Inspired by the theory of P -and Q-polynomial association schemes we consider the following situation in linear algebra. Let F denote a field, and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A : V → V and A * : V → V satisfying the following four conditions. θ * i−1 − θ * i both equal β + 1, for 2 ≤ i ≤ d − 1. We hope these results will ultimately lead to a complete classification of the TD pairs.
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