# ICA and orthogonality of Independent Components

In the book by Aapo Hyvärinen, it is shown that:

Where z is the white vector of a data matrix x, s are the IC's and Ã is the mixing matrix of the whitened data matrix z. My question is: If the matrix Ã is orthonormal as show above, does it implies that s is orthogonal as $$\\E(ss^T)$$ has to be $$\\I$$? If so should the IC's in this case be orthogonal?

$$E[ss^T]$$ is $$I$$ via the assumptions of ICA, i.e. $$s_i$$ are statistically independent basis signals. $$E[zz^T]$$ is also $$I$$ since we know that it is white. Knowing the two, we can say that $$\tilde A$$ is orthonormal. Conversely, if we know that $$\tilde A$$ is orthonormal, we can multiply the equation with $$\tilde A^T$$ and $$\tilde A$$ from left and right respectively and conclude that $$E[ss^T]=I$$: $$\tilde AE[ss^T]\tilde A^T=I\rightarrow \underbrace{\tilde A^T\tilde A}_IE[ss^T]\underbrace{\tilde A^T\tilde A}_I=\underbrace{\tilde A^T\tilde A}_I\rightarrow E[ss^T]=I$$