Prove that an alternate computing formula for \(\mathrm{dCor}_{n}^{2}\) (Definitions \(12.5-12.7)\) is

\[
\mathrm{dCor}_{n}^{2}(X, Y)=\frac{\left\langle C \Delta_{X} C, C \Delta_{Y} Cightangle}{\left\|C \Delta_{X} Cight\|\left\|C \Delta_{Y} Cight\|}
\]

where \(\Delta_{X}, \Delta_{Y}\) are the Euclidean distance matrices of the \(X\) and \(Y\) samples, and \(C=\mathbf{I}_{n}-\frac{1}{n} \mathbf{1 1} \mathbf{1}^{\top}\). See Example 22.3.