Constrained generalized inverse explained

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 equationsAx = b ( withgiven A\in\R ^{m x} and b\in\R ^{m)} is acceptable only when it is in a certain linear subspace L of \R ^{n} .

In the following, the orthogonal projection on

L will be denoted by P _{L} .Constrained system of linear equationsAx = b x\in L has a solution if and only if the unconstrained system of equations (A P _{L)} x = b x\in\R ^{n} is solvable. If the subspace L is a proper subspace of \R ^{n} , 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 equationA ^{( -1 )} _{L} : = P _{L(A} P _{L} + P _{L ^{\perp}} ) ^{-1} , if the inverse on the right-hand-side exists.