Woodbury matrix identity

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, specifically <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, the Woodbury matrix identity – named after <a href="/facts/Max_A._Woodbury/bcmjY73W">Max A. Woodbury</a> – says that the inverse of a rank-k correction of some <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> can be computed by doing a rank-k correction to the inverse of the original matrix. Alternative names for this formula are the matrix inversion lemma, Sherman–Morrison–Woodbury formula or just Woodbury formula. However, the identity appeared in several papers before the Woodbury report.
The Woodbury matrix identity is

(
            
              A
              +
              U
              C
              V
            
            )
          
          
            −
            1
          
        
        =
        
          A
          
            −
            1
          
        
        −
        
          A
          
            −
            1
          
        
        U
        
          
            (
            
              
                C
                
                  −
                  1
                
              
              +
              V
              
                A
                
                  −
                  1
                
              
              U
            
            )
          
          
            −
            1
          
        
        V
        
          A
          
            −
            1
          
        
        ,
      
    
    {\displaystyle \left(A+UCV\right)^{-1}=A^{-1}-A^{-1}U\left(C^{-1}+VA^{-1}U\right)^{-1}VA^{-1},}

where A, U, C and V are <a href="/facts/Conformable_matrix/fTT4EvG8">conformable matrices</a>: A is n×n, C is k×k, U is n×k, and V is k×n. This can be derived using <a href="/facts/Invertible_matrix/fPqXk3V8">blockwise matrix inversion</a>.
While the identity is primarily used on matrices, it holds in a general <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> or in an <a href="/facts/Ab-category/l308bN5A">Ab-category</a>.
The Woodbury matrix identity allows cheap computation of inverses and solutions to linear equations. However, little is known about the numerical stability of the formula. There are no published results concerning its error bounds. Anecdotal evidence suggests that it may diverge even for seemingly benign examples (when both the original and modified matrices are <a href="/facts/Well-conditioned/9Tjtk6wh">well-conditioned</a>).

Woodbury matrix identity open-in-new

Woodbury matrix identity