Woodbury matrix identity

woodbury matrix identity

 

2014/6/20

 

【转载请注明出处】http://www.cnblogs.com/mashiqi

 

 

http://en.wikipedia.org/wiki/Woodbury_matrix_identity

Today I'm going to write down a proof of this Woodbury matrix identity, which is very important in some practical situation. For instance, the 40^th equation of this paper" bayesian compressive sensing using Laplace priors" applied this identity. Now let me give the details of it.

 

The Woodbury matrix identity is:

 

${(A + UCV)^{ - 1}} = {A^{ - 1}} - {A^{ - 1}}U{({C^{ - 1}} + V{A^{ - 1}}U)^{ - 1}}V{A^{ - 1}}$

where , , and are both assumed reversible.

 

Proof:

We denote with , namely .So:

\[M{A^{ - 1}} = I + UCV{A^{ - 1}}\]       

By multiply U with both side we get:

 

\[\begin{array}{l}
M{A^{ - 1}}U = U + UCV{A^{ - 1}}U = U(I + CV{A^{ - 1}}U)\\
{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = UC({C^{ - 1}} + V{A^{ - 1}}U)
\end{array}\]

is reversible, we get:

        

But how could we deal with this nasty term? We should notice that this term, which may not square, is coming from itself, which is right a square and reversible matrix. So, from formula , we make up a pleasant with is nasty :

 

\[\begin{array}{l}
M{A^{ - 1}}U{({C^{ - 1}} + V{A^{ - 1}}U)^{ - 1}}V + A = UCV + A = M\\
\Rightarrow M = M{A^{ - 1}}U{({C^{ - 1}} + V{A^{ - 1}}U)^{ - 1}}V + A\\
\Rightarrow I - {A^{ - 1}}U{({C^{ - 1}} + V{A^{ - 1}}U)^{ - 1}}V = {M^{ - 1}}A
\end{array}\]     

And finally due to the reversibility of , we get the Woodbury matrix identity:

\[{M^{ - 1}} = {(A + VCU)^{ - 1}} = {A^{ - 1}} - {A^{ - 1}}U{({C^{ - 1}} + V{A^{ - 1}}U)^{ - 1}}V{A^{ - 1}}\]        

Done.

 

We should notice that if and are identity matrix, then Woodbury matrix identity can be reduced to this form:

\[{(A + C)^{ - 1}} = {A^{ - 1}} - {A^{ - 1}}{({C^{ - 1}} + {A^{ - 1}})^{ - 1}}{A^{ - 1}}\]        

,which is equivalent to:

\[{(A + C)^{ - 1}} = {C^{ - 1}}{({C^{ - 1}} + {A^{ - 1}})^{ - 1}}{A^{ - 1}}\]        

This is because:

 

\[\begin{array}{l}
{(A + C)^{ - 1}} = {A^{ - 1}} - ( - {C^{ - 1}} + {C^{ - 1}} + {A^{ - 1}}){({C^{ - 1}} + {A^{ - 1}})^{ - 1}}{A^{ - 1}}\\
{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = {A^{ - 1}} + {C^{ - 1}}{({C^{ - 1}} + {A^{ - 1}})^{ - 1}}{A^{ - 1}} - ({C^{ - 1}} + {A^{ - 1}}){({C^{ - 1}} + {A^{ - 1}})^{ - 1}}{A^{ - 1}}\\
{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = {A^{ - 1}} + {C^{ - 1}}{({C^{ - 1}} + {A^{ - 1}})^{ - 1}}{A^{ - 1}} - {A^{ - 1}}\\
{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = {C^{ - 1}}{({C^{ - 1}} + {A^{ - 1}})^{ - 1}}{A^{ - 1}}
\end{array}\]