- Constrained generalized inverse
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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 x of a linear system of equations
is admissible only when it is in a certain linear subspace L of
.
In the following, the orthogonal projection on L will be denoted by PL. Constrained system of linear equations
has a solution if and only if the unconstrained system of equations
is solvable. If the subspace L is a proper subspace of
, then the matrix of the unconstrained problem (APL) 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 (APL) 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
if the inverse on the right-hand-side exists.
Categories:- Mathematics stubs
- Matrices
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