If $M$ is a symmetric positive-definite 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 $M$? E.g., I want $$\lambda_{\text{min}} \geq f(\lVert M \rVert)$$ or something like that. $M$ is a Gram matrix, if that helps.
[Math] Lower bound on smallest eigenvalue of (symmetric positive-definite) matrix
eigenvalues-eigenvectorslinear algebramatricespositive definitesymmetric matrices
Best Answer
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 Frobenius 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.
[reference]: K. H. Schindler, "A New Lower Bound for the Minimal Singular Value for Real Non-Singular Matrices by a Matrix Norm and Determinant", Journal of Applied Mathematical Sciences, Vol. 4, No. 64, 2010.