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

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

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

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

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