Constrained generalized inverse

In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, a constrained generalized inverse is obtained by solving a <a href="/facts/System_of_linear_equations/fk9CFdgp">system of linear equations</a> with an additional constraint that the solution is in a given subspace. One also says that the problem is described by a system of <a href="/facts/Linear_least_squares_(mathematics)/dKctFUCV">constrained linear equations</a>.
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 <a href="/facts/Linear_subspace/2wtKTgHL">linear subspace</a> 
 
 
 
 L
 
 
 {\displaystyle L}
 
 of 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
.
In the following, the <a href="/facts/Orthogonal_projection/sElXGkxD">orthogonal projection</a> 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.

Constrained generalized inverse open-in-new

Constrained generalized inverse