The matrix-valued numerical range over finite fields

Abstract

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