Constrained Generalized Inverse

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...

, a constrained generalized inverse is obtained by solving a

system of linear equations In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variable (math), variables. For example, : \begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of th ...

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 (\textA\in\R^\text b\in\R^m)

is acceptable only when it is in a certain

linear subspace In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a ''function (mathematics), function'' (or ''mapping (mathematics), mapping''); * linearity of a ''polynomial''. An example of a li ...

L

\R^n

. In the following, the

orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it we ...

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 :

(A P_L) x = b\qquad x\in\R^n

is solvable. If the subspace

L

is a proper subspace of

\R^n

, then the matrix of the unconstrained problem

(A P_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

(A P_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^:=P_L(A P_L + P_)^,

if the inverse on the right-hand-side exists. Matrices (mathematics) {{matrix-stub