Why are the eigenvectors of this hermitian matrix not orthogonal?

In summary, the eigenvectors of a Hermitian matrix are typically orthogonal, but if the matrix has degenerate eigenvalues (i.e., multiple eigenvectors corresponding to the same eigenvalue), the eigenvectors associated with these degenerate values may not be orthogonal unless specifically orthogonalized. Therefore, the lack of orthogonality arises from the presence of degeneracy in the eigenvalues, allowing for non-orthogonal eigenvectors within the same eigenspace.
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rghurst
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TL;DR Summary
I am unable to show that the eigenvectors are orthogonal.
Why are the eigenvectors of this hermitian matrix not checking out as orthogonal? The eigenvalues are certainly distinct. ChatGPT also is miscalculating repeatedly. I have checked my work many times and cannot find the error. Kindly assist.
 

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  • #2
rghurst said:
TL;DR Summary: I am unable to show that the eigenvectors are orthogonal.

Why are the eigenvectors of this hermitian matrix not checking out as orthogonal? The eigenvalues are certainly distinct. ChatGPT also is miscalculating repeatedly. I have checked my work many times and cannot find the error. Kindly assist.
Don't rely on ChatGPT and please repost your calculation attempt in LaTeX so it's readable and quotable.
 
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