# Making two vectors uncorrelated in terms of Kendall Tau correlation

Assume that we have two normalized $n\times 1$ vectors $\bf x$ and $\bf y$. In terms of Pearson correlation, these two signals are uncorrelated if ${\bf x}^T {\bf y} = 0$. Now, assume that ${\bf x}^T {\bf y} \neq 0$. If we perform the following projection: $$\widehat{\bf x} = {\bf x} - ({\bf x}^T {\bf y}) {\bf y},$$ we get $\widehat{\bf x}^T {\bf y} = 0$.

In all the above computations, we use Pearson correlation.

My questions is if we can perform such projection for Kendall Tau correlation. In other words, if we have two normalized $n\times 1$ vectors $\bf x$ and $\bf y$ such that their Kendall Tau correlation is not zero, can we perform a projection of $\bf x$ to get $\widehat{\bf x}$ such that the Kendall Tau correlation of $\widehat{\bf x}$ and $\bf y$ is zero?

Specifically, if $\hat{\beta}^K$ is the slope estimate for the Theil-Sen regression of $y$ on $x$, then the Kendall correlation of $x$ and $r=y-\hat{\beta}^K x$ will be 0.