• Source: 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


{\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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