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 $$x$$ of a linear system of equations

Ax=b\qquad (\text{with given }A\in\R^{m\times n}\text{ and } b\in\R^m) $$ is acceptable only when it is in a certain linear subspace $$L$$ of $$\R^m$$.

In the following, the orthogonal projection on $$L$$ will be denoted by $$P_L$$. Constrained system of linear equations
 * $$Ax=b\qquad x\in L$$

has a solution if and only if the unconstrained system of equations
 * $$(A P_L) x = b\qquad x\in\R^m$$

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

An example of a pseudoinverse that can be used for the solution of a constrained problem is the Bott–Duffin inverse of $$A$$ constrained to $$L$$, which is defined by the equation
 * $$A_L^{(-1)}:=P_L(A P_L + P_{L^\perp})^{-1},$$

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