In mathematics, Brandt matrices are matrices, introduced by , that are related to the number of ideals of given norm in an ideal class of a definite quaternion algebra over the rationals, and that give a representation of the Hecke algebra.
calculated the traces of the Brandt matrices.
Let O be an order in a quaternion algebra with class number H, and I<sub>i</sub>,...,I<sub>H</sub> invertible left O-ideals representing the classes. Fix an integer m. Let e<sub>j</sub> denote the number of units in the right order of I<sub>j</sub> and let B<sub>ij</sub> denote the number of ñ in I<sub>j</sub><sup>âÂÂ1</sup>I<sub>i</sub> with reduced norm N(ñ) equal to mN(I<sub>i</sub>)/N(I<sub>j</sub>). The Brandt matrix B(m) is the HÃÂH matrix with entries B<sub>ij</sub>. Up to conjugation by a permutation matrix it is independent of the choice of representatives I<sub>j</sub>; it is dependent only on the level of the order O.