where , , and . This assumes that so that .

**For
**

implies that and . Consequently, using the SVD decompositions and gives . Therefore and so , so that the integral is given by:

(17) |

**For
,**

is taken to have full rank, but
is
not, as is not a square matrix, so that does not
exist. Instead,
where
by SVD decomposition. As has dimensions by (same
as ) then
is by and hence invertible (by virtue
of
being full rank). Therefore,
as desired, but
which is a projection matrix. Consequently,
is not zero, but is the residual projection
matix (onto the null space of
- the prewhitened version
of ). The integral can then be written as:

(18) |

where is the residual projection matrix in the coloured space, and is the residual projection matrix in the whitened space.

**For
,**

In this case the matrix will have many linearly dependent columns and cannot be achieved. Instead, let the number of independent columns be . Furthermore, let by SVD, where

(19) |

where the null parameters are integrated over , and , with