Matrix Identities

From GHER

(Difference between revisions)
Jump to: navigation, search
Line 34: Line 34:
<math>  
<math>  
-
\left(A+UCV \right)^{-1} = A^{-1} - A^{-1}U \left(C^{-1}+VA^{-1}U \right)^{-1} VA^{-1},  
+
\left(\mathbf{A}+\mathbf{U} \mathbf{C} \mathbf{V} \right)^{-1} = \mathbf{A}^{-1} - \mathbf{A}^{-1}\mathbf{U} \left(\mathbf{C}^{-1}+\mathbf{V}\mathbf{A}^{-1}\mathbf{U} \right)^{-1} \mathbf{V}\mathbf{A}^{-1},  
</math>
</math>
 +
 +
= Other useful matrix properties =

Revision as of 01:42, 4 February 2013

Inverse of block matrix


\begin{bmatrix} 
\mathbf{A} & \mathbf{B} \\ 
\mathbf{C} & \mathbf{D} 
\end{bmatrix}^{-1} 
= 
\begin{bmatrix} 
\mathbf{A}^{-1}+\mathbf{A}^{-1}\mathbf{B}(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\mathbf{CA}^{-1} & 
-\mathbf{A}^{-1}\mathbf{B}(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1} \\ 
-(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\mathbf{CA}^{-1} & 
(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1} 
\end{bmatrix}


\begin{bmatrix} 
\mathbf{A} & 
\mathbf{B} \\ 
\mathbf{C} & 
\mathbf{D} 
\end{bmatrix}^{-1} 
= 
\begin{bmatrix} 
(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1} & 
-(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1}\mathbf{BD}^{-1} \\ 
-\mathbf{D}^{-1}\mathbf{C}(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1} & 
\mathbf{D}^{-1}+\mathbf{D}^{-1}\mathbf{C}(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1}\mathbf{BD}^{-1}
\end{bmatrix}.

Sherman–Morrison–Woodbury

 
\left(\mathbf{A}+\mathbf{U} \mathbf{C} \mathbf{V} \right)^{-1} = \mathbf{A}^{-1} - \mathbf{A}^{-1}\mathbf{U} \left(\mathbf{C}^{-1}+\mathbf{V}\mathbf{A}^{-1}\mathbf{U} \right)^{-1} \mathbf{V}\mathbf{A}^{-1},

Other useful matrix properties

Personal tools