In
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 matrices ...
, 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
of a linear system of equations
:
is acceptable only when it is in a certain
linear subspace of
.
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 wer ...
on
will be denoted by
.
Constrained system of linear equations
:
has a solution if and only if the unconstrained system of equations
:
is solvable. If the subspace
is a proper subspace of
, then the matrix of the unconstrained problem
may be singular even if the system matrix
of the constrained problem is invertible (in that case,
). This means that one needs to use a generalized inverse for the solution of the constrained problem. So, a generalized inverse of
is also called a
-''constrained pseudoinverse'' of
.
An example of a pseudoinverse that can be used for the solution of a constrained problem is the Bott–Duffin inverse of
constrained to
, which is defined by the equation
:
if the inverse on the right-hand-side exists.
Matrices
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