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 $\mathbb{R} ^{m}$ .

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

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

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

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