Abstract: Given an n×n complex matrix A and an n-dimensional complex vector y=(ν₁ , ··· , ν_n ), the y-numerical radius of A is the nonnegative quantity r_y(A)=max{n∑_j=1,..., x_j ∈C_n}. Here C_n is an n-dimensional linear space over the complex field C. For y = (1, 0, ··· , 0) it reduces to the classical radius r(A) =max {|x*Ax|: x*x=1}. We show that r_y is a generalized matrix norm if and only if n∑_j=1ν_j≠ 0. Next, we study some properties of the y-numerical radius of matrices and vectors with non-negative entries. 

Key words: numerical range, numerical radius, generalized matrix norm