Lower bounds for the numerical radius

We show that if A = [a_ij]ⁿ_{i, j=1} is an n-by-n complex matrix and A′ = [a′_i,j]ⁿ_{i, j=1}, where (eqution found), then w(A)≥ w(A′), where w(·) denotes the numerical radius of a matrix. Moreover, if n is odd and a₁₂,…,a_n−1,n,a_n1 are all nonzero, then w(A) = w(A′) if and only if A = A′. For an even n, under the same nonzero assumption, we have W(A) =W(A′) if and only if A = A′, where W(·) is the numerical range of a matrix.