A query about the reciprocals of condition numbers of nonsingular matrices The condition number of a nonsingular matrix $A$ is defined as
$$
\kappa(A) = \mbox{cond}(A) = \Vert A \Vert  \Vert  A^{-1} \Vert  \tag{1}
$$
and its value is dependent on the choice of the norm in (1).
However, as all $p$-norms are equivalent in a finite-dimensional space, if a matrix $A$ is ill-conditioned in any $p$-norm, it will continue to be ill-conditioned in other $p$-norms also, where $p = 1, 2, \ldots..$ In fact, I also saw some results giving equivalence of $\Vert \cdot  \Vert_1$ and $\Vert \cdot \Vert_\infty$ with some bounds.
My query is on numerical linear algebra, where I saw some results giving an estimate for
$$
{1 \over \kappa(A)}
$$
rather than $\kappa(A)$.
As we use ill-conditioned matrices (like Hilbert's matrix) for test matrices, $\kappa(A)$ will be quite large. Is my reasoning correct that numerical estimates are given for ${1 \over \kappa(A)}$ rather than $\kappa(A)$ as ${1 \over \kappa(A)}$ (reciprocals) will be closer to zero for an ill-conditioned matrix $A$, while $\kappa(A)$ can get quite large. This helps the accuracy of the numerical estimate.
I checked the available commands in MATLAB and I find that MATLAB uses "rcond(A)" for the reciprocal of condition number of a matrix $A$ using $1$-norm. As MATLAB does not give the paper used for finding this estimate, I like to know why specially $1$-norm is used in this estimate, as the default value of "cond(A)" in MATLAB uses the spectral norm or the $2$-norm. [In MATLAB, if the value of "rcond(A)" is closer to zero, it indicates that $A$ is very close to a singular matrix, while if the value of "rcond(A)" is closer to one, it indicates that $A$ is well-conditioned.]
Your comments are welcome!
 A: William W. Hager (1984) developed an algorithm for 1-norm condition number estimator, and his algorithm essentially gives an estimate for $\Vert A^{-1} \Vert_1$ from which $\kappa_1(A)$ can be directly computed. This is a direct method for finding $\kappa_1(A)$ and it does not proceed like finding the reciprocal of $\kappa_1(A)$.
References:

*

*Hager WW (1984) Condition estimates. SIAM J Sci Stat Comput 5(2):311–316


*Hager WW (1998) Applied numerical linear algebra. Prentice Hall, New Jersey
A: If $y$ is any non-zero vector in $\mathbb{R}^n$, then
$$
A z = y
$$
yields
$$
z = A^{-1} \ y
$$
which shows that
$$
\Vert z \Vert = \Vert A^{-1} \ y \Vert \leq \Vert A^{-1} \Vert \Vert y \Vert
$$
Thus,
$$
\Vert A^{-1} \Vert \geq {\Vert z \Vert \over \Vert y \Vert}, \ \ \mbox{where} \ \ y \neq 0. \tag{1}
$$
For an ill-conditioned matrix $A$, which is usually taken as a test matrix for estimating $\kappa(A)$, $\Vert A^{-1} \Vert$ will be very large.
If we choose a $y$ such that
$$
{\Vert z \Vert \over \Vert y \Vert} \tag{2}
$$
is large, than we may get a reasonably good estimate for
$$
\Vert A^{-1} \Vert
$$
(using the bound given in (1)).
This is the basic idea for the condition number estimatator given in the classical paper:
"An Estimate for the Condition Number of a Matrix" (1979)
A. K. Cline, C. B. Moler,  G. W. Stewart and J. H. Wilkinson
(SIAM Journal on Numerical Analysis)
To avoid overflow, the LINPACK condition number estimator computes an estimate for the reciprocal of $\kappa(A)$, namely
$$
RCOND = {\Vert y \Vert \over \Vert A \Vert \Vert z \Vert}
$$
Cline's paper in SIAM J. Numerical Analysis (1979)
