Geodesic
Estimates for the Eigenvalues of Matrices
Matematičeskie zametki, Tome 79 (2006) no. 2, pp. 169-177

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For matrices whose eigenvalues are real (such as Hermitian or real symmetric matrices), we derive simple explicit estimates for the maximal $(\lambda_{\max})$ and the minimal $(\lambda_{\min})$ eigenvalues in terms of determinants of order less than 3. For $3\times3$ matrices, we derive sharper estimates, which use $\det A$ but do not require to solve cubic equations. 
@article{MZM_2006_79_2_a1,
     author = {R. I. Alidema and A. F. Filippov},
     title = {Estimates for the {Eigenvalues} of {Matrices}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {169--177},
     publisher = {mathdoc},
     volume = {79},
     number = {2},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_79_2_a1/}
}
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R. I. Alidema; A. F. Filippov. Estimates for the Eigenvalues of Matrices. Matematičeskie zametki, Tome 79 (2006) no. 2, pp. 169-177. http://geodesic.mathdoc.fr/item/MZM_2006_79_2_a1/