Abstract
We give the complete classification of quadratic forms tr( AX 2) obtained by taking the trace of n × n matrices AX 2 for n = 2,3, where A ϵ M n ( F q ), X is a square matrix of n 2 variables, and F q is a finite field of order q and of characteristic different from 2. As an application of the classification we can compute the number of solutions of a matrix equation X 2 1 + X 2 2 + … + X 2 m = B for a 2 × 2 matrix B over F q .
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