Matrix Identities

(Difference between revisions)
 Revision as of 01:47, 4 February 2013 (view source)Jmb (Talk | contribs) (→Other useful matrix properties)← Older edit Revision as of 12:55, 4 July 2013 (view source)Jmb (Talk | contribs) Newer edit → Line 30: Line 30: \end{bmatrix}. \end{bmatrix}. [/itex] [/itex] + = Sherman–Morrison–Woodbury = = Sherman–Morrison–Woodbury = Line 36: Line 37: \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}, \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}, [/itex] [/itex] + + = Other useful matrix properties = = Other useful matrix properties =

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

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