When there is a double root for the eigenvalue, how many eigenvectors?

In summary, when a eigenvalue has a double root, there will always be at least one eigenvector, but there may or may not be a second, linearly independent, eigenvector.
  • #1
Petrus
702
0
Hello MHB,
I got one question. If I want to find basis ker and it got double root in eigenvalue but in that eigenvalue i find one eigenvector(/basis) what kind of decission can I make? Is it that if a eigenvalue got double root Then it Will ALWAYS have Two eigenvector(/basis)?

Regards,
\(\displaystyle |\pi\rangle\)
 
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  • #2
Re: 1 basis or Two basis for double root to ker?

Petrus said:
If I want to find basis ker and it got double root in eigenvalue but in that eigenvalue i find one eigenvector(/basis) what kind of decission can I make? Is it that if a eigenvalue got double root Then it Will ALWAYS have Two eigenvector(/basis)?
Not necessarily. When there is a double root for the eigenvalue there will always be at least one eigenvector. There may or may not be a second, linearly independent, eigenvector. For example, the matrices $\begin{bmatrix}1&0\\ 0&1 \end{bmatrix}$ and $\begin{bmatrix}1&1\\ 0&1 \end{bmatrix}$ both have a repeated eigenvalue $1$, but the first one has two linearly independent eigenvectors and the second one only has one.
 

FAQ: When there is a double root for the eigenvalue, how many eigenvectors?

What is a double root for an eigenvalue?

A double root for an eigenvalue is when a polynomial equation has two identical solutions for the eigenvalue. This means that the characteristic polynomial has a repeated root, which can occur when the matrix has repeated eigenvalues.

How do you determine the number of eigenvectors when there is a double root for the eigenvalue?

When there is a double root for the eigenvalue, there are two possible scenarios. The first scenario is that there is only one linearly independent eigenvector for the repeated eigenvalue. In this case, the double root represents a single eigenvalue with only one corresponding eigenvector. The second scenario is that there are two linearly independent eigenvectors for the repeated eigenvalue. In this case, the double root represents two eigenvalues with two corresponding eigenvectors.

Can a matrix have more than two eigenvectors for a double root of the eigenvalue?

Yes, a matrix can have more than two eigenvectors for a double root of the eigenvalue. In fact, the maximum number of eigenvectors for a double root is equal to the degree of the characteristic polynomial. This means that a matrix can have up to n eigenvectors for a double root, where n is the size of the matrix.

How does a double root for the eigenvalue affect the diagonalization of a matrix?

When there is a double root for the eigenvalue, the matrix may not be diagonalizable. This is because diagonalization requires the matrix to have n linearly independent eigenvectors, where n is the size of the matrix. If the double root only has one corresponding eigenvector, the matrix cannot be diagonalized. However, if there are two linearly independent eigenvectors for the double root, the matrix can still be diagonalized.

Can a double root for the eigenvalue occur for any matrix?

Yes, a double root for the eigenvalue can occur for any matrix. However, it is more common for matrices with repeated eigenvalues to have a double root. This can happen when the matrix has a symmetry or a repeated pattern, which results in repeated eigenvalues and a double root for the eigenvalue.

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