Can we still diagnolize a matrix if its eigenvectors matrix is singular?

In summary, if a matrix has n linearly independent eigen-vectors, it can be easily diagonalized. However, if it is not diagonalizable in that way, there are still other methods to diagonalize it, such as using Jordan normal forms. Additionally, even if a matrix is not diagonalizable at all, the Jordan-Chevalley decomposition allows for the computation of its exponential matrix. A theorem states that a matrix is diagonalizable if and only if the sum of the dimensions of its eigenspaces is equal to n.
  • #1
AdrianZ
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well, if a matrix has n linearly independent eigen-vectors then it's easy, what if a matrix is not diagnolizable in that way? Can we still diagnolize it by other means?
And what if a matrix is not diagonalizable at all? Are there still ways to find its exponential matrix?
 
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  • #2
If you don't have enough eigenvectors you can't diagonalize it. There is a theorem that says: An n-by-n matrix A is diagonalizable if and only if the sum of the dimensions of its eigenspaces is equal to n.

However, you can 'almost' diagonalize any matrix you want. One way to do this is to use something called Jordan normal forms (http://en.wikipedia.org/wiki/Jordan_normal_form).

Since you are asking about exponential matrices, you might be interested in the Jordan–Chevalley decomposition which allows you to compute the exponential of a matrix by writing it as the sum of a diagonalizable matrix and nilpotent matrix.
 

1. How is a matrix diagnosed?

A matrix is diagnosed by finding its eigenvalues and eigenvectors. The eigenvalues are the numbers that are multiplied by the eigenvectors to produce the original matrix.

2. What does it mean if a matrix's eigenvector matrix is singular?

If a matrix's eigenvector matrix is singular, it means that the matrix does not have enough linearly independent eigenvectors to diagonalize the matrix.

3. Can a matrix be diagonalized if its eigenvector matrix is singular?

No, if a matrix's eigenvector matrix is singular, it cannot be diagonalized. This is because the eigenvectors are used to form the diagonalizing matrix, and if they are not linearly independent, the diagonalizing matrix cannot be formed.

4. What are the consequences of not being able to diagonalize a matrix?

If a matrix cannot be diagonalized, it means that it cannot be simplified into a simpler form. This can make computations and calculations involving the matrix more difficult and complex.

5. Is there any other way to simplify a matrix if its eigenvector matrix is singular?

Yes, other methods such as Jordan decomposition or singular value decomposition can be used to simplify a matrix even if its eigenvector matrix is singular. However, these methods may not always be applicable and may not result in a diagonalized matrix.

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