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We consider a pair of linear transformations $A:V\\to V$ and $A^*:V\\to V$ that satisfy the following two conditions:\n  There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal.\n  There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal.\n  We call such a pair a Leonard pair on $V$. 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