Matrix inversion formulae

Throughout the thesis we are using the matrix inversion lemma: having quadratic matrices A and B and a matrix X of corresponding number of columns and rows, we have the following equality (see for example [43, Appendix B] or [64])

$$\left(\vphantom{ {\boldsymbol { A } }+{\boldsymbol { X } }{\boldsymbol { B } }{\boldsymbol { X } }^T}\right. \left.\vphantom{ {\boldsymbol { A } }+{\boldsymbol { X } }{\boldsymbol { B } }{\boldsymbol { X } }^T}\right)^{-1}_{}$$ (180)

The matrix inversion formula for a symmetric matrix with block sub-matrices is:

$$\begin{bmatrix} {\boldsymbol { A } }& {\boldsymbol { B } }\\ {\boldsymbol { B } }^T &{\boldsymbol { C } } \end{bmatrix}^{-1}_{} \begin{bmatrix} {\boldsymbol { D } }^{-1} & -{\boldsymbol { A } }... ...ol { B } }^T{\boldsymbol { A } }^{-1} & {\boldsymbol { E } }^{-1} \end{bmatrix}$$ = (181)

$$\begin{bmatrix} {\boldsymbol { A } }^{-1} + {\boldsymbol { A } }^... ...dsymbol { D } }^{-1}{\boldsymbol { B } }{\boldsymbol { C } }^{-1} \end{bmatrix}$$ = (182)

where    D = A - BC^-1B^T

E = C - B^TA^-1B (183)

The formulae for the determinants of block matrices are also used in the thesis:

$$\begin{vmatrix}{\boldsymbol { A } }& {\boldsymbol { B } }\\ {\boldsymbol { B } }^T &{\boldsymbol { C } } \end{vmatrix}$$ (184)

A useful short guide to manipulating block matrices and computing determinants is provided by Sam Roweis, available at http://www.gatsby.ucl.ac.uk/~roweis/notes.html.