due to the invariance of the Lagrangian under the so-called Weak Basis Transformations (WBT) [WBT,quico]{}. The theory is invariant if we transform the fields as, $$q_{L}=R_{q}q_{L}^{\prime }\quad ;\quad u_{R}=R_{u}u_{R}^{\prime }\quad
;\quad d_{R}=R_{d}d_{R}^{\prime }$$where $R_{q}$, $R_{u}$ and $R_{d}$ are transformations from the global unitary groups $U(3)_{L}$, $U(3)_{u_{R}}$ and $U(3)_{d_{R}}$ respectively, while simultaneously the Yukawa couplings are transformed as, $$Y_{u}^{\prime }=R_{q}^{\dagger }Y_{u}R_{u}\qquad Y_{d}^{\prime
}=R_{q}^{\dagger }Y_{d}R_{d}  \label{yukawa:wbt}$$Therefore when we choose the different $R_{a}$ all we are doing is to choose a particular weak basis where we write our theory and the physical results are absolutely independent of this choice. However, it is very useful to choose the unitary transformation $R$ in the definition of $N=K^{-1/2}$ to get a simpler form for this transformation. The form that proves to be useful is the Cholesky decomposition of an Hermitian matrix. It is always possible to write an Hermitian matrix as $K=U^{\dagger }U$ in terms of an upper $U$ triangular matrix, $$K=\begin{pmatrix}
K_{11} & K_{12} & K_{13} \\ 
K_{12}^{\ast } & K_{22} & K_{23} \\ 
K_{13}^{\ast } & K_{23}^{\ast } & K_{33}\end{pmatrix}=U^{\dagger }U=\begin{pmatrix}
u_{11} & 0 & 0 \\ 
u_{12}^{\ast } & u_{22} & 0 \\ 
u_{13}^{\ast } & u_{23}^{\ast } & u_{33}\end{pmatrix}\begin{pmatrix}
u_{11} & u_{12} 