Matrix Identities

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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},


in particular

 
\left(\mathbf{R}+\mathbf{H} \mathbf{P} \mathbf{H}^T \right)^{-1} = \mathbf{R}^{-1} - \mathbf{R}^{-1}\mathbf{H} \left(\mathbf{P}^{-1}+\mathbf{H}^T \mathbf{R}^{-1}\mathbf{H} \right)^{-1} \mathbf{H}^T \mathbf{R}^{-1},


or

 
\mathbf{A} 
\left(\mathbf{I}+\mathbf{A}^T \mathbf{A} \right)^{-1} =
\left(\mathbf{I}+\mathbf{A} \mathbf{A}^T \right)^{-1} 
\mathbf{A}

Other useful matrix properties

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