Constrained generalized inverse

In linear algebra, a constrained generalized inverse is obtained by solving a system of linear equations with an additional constraint that the solution is in a given subspace. One also says that the problem is described by a system of constrained linear equations.

In many practical problems, the solution [math]\displaystyle{ x }[/math] of a linear system of equations

[math]\displaystyle{ Ax=b\qquad (\text{with given }A\in\R^{m\times n}\text{ and } b\in\R^m) }[/math]

is acceptable only when it is in a certain linear subspace [math]\displaystyle{ L }[/math] of [math]\displaystyle{ \R^m }[/math].

In the following, the orthogonal projection on [math]\displaystyle{ L }[/math] will be denoted by [math]\displaystyle{ P_L }[/math]. Constrained system of linear equations

[math]\displaystyle{ Ax=b\qquad x\in L }[/math]

has a solution if and only if the unconstrained system of equations

[math]\displaystyle{ (A P_L) x = b\qquad x\in\R^m }[/math]

is solvable. If the subspace [math]\displaystyle{ L }[/math] is a proper subspace of [math]\displaystyle{ \R^m }[/math], then the matrix of the unconstrained problem [math]\displaystyle{ (A P_L) }[/math] may be singular even if the system matrix [math]\displaystyle{ A }[/math] of the constrained problem is invertible (in that case, [math]\displaystyle{ m=n }[/math]). This means that one needs to use a generalized inverse for the solution of the constrained problem. So, a generalized inverse of [math]\displaystyle{ (A P_L) }[/math] is also called a [math]\displaystyle{ L }[/math]-constrained pseudoinverse of [math]\displaystyle{ A }[/math].

An example of a pseudoinverse that can be used for the solution of a constrained problem is the Bott–Duffin inverse of [math]\displaystyle{ A }[/math] constrained to [math]\displaystyle{ L }[/math], which is defined by the equation

[math]\displaystyle{ A_L^{(-1)}:=P_L(A P_L + P_{L^\perp})^{-1}, }[/math]

if the inverse on the right-hand-side exists.