Matrix Identities

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= Other useful matrix properties =
= Other useful matrix properties =
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[http://orion.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf Matrix cookbook]

Revision as of 01:47, 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

Matrix cookbook

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