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We consider an ordered pair of linear transformations $A:V\\to V$ and $A^*:V\\to V$ that satisfy conditions (i), (ii) below.\n  (i) 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  (ii) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal.\n  We call such a pair a {\\it Leonard pair} on $V$. 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We consider an ordered pair of linear transformations $A:V\\to V$ and $A^*:V\\to V$ that satisfy conditions (i), (ii) below.\n  (i) 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  (ii) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal.\n  We call such a pair a {\\it Leonard pair} on $V$. 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