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

Totally bipartite tridiagonal pairs