Why Can't I Find All Eigenvectors for My 3x3 Matrix?

In summary, eigenvectors for a matrix are special vectors that remain in the same direction after being multiplied by the matrix. They are associated with a corresponding eigenvalue, which represents the scaling factor of the eigenvector. Eigenvectors and eigenvalues are related in that each eigenvector is associated with a unique eigenvalue. The eigenvalue represents the scaling factor by which the eigenvector is multiplied when multiplied by the matrix. Eigenvectors and eigenvalues have many practical applications, such as in image and signal processing, data compression, and machine learning. They are also used in physics, particularly in quantum mechanics, to describe the behavior of physical systems. To find the eigenvectors for a matrix, you first need to find
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haz
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Hey everyone.
I have a matrix A = {{7,-5,0},{-5,7,0},{0,0,-6}}
I have found the Eigenvalues, 2,12,-6 but I'm only getting one Eigenvector, (0,0,1)..
I know there is 2 others (-1,1,0) and (1,1,0) but I am unable to get them by hand.
Once I get the matrix in the form (A-λI)*v = o, I just get the one vector..
Any help is much appreciated.
 
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Related to Why Can't I Find All Eigenvectors for My 3x3 Matrix?

What are eigenvectors for a matrix?

Eigenvectors for a matrix are special vectors that remain in the same direction after being multiplied by the matrix. They are associated with a corresponding eigenvalue, which represents the scaling factor of the eigenvector.

How are eigenvectors and eigenvalues related?

Eigenvectors and eigenvalues are related in that each eigenvector is associated with a unique eigenvalue. The eigenvalue represents the scaling factor by which the eigenvector is multiplied when multiplied by the matrix.

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

Eigenvectors and eigenvalues have many practical applications, such as in image and signal processing, data compression, and machine learning. They are also used in physics, particularly in quantum mechanics, to describe the behavior of physical systems.

How do you find eigenvectors for a matrix?

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

Can a matrix have more than one eigenvector?

Yes, a matrix can have multiple eigenvectors associated with the same eigenvalue. In fact, the number of eigenvectors for a given eigenvalue is equal to the geometric multiplicity of that eigenvalue, which is the dimension of the eigenspace associated with that eigenvalue.

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