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If $M$ is a positive-definite symmetric matrix, is it possible to get a positive lower bound on the smallest eigenvalue of $M$ in terms of a matrix norm of $M$ or elements of $M$? Eg. I want $$\lambda_{\text{min}} \geq f(\lVert M \rVert)$$ or something like that.

$M$ is a Gram matrix if that helps. Thanks.

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Hi @ampeo, have you found the answer? I have the same question and all the bounds I can get are negative, which doesn't make any sense for a PD matrix. –  D. Chen Nov 7 '13 at 7:07
    
There is an obvious bound in terms of the operator norm of $M^{-1}$, of course. –  user7530 Dec 21 '13 at 16:52
    
As user7530 wrote, $\lambda_{min}$ depends essentially on $M^{-1}$. Moreover to find an inequality in the form $\lambda_{min}\geq f(||M||)$ is beyond all hope. For instance let $A_{\epsilon}=diag(1,\epsilon)$. We should obtain, for every $\epsilon >0$, $\epsilon\geq f(1)$ (for $||.||_2$). –  loup blanc Jan 24 at 16:59
    
Lower bounds on the smallest eigenvalue of a positive definite matrix are related to estimates of the condition number, for which see my Answer to a SciComp.SE Question. –  hardmath Jan 27 at 2:36
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2 Answers 2

There is one lower bound on minimum eigenvalue of symmetric p.d. matrix given in [Applied Math. Sc., vol. 4, no, 64] which is based on Forbenius norm (F) and Euclidean norm (E)

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

if it helps.

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Rather than reposting (with the same "typos" as best I can tell), you could have edited this, your original post. See "edit" link just under the Answer. I took a guess at the mismatched parenthesis in the denominator; please check it is correct. –  hardmath Jan 27 at 2:40
    
It is all correct, Thanks! –  Saeed Manaffam Jan 27 at 6:16
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A quick comment: If you have diagonal dominance, then Gerhsgorin's circle theorem for eigenvalues will get you at least something. So for each row, subtract the diagonal term from the sum of the absolute values of the off-diagonal terms, and take the minimum over the rows. That is a bound on the eigenvalue that will be positive (again, if you have diagonal dominance, which may not hold for all Gram matrices).

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