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### Case 3:

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

where is an by diagonal matrix with all diagonal entries being non-zero. Integration of the parameters associated with the zero singular values can be carried out if they have finite extents. Letting , and gives
 (19)

where the null parameters are integrated over , and , with

which is an by matrix.

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