Some lower bounds for the spectral radius of matrices using traces

Let A be an n×n matrix with eigenvalues λ1,λ2,…,λn, and let m be an integer satisfying rank(A)⩽m⩽n. If A is real, the best possible lower bound for its spectral radius in terms of m, trA and trA2 is obtained. If A is any complex matrix, two lower bounds for are compared, and furthermore a new lower bound for the spectral radius is given only in terms of trA,trA2,‖A‖,‖A∗A-AA∗‖,n and m.