Decorrelation Manifold

Let the matrix $S_1$ be mixed by a matrix $Q^{-1}$. The reduced data can now be represented as

$$Y_R = Q Q^{-1} S_1 = Q S_2$$ (5)

where $S_2 = Q^{-1} S_1$ is a new set of spatial maps.

The correlation of these spatial maps is given by $S_2 S_2^T = Q^{-1} S_1 S_1^T Q^{-T} = Q^{-1} Q^{-T}$. Therefore, in order to keep the maps uncorrelated it is necessary to impose the condition that $Q^T Q = I$, which states that $Q$ must be an orthogonal matrix. The set of all such matrices $Q$ represents a set called a manifold, and since it maintains decorrelation it is known as the decorrelation manifold.