I am studying canonical correlation analysis. And I'm completely stumped for the last few days at the following manipulation.
How does the following change of basis works?
The equation doesn't even look like a linear equation. And I'm not even seeing the inverse of the basis matrix.
Original article here
The notation $\Sigma^{1/2}$ means the square root of the matrix $\Sigma$, which exists because your covariance matrix $\Sigma$ is symmetric positive definite, and thus $\Sigma = MDM^{-1}$ where $D$ is a diagonal matrix with all positive diagonal entries, so then $\Sigma^{1/2}$ is obtained by replacing each diagonal entry in $D$ with the square root of the entry. Since $\Sigma^{1/2}$ is a square matrix, the equations for $c$ and $d$ are indeed linear transformations. Then, in the last expression, the matrix $\Sigma^{-1/2}$ is the inverse of $\Sigma^{1/2}$. So you can verify e.g. that $\Sigma_{XX}^{-1/2}c = a$ and so the last expression is the same as the original expression in terms of $a$ and $b$.