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Recently I encountered a lower bound on the minimum eigenvalue of positive Hermitian matrices in (Lower bound on smallest eigenvalue of (symmetric positive-definite) matrix). The lower bound is stated as:

$$ \lambda_{min} \gt \sqrt{\frac{||A||_F^2-n||A||_E^2}{n(1-||A||_E^2/|det(A)|^{2/n})}} $$

My question is if this bound exists in the first place, and if it does, is it only for real matrices or does it include complex ones too. Also, what is the difference between the Frobenius (F) and the Euclidean norm (E) here? The referenced page is not clear either. I would really appreciate your help. Thank you.

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  • $\begingroup$ You should consider your textbooks for the definition of the Frobenius and of the Euclidean 2-norm... $\endgroup$ – daw Apr 3 '14 at 6:49
  • $\begingroup$ Is there a similar bound for non-symmetric matrices? $\endgroup$ – Amir Sagiv Mar 6 '18 at 11:19
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Frobenius norm is the same as Euclidean norm and their squares is the sum of the squares of matrix entries. Therefore the bound you stated is wrong.

I think the bound you have encountered has the square of the maximum eigenvalue in the numerator instead of the Euclidean norm (Schindler's publication):http://library.utia.cas.cz/separaty/2009/AS/schindler-tikhonov%20regularization%20parameter%20in%20reproducing%20kernel%20hilbert%20spaces%20with%20respect%20to%20the%20sensitivity%20of%20the%20solution.pdf

But this bound is also wrong and should be corrected by the author. If you look into the proof you will notice preliminary mistakes.

If you're looking for a lower bound in terms of trace and determinant there are some publications available that you can trust. I can recommend this one for instance: https://www.researchgate.net/publication/242985986_Bounds_for_eigenvalues_using_the_trace_and_determinant

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