TITLE: smallest singular value
QUESTION [3 upvotes]: I know this question is a difficult one, but any advice/tip/reference/heuristic is welcome. Is there any good lower bound (other than $0$) on the smallest singular value of a matrix? It is easy to get an upper bound, but any ideas for a lower bound?
An answer for the smallest eigenvalue of a positive matrix is also appreciated.

REPLY [3 votes]: Suppose you are able to obtain a bound on the largest SV of a matrix, called $A$.  Then one can compute an $LU$ factorization of $A$, with pivots, and compute $A^{-1}$ with these factors.  The smallest SV of $A$ is the reciprocal of the largest one for $A^{-1}$.  This
works for square matrices.  For rectangular matrices, note that you can preserve the singular values by multiplication by an orthogonal matrix that has a square matrix factor with the same singular values as $A$.
During the $LU$ factorization you may discover rank deficiency.  But then the smallest SV
of $A$ is zero.
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