In

matrix notation let

**A** be some
matrix. The complex conjugate (c.c.) of

**A** is denoted
with a "*" i.e.

c.c.(**A**) defined **A**^{*}.

You generate it by replacing each element of the matrix a_{ij} with its complex conjugate a^{*}_{ij}

In quantum mechanics wavefunctions
can be represented in matrix form.
The proabaility of a state is the integral over
space of the wavefunction multiplied by its c.c.
This gauranties a positive definite value for the
probability.

A real number is equal to it's complex conjugate.