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 formula_1 of a linear system of equations
is acceptable only when it is in a certain linear subspace formula_3 of formula_4.
In the following, the orthogonal projection on formula_3 will be denoted by formula_6.
Constrained system of linear equations
has a solution if and only if the unconstrained system of equations 
is solvable. If the subspace formula_3 is a proper subspace of formula_4, then the matrix of the unconstrained problem formula_11 may be singular even if the system matrix formula_12 of the constrained problem is invertible (in that case, formula_13). This means that one needs to use a generalized inverse for the solution of the constrained problem. So, a generalized inverse of formula_11 is also called a formula_3-"constrained pseudoinverse" of formula_12.
An example of a pseudoinverse that can be used for the solution of a constrained problem is the Bott–Duffin inverse of formula_12 constrained to formula_3, which is defined by the equation
if the inverse on the right-hand-side exists.