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

$Ax=b\qquad ({\text{with given }}A\in \mathbb {R} ^{m\times n}{\text{ and }}b\in \mathbb {R} ^{m})$

is acceptable only when it is in a certain linear subspace $L$ of $\mathbb {R} ^{n}$.

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

$(AP_{L})x=b\qquad x\in \mathbb {R} ^{n}$

is solvable. If the subspace $L$ is a proper subspace of $\mathbb {R} ^{n}$, 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

$A_{L}^{(-1)}:=P_{L}(AP_{L}+P_{L^{\perp }})^{-1},$

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