Linear independence of eigenvectors

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Discussion Overview

The discussion centers around the linear independence of eigenvectors of matrices, particularly focusing on the conditions under which an arbitrary n x n matrix may have n linearly independent eigenvectors. The context includes considerations of positive-definite and semi-positive-definite matrices, as well as the implications of repeated eigenvalues.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant questions how to demonstrate that an arbitrary n x n matrix has n linearly independent eigenvectors, particularly when the matrix is the product of a positive-definite and a semi-positive-definite matrix.
  • Another participant asserts that it is not generally true that eigenvectors are linearly independent, noting that this holds only for eigenvectors corresponding to distinct eigenvalues.
  • The same participant provides examples illustrating that with repeated eigenvalues, it may not be possible to find n independent eigenvectors, citing the identity matrix and a specific nilpotent matrix as cases.
  • In contrast, it is stated that for positive-definite matrices, one can guarantee a linearly independent set of eigenvectors, referencing the spectral theorem applicable to Hermitian matrices.
  • There is a query regarding whether Hermitian matrices can have repeated eigenvalues, with a participant noting that the identity matrix, which is Hermitian, has a repeated eigenvalue.

Areas of Agreement / Disagreement

Participants express disagreement regarding the generality of the statement about linear independence of eigenvectors, with some asserting that it depends on the distinctness of eigenvalues. The discussion remains unresolved regarding the conditions under which Hermitian matrices may have repeated eigenvalues.

Contextual Notes

The discussion highlights limitations in understanding the implications of eigenvalues and eigenvectors, particularly in cases of repeated eigenvalues and the specific properties of positive-definite and Hermitian matrices.

AxiomOfChoice
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How can you show that an arbitrary [itex]n \times n[/itex] matrix has [itex]n[/itex] linearly independent eigenvectors? What if all you know about the matrix is that it's the product of a positive-definite matrix and a semi-positive-definite matrix?
 
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It is NOT true in general that the eigenvectors are linearly independent. It's only true for those eigenvectors corresponding to DISTINCT eigenvalues.

In the case of repeated eigenvalues, it may or may not be possible to find independent eigenvectors. For example, the identity matrix has only one eigenvalue, 1, repeated n times. In this case, EVERY nonzero vector is an eigenvector, and if you simply choose n of them at random, most likely they won't be linearly independent. (However, in this case at least it's POSSIBLE to choose n independent eigenvectors.)

For some matrices with repeated eigenvectors, it's not even possible to choose n independent eigenvectors: for example, the matrix

[0 1]
[0 0]

has only one (repeated) eigenvalue, 0. Any eigenvector [x y]^T must satisfy y = 0, so this restricts the eigenvectors to a one-dimensional subspace, and therefore you can't have two independent eigenvectors.

On the other hand, for a positive definite matrix, you are guaranteed to be able to find a linearly independent (even orthonormal) set of eigenvectors. This is even true for any Hermitian matrix by the spectral theorem (see, e.g., Horn and Johnson, "Matrix Analysis," theorem 2.5.6).

Since a positive-definite or positive-semidefinite matrix is by definition Hermitian, and so therefore is their product, then the answer to your second question is yes, it's true, but the proof is not so elementary.
 
Thanks - that's helpful.

Is it possible for a Hermitian matrix to have a repeated eigenvalue, or several repeated eigenvalues? Or do we know that [itex]n \times n[/itex] Hermitian matrices will have [itex]n[/itex] distinct eigenvalues?
 
AxiomOfChoice said:
Thanks - that's helpful.

Is it possible for a Hermitian matrix to have a repeated eigenvalue, or several repeated eigenvalues? Or do we know that [itex]n \times n[/itex] Hermitian matrices will have [itex]n[/itex] distinct eigenvalues?

Sure, the identity matrix is Hermitian and all of its eigenvalues are 1.
 

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