Editing Cabibbo–Kobayashi–Maskawa matrix (section)

==General case construction==
To generalize the matrix, count the number of physically important parameters in this matrix {{mvar|V}} which appear in experiments. If there are {{sc|{{mvar|N}}}} generations of quarks (2{{sc|{{mvar|N}}}} [[flavour (particle physics)|flavour]]s) then

* An {{sc|{{mvar|N}}}}&nbsp;×&nbsp;{{sc|{{mvar|N}}}} unitary matrix (that is, a matrix {{mvar|V}} such that {{mvar|V{{sup|†}}V&nbsp;{{=}}&nbsp;I}}, where {{mvar|V{{sup|†}}}} is the [[conjugate transpose]] of {{mvar|V}} and {{mvar|I}} is the identity matrix) requires {{sc|{{mvar|N}}}}<sup>2</sup> real parameters to be specified.
* 2{{sc|{{mvar|N}}}}&nbsp;−&nbsp;1 of these parameters are not physically significant, because one phase can be absorbed into each quark field (both of the mass eigenstates, and of the weak eigenstates), but the matrix is independent of a common phase. Hence, the total number of free variables independent of the choice of the phases of basis vectors is {{sc|{{mvar|N}}}}<sup>2</sup>&nbsp;−&nbsp;(2{{sc|{{mvar|N}}}}&nbsp;−&nbsp;1) = ({{sc|{{mvar|N}}}}&nbsp;−&nbsp;1)<sup>2</sup>.
** Of these, {{sfrac|1|2}}{{sc|{{mvar|N}}}}({{sc|{{mvar|N}}}}&nbsp;−&nbsp;1) are rotation angles called ''quark mixing angles''.
** The remaining {{sfrac|1|2}}({{sc|{{mvar|N}}}}&nbsp;−&nbsp;1)({{sc|{{mvar|N}}}}&nbsp;−&nbsp;2) are complex phases, which cause [[CP violation]].

=== {{sc|{{mvar|N}}}} = 2 ===
For the case {{sc|{{mvar|N}}}}&nbsp;=&nbsp;2, there is only one parameter, which is a mixing angle between two generations of quarks. Historically, this was the first version of CKM matrix when only two generations were known. It is called the '''Cabibbo angle''' after its inventor [[Nicola Cabibbo]].

=== {{sc|{{mvar|N}}}} = 3 ===
For the [[Standard Model]] case ({{sc|{{mvar|N}}}}&nbsp;=&nbsp;3), there are three mixing angles and one CP-violating complex phase.<ref>
{{cite web
 |first=J.C. |last=Baez
 |date=4 April 2011
 |title=Neutrinos and the mysterious Pontecorvo-Maki-Nakagawa-Sakata matrix
 |url=http://math.ucr.edu/home/baez/neutrinos.html |access-date=2016-02-13
 |df=dmy-all
 |quote=In fact, the [[Pontecorvo–Maki–Nakagawa–Sakata matrix]] actually affects the behavior of all leptons, not just neutrinos. Furthermore, a similar trick works for quarks – but then the matrix ''U'' is called the Cabibbo–Kobayashi–Maskawa matrix.
}}
</ref>