# Lower bound on the smallest eigenvalue

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.

• You should consider your textbooks for the definition of the Frobenius and of the Euclidean 2-norm... – daw Apr 3 '14 at 6:49
• Is there a similar bound for non-symmetric matrices? – Amir Sagiv Mar 6 '18 at 11:19