The general form of the ShermanMorrisonWoodbury formula [8] is
Also, the inverse for a matrix in block form is given by
Theorem A:
Any model of the form
That is, the signals of interest can be made orthogonal to the confounds without affecting the estimation of the parameters or the residuals.
Proof:
The proof is by construction, where we show that orthogonalising with respect to gives the desired results. Let
These equations give . Also, the combined span of and is clearly the same as that of and .
Now consider the covariances
For the first model, the parameter estimates, given by
equation 5, can be written using the matrix block inversion
formula, giving
Applying the ShermanMorrisonWoodbury formula to the second term
in equation 12 gives
Substituting this into equation 12 gives

Theorem B:
Given the standard GLM, , and a set of linearly independent contrasts specified by such that , then an equivalent model without contrasts, but with confounds, exists in the form
Proof:
The proof is, again, by construction. Firstly, let be a set of contrasts that when combined with form a complete linearly independent set of contrasts. That is, the matrix will be full rank (and hence invertible). Then let
From these definitions it is easy to see that , which represents an orthogonality condition. As before, it is straightforward to verify that the combined span of and is equal to the span of . Consequently,
The estimation equations for the model become
Thus
