Since the set of real numbers is a subset of the complex numbers, even though <u, v> is defined to be a real number, you can think of it as a function from pairs of vectors to a

**complex number**.

For example,

with inner product <(a, b), (c, d)>= ac+ bd has inner product always real but the matrix

has characteristic equation

which has i and -i as roots. Since we

**can't** have Av= iv in

, the corresponding linear transformation does NOT have any eigevalues. But we can, temporarily switch to "over

so that any linear transformation has eigenvalues, show, as you did, that, for a self-adjoint linear transformation, they must be real and so assert that any self-adjoint linear transformation on a vector space over the

**real** numbers has eigenvalues.