Lower bounds for the numerical radius

Let A be a complex n × n matrix. We find lower bounds for its numerical radius r(A)=max{|x∗Ax|∣x∈Cn,x∗x=1}. First we choose x satisfying x∗x = 1 and compute ∣x∗Ax∣. We also improve the simple bound so obtained. Second, we applyr(A)=maxλzA+z¯A∗2z∈C,|z|=1where λ denotes the largest eigenvalue.