.. _condSVD:
.. _condition-number:

Condition Number
================

.. index:: rcond

As described in :ref:`conditionNum`, the problem with a poorly
conditioned matrix equation is sensitive to perturbations of the
elements of :math:`\bm{b}`. Viewing the solution to
:math:`\mathbf{A}\,\bm{x} = \bm{b}` from the perspective of the outer
product of the SVD gives us an intuition into the sensitivity of
:math:`\bm{x}` to perturbations in :math:`\bm{b}` [GOLUB13]_.

.. math::

   \begin{aligned}
   \bm{x} &= \mathbf{A}^{+}\,\bm{b}
               = \left(\mathbf{V\,\Sigma}^+\,\mathbf{U}^T\right)\, \bm{b} \\
              &= \sum_{i=1}^{n} \frac{\bm{u}_i^T \bm{b}}{\sigma_i}\bm{v}_i
   \end{aligned}

The scalar fractions :math:`\left(\bm{u}_i^T \bm{b}\right)/\sigma_i` are
dot products divided by singular values. Thus, the magnitude of the
singular values has a significant impact on the sensitivity of the
:math:`\mathbf{A}\,\bm{x} = \bm{b}` problem. 
A matrix with singular values close to zero is poorly conditioned.

The condition number is defined by equation :eq:`eq-conNum` in
:ref:`conditionNum` as a product of matrix norms.

.. math::

   \kappa(\mathbf{A}) = \text{cond}(\mathbf{A}) = 
   \norm{\mathbf{A}}_2\norm{\mathbf{A}^{-1}}_2

From :ref:`mat2norm`, we learn that the :math:`\norm{\cdot}_2` norm
focuses on the ability of a matrix to stretch a vector, and is thus the
largest singular value of the matrix. The singular values of a matrix’s
inverse are the reciprocals of the matrix’s singular values. Refer to
equation :eq:`eq-eigInv` in :ref:`eigProperties`. So the
:math:`\norm{\cdot}_2` condition number of a matrix is the ratio of the
largest and smallest singular values.

.. math:: \kappa(\mathbf{A}) = \frac{\sigma_{max}}{\sigma_{min}}

A full rank matrix will have a fairly small condition number. Singular
and near-singular matrices will have condition numbers of infinity or
very large (several thousand). Thus, the condition number is an
invertible test. To avoid division by zero, MATLAB uses the reciprocal
of the condition number, ``rcond``. If :math:`\bf{A}` is
well-conditioned, ``rcond(A)`` is near 1, and near 0 if :math:`\bf{A}`
is poorly conditioned.

.. index:: condition number
.. index:: invertible test