Erik, note that a function of a matrix, like eigs (eigenvalues)
interprets a matrix as input as a single input value, i.e. the matrix
itself. But sometimes we want a user defined function to understand
that a matrix input represents a collection of input values where the
function should be evaluated, this is the "element-wise" case.
For example, you can create a discrete Fourier basis quite easily
(warning:this exploits automatic broadcasting, matlab doesn't do it)
t = linspace (0,2*pi,100).';
f = 1:10;
F = [sin(f.*t) cos(f.*t)];
subplot(2,1,1)
plot(t,F(:,1:10));
subplot(2,1,2)
plot(t,F(:,11:end));
Since the functions "sin" and "cos" interprets the input matrix "f.*t"
as a collection of arguments it evaluates on each entry of the matrix.