such as data analysis, computer graphics, and quantum mechanics.

2. How do you calculate the eigenvectors of a matrix?

To calculate the eigenvectors of a matrix, you will need to first find the eigenvalues of the matrix. This can be done by solving the characteristic equation for the matrix, which is det(A-λI)=0, where A is the matrix, λ is the eigenvalue, and I is the identity matrix. Once you have the eigenvalues, you can find the corresponding eigenvectors by solving the equation (A-λI)x=0, where x is the eigenvector.

3. What is the relationship between eigenvectors and eigenvalues?

Eigenvectors and eigenvalues are closely related. Eigenvectors are the vectors that do not change direction when multiplied by a matrix, and eigenvalues represent the scaling factor by which the eigenvectors are multiplied. In other words, the eigenvalues determine the amount and direction of scaling that occurs when the matrix is applied to the eigenvectors.

4. Can a matrix have multiple eigenvectors?

Yes, a matrix can have multiple eigenvectors. In fact, most matrices have multiple eigenvectors corresponding to different eigenvalues. However, there are some cases where a matrix may have only one eigenvector, such as when the matrix is a scalar multiple of the identity matrix.

5. Are eigenvectors unique for a given matrix?

No, eigenvectors are not unique for a given matrix. A matrix can have multiple eigenvectors with the same eigenvalue, and even if the eigenvalues are different, there can be multiple eigenvectors with the same eigenvalue. However, the set of eigenvectors for a matrix will always be linearly independent, meaning no eigenvector can be expressed as a linear combination of the others.

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