Constrained generalized inverse

A constrained generalized inverse 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 admissible 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 (AP_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 (AP_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.