Matrix representations of Clifford algebras

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As is well known, Clifford algebras can be faithfully realized as certain matrix algebras, the matrix entries being real numbers, complex numbers, or quaternions, depending on the particular Clifford algebra. We show that the matrix representations of the basis elements of a Clifford algebra can be chosen to satisfy a certain additional trace condition; we then use this trace condition to establish optimal inequalities involving norms in Clifford algebras.