# Matrix Identities

(Difference between revisions)
 Revision as of 09:03, 22 May 2014 (view source)Alex (Talk | contribs) (→Sherman–Morrison–Woodbury)← Older edit Latest revision as of 11:31, 8 August 2018 (view source)Alex (Talk | contribs) (→Sherman–Morrison–Woodbury) Line 45: Line 45: [/itex] [/itex] + + + $+ \mathbf{P} \mathbf{H}^T + \left(\mathbf{R}+\mathbf{H} \mathbf{P} \mathbf{H}^T \right)^{-1} = + \mathbf{P} \mathbf{H}^T + \mathbf{R}^{-1} - \mathbf{P} \mathbf{H}^T \mathbf{R}^{-1}\mathbf{H} \left(\mathbf{P}^{-1}+\mathbf{H}^T \mathbf{R}^{-1}\mathbf{H} \right)^{-1} \mathbf{H}^T \mathbf{R}^{-1}, +$ + + + $+ = \mathbf{P} \mathbf{H}^T + \mathbf{R}^{-1} - \mathbf{P} + ( \mathbf{H}^T \mathbf{R}^{-1}\mathbf{H} + \mathbf{P}^{-1} - \mathbf{P}^{-1}) + \left(\mathbf{P}^{-1}+\mathbf{H}^T \mathbf{R}^{-1}\mathbf{H} \right)^{-1} \mathbf{H}^T \mathbf{R}^{-1}, +$ + + + $+ =\left(\mathbf{P}^{-1}+\mathbf{H}^T \mathbf{R}^{-1}\mathbf{H} \right)^{-1} \mathbf{H}^T \mathbf{R}^{-1}, +$ or or

# 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

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

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

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

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

or

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