The discussion revolves around the Gram-Schmidt orthogonalization process in the context of linear algebra, specifically regarding the diagonalization of a matrix A through its eigenvectors. Participants explore whether orthonormalizing a basis of eigenvectors can lead to a matrix P such that PTAP is diagonal.
The conversation is ongoing, with some participants expressing uncertainty about the relationship between orthonormalization and eigenvectors. There is a mix of perspectives, with some suggesting that the Gram-Schmidt process may not preserve the eigenvector property, while others seek clarification on the definitions and processes involved.
Participants are grappling with the assumptions related to the orthogonality of eigenvectors, particularly when they correspond to different eigenvalues. The discussion highlights the distinction between normalizing vectors and applying the Gram-Schmidt process.
nautolian said:Homework Statement
in light of the gram-schmidt orthogonalization process, if A: Rn -> Rn and we have a basis of Rn of eigenvectors of A, can't we just orthonormalize them and get a matrix P such that P-1=PT and thus PTAP is diagonal?
Homework Equations
The Attempt at a Solution
I believe the answer to this is yes, but I'm not sure how to prove this. Any ideas would be appreciated, thanks!
nautolian said:Why aren't those eigenvectors? Because you've changed the vectors too much?
nautolian said:I thought it might be an eigenvector because i was under the impression for some reason that the orthonormalization would no necessarily mean using the Gram-Shmidt process, but rather just normalizing it (dividing by sqrt(sum of squares of eigenvector values)). Is this not the case?