Constrained generalized inverse

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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.