The matrix-valued numerical range over finite fields

In this paper we define and study the matrix-valued k × k numerical range of n × n matrices using the Hermitian product and the product with n × k unitary matrices U (on the right with U , on the left with its adjoint U† = U−1 ). For all i, j = 1, . . . , k we study the possible (i, j) -entries of these k × k matrices. Our results are for the case in which the base field is finite, but the same definition works over C . Instead of the degree 2 extension R ↪→ C we use the degree 2 extension Fq ↪→ Fq2 , q a prime power, with the Frobenius map t 7→ t as the nonzero element of its Galois group. The diagonal entries of the matrix numerical ranges are the scalar numerical ranges, while often the nondiagonal entries are the entire Fq2 . We also define the matrix-valued numerical range map.