A Why are the eigenvectors of this hermitian matrix not orthogonal?

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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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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