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I am currently trying to study the mathematics of quantum mechanics. Today I cam across the theorem that says that a Hermitian matrix of dimensionality ##n## will always have ##n## independent eigenvectors/eigenvalues. And my goal is to prove this. I haven't taken any linear algebra classes so my knowledge in this field is quite limited.

Now, I have realized that if I were to prove this statement, I would basically have to show that there is a polynomial associated with the matrix and that this polynomial can be factored into a product of distinct linear factors. Mathematically, if I let ##M## to be the Hermitian matrix, ##I## be the identity matrix, and ##\lambda## be the placeholder for the eigenvalues, then I have to show that the following is true for all ##n##:

##det(M-I\lambda) = (\lambda - \lambda_1)(\lambda - \lambda_2)......(\lambda - \lambda_n)## (where ##\lambda_i## are independent eigenvalues)

Here, I’m just stuck. I really don’t know how to tackle this at all. If someone could help me out on this problem, it would be really great. Or if you guys can point me to some sort of literature, that would also be good for me. Thank you very much in advance! :)

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# Number of eigenvectors for Hermitian matrices

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