Overview
When the matrix {% A %} is a square, nosingular matrix, the equation
{% A\vec{x} = \vec{y} %}
has a solution
{% \vec{x} = A^{-1}y %}
where {% A^{-1} %} is the
matrix inverse
of {% A %}.
When {% A %} is an {% n \times m %} matrix, this equation still has a solution if {% \vec{y} \in Span(\vec{x}) %}. IN that case, there is at least one matrix, labeled {% A^{-} %} for which the following equation holds.
{% \vec{x} = A^{-}\vec{y} %}
If {% Kernel(A) \neq \{0 \} %} then
for any {% \vec{x}_0 \in Kernel(A) %} we have
{% A(\vec{x} + \vec{x}_0) = y %}
that, is {% \vec{x}_0 %} can be added to any solution of the equation, and a new solution is obtained.
Properties of the Generalized Inverse
The generalized inverse, as defined above, satifies the following property.
{% AA^{-}A = A %}
This property is often taken as the definition of a generalized inverse.