Totally bipartite tridiagonal pairs

There is a concept in linear algebra called tridiagonal pair. The was motivated by the theory of $Q$-polynomial distance-regular graphs. We give tutorial introduction to pairs, working with special case as concrete example. totally bipartite, or bipartite (TB). Starting from first principles, we an elementary but comprehensive account TB pairs. following topics are discussed: (i) notion system; (ii) eigenvalue array; (iii) standard basis and matrix representations; (iv) intersection numbers; (v) Askey--Wilson relations; (vi) recurrence involving (vii) classification systems; (viii) self-dual pairs (ix) $\mathbb{Z}_3$-symmetric (x) some automorphisms antiautomorphisms associated pair; (xi) action modular group ${\rm PSL}_2(\mathbb{Z})$