 Solution

The Yukawa matrices for up-type and down-type quarks are diagonalized by separate unitary changes of basis. Let $u_L^{\mathrm{weak}}=U_u{u}_L^{\mathrm{mass}}$ and $d_L^{\mathrm{weak}}=U_d{d}_L^{\mathrm{mass}}$. The generation-diagonal weak charged current in the weak basis becomesThus the CKM matrix records the mismatch between the two left-handed mass bases, not an additional non-unitary interaction.

An $N\times N$ unitary matrix has $N^2$ real parameters. The $2N$ quark-field phases can remove $2N-1$ phases from $V$: the common phase changes neither current. This leaves $(N-1{)}^2$ physical real parameters, comprising $N(N-1)/2$ mixing angles and $(N-1)(N-2)/2$ CP-violating phases. Hence three generations give four parameters: three angles and one phase; four generations give nine: six angles and three phases. This count assumes the usual nondegenerate masses, so that extra mass degeneracies do not enlarge the allowed basis transformations.