  The problem of extracting a well conditioned submatrix from any rectangular
matrix (with normalized columns) has been studied for some time in functional
and harmonic analysis; see
\cite{BourgainTzafriri:IJM87,Tropp:StudiaMath08,Vershynin:IJM01} for methods
using random column selection. More constructive approaches have been proposed
recently; see the recent contributions of
\cite{SpielmanSrivastava:IJM12,Youssef:IMRN14}. The column selection problem we
consider in this paper is concerned with extracting a well conditioned
submatrix, i.e. a matrix whose singular values all lie in
$[1-\epsilon,1+\epsilon]$. We provide individual lower and upper bounds for
each singular value of the extracted matrix at the price of conceding only one
log factor in the number of columns, when compared to the Restricted
Invertibility Theorem of Bourgain and Tzafriri. Our method is fully
constructive and the proof is short and elementary.
