Characterizations and lower bounds for the spread of a normal matrix

The spread of an n x n matrix A with eigenvalues lambda(1),..., lambda(n) is defined by spr A = max(j,k) \lambda(j) - lambda(k)\. We prove that if A is normal, then sprA = max {\x*Ax - y*Ay\\ x, y is an element of C-n, parallel toxparallel to = parallel toyparallel to = 1} = max {\x*Ay + y*Ax\\x, y is an element of C-n, parallel toxparallel to = parallel toyparallel to = 1, re x*y = 0} = max {\x*Ay + y*Ax\\x, y is an element of C-n, parallel toxparallel to = parallel toyparallel to = 1, x*y = 0} = max{spr zA + (z) over bar*/2 \z is an element of C, \z\ = 1} = root2 max {(\x*A(2)x - (x*Ax)(2)\ + x*Ax\(2))(1/2) \x is an element of C-n, parallel toxparallel to = 1}. We also present several lower bounds for spr A, given by these characterizations. (C) 2003 Elsevier Science Inc. All rights reserved.