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 ${\displaystyle x}$ of a linear system of equations

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

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

In the following, the orthogonal projection on ${\displaystyle L}$ will be denoted by ${\displaystyle P_{L}}$. Constrained system of linear equations

${\displaystyle Ax=b\qquad x\in L}$

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

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

is solvable. If the subspace ${\displaystyle L}$ is a proper subspace of ${\displaystyle \mathbb {R} ^{n}}$, then the matrix of the unconstrained problem ${\displaystyle (AP_{L})}$ may be singular even if the system matrix ${\displaystyle A}$ of the constrained problem is invertible (in that case, ${\displaystyle m=n}$). This means that one needs to use a generalized inverse for the solution of the constrained problem. So, a generalized inverse of ${\displaystyle (AP_{L})}$ is also called a ${\displaystyle L}$-constrained pseudoinverse of ${\displaystyle A}$.

An example of a pseudoinverse that can be used for the solution of a constrained problem is the Bott–Duffin inverse of ${\displaystyle A}$ constrained to ${\displaystyle L}$, which is defined by the equation

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

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