Condition numbers are a central concept in numerical analysis.
They provide a natural parameter for studying the behavior of
algorithms, as well as sensitivity and geometric properties of a problem.
The condition number of a problem instance is usually a measure
of the distance to the set of ill-posed instances. For instance, the
classical Eckart and Young identity characterizes the condition
number of a square matrix as the reciprocal of its relative distance
to singularity.
\\
\\
We present concepts of conditioning for optimization problems and
for more general variational problems. We show that the Eckart and
Young identity has natural extension to much wider contexts. We also
discuss conditioning under the presence of block-structure, such as
that determined by a sparsity pattern. The latter has interesting
connections with the mu-number in robust control and with the sign-real
spectral radius.