Eigenvector of 2x2 matrix

In summary, an eigenvector of a 2x2 matrix is a vector that remains in the same direction after being multiplied by the matrix. To find the eigenvectors, you first need to find the eigenvalues and then solve a system of equations. The significance of eigenvectors lies in their representation of simple effects and providing information about stability and behavior of the system. A 2x2 matrix can have complex eigenvectors, as the eigenvalues and eigenvectors do not have to be real numbers. The eigenvectors of a 2x2 matrix are related to its determinant through the fact that the determinant is equal to the product of the eigenvalues.
  • #1
565
2
for the matrix
{5,0}
{2,-2}

when determining the eigenvector for its 2nd eigenvalue, -2, you would start out by doing


{5--2 ,0}
{2 ,-2--2}

giving

{7,0}
{2,0}

In equation form this is

7u + 0v = 0
2u + 0v = 0

Ordinarily I would set u or v to a value and solve for the other letter.


But in this case this can't be done since in both equations v has a coefficient of 0.


So what would you do in this situation?
 
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  • #2
Clearly u=0, and the equations will both be satisfied for any value of v, so the normalized eigenvector is (0,1).
 

What is an eigenvector of a 2x2 matrix?

An eigenvector of a 2x2 matrix is a vector that does not change direction when multiplied by the matrix. In other words, when the matrix is multiplied by the eigenvector, the resulting vector is a scalar multiple of the original eigenvector.

How do you find the eigenvectors of a 2x2 matrix?

To find the eigenvectors of a 2x2 matrix, you first need to find the eigenvalues of the matrix. Then, you can plug each eigenvalue into the equation (A - λI)v = 0, where A is the matrix, λ is the eigenvalue, and v is the eigenvector. This will give you a system of equations that you can solve to find the eigenvectors.

What is the significance of the eigenvectors of a 2x2 matrix?

The eigenvectors of a 2x2 matrix are important because they represent the directions in which the matrix has a simple effect. They also provide information about the stability and behavior of a system described by the matrix.

Can a 2x2 matrix have complex eigenvectors?

Yes, a 2x2 matrix can have complex eigenvectors. This is because the eigenvalues and eigenvectors of a matrix do not have to be real numbers, they can be complex numbers. In fact, if the matrix has complex eigenvalues, then it will also have complex eigenvectors.

How are the eigenvectors of a 2x2 matrix related to its determinant?

The eigenvectors of a 2x2 matrix are related to its determinant in that the determinant is equal to the product of the eigenvalues. This means that if the determinant is zero, then at least one of the eigenvalues must also be zero, and therefore there will be at least one eigenvector with a zero eigenvalue.

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