Constrained generalized inverse
Encyclopedia
A constrained generalized inverse 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 admissible only when it is in a certain linear subspace
of 
.
In the following, the orthogonal projection 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.
In many practical problems, the solution
of a linear system of equationsis admissible only when it is in a certain linear subspace
Linear subspace
The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
of .In the following, the orthogonal projection 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 equationif the inverse on the right-hand-side exists.