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Constrained generalized inverse

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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 {\displaystyle x} of a linear system of equations

A x = b ( with given  A ∈ R m × n  and  b ∈ R m ) {\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 L {\displaystyle L} of R n {\displaystyle \mathbb {R} ^{n}} .

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

A x = b x ∈ L {\displaystyle Ax=b\qquad x\in L}

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

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

is solvable. If the subspace L {\displaystyle L} is a proper subspace of R n {\displaystyle \mathbb {R} ^{n}} , then the matrix of the unconstrained problem ( A P L ) {\displaystyle (AP_{L})} may be singular even if the system matrix A {\displaystyle A} of the constrained problem is invertible (in that case, m = n {\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 ( A P L ) {\displaystyle (AP_{L})} is also called a L {\displaystyle L} -constrained pseudoinverse of A {\displaystyle A} .

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

A L ( − 1 ) := P L ( A P L + P L ⊥ ) − 1 , {\displaystyle A_{L}^{(-1)}:=P_{L}(AP_{L}+P_{L^{\perp }})^{-1},}

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


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