To make a complex conjugate, replace every instance of i with -i.

The complex conjugate of a complex number a + bi is a - bi, where i is the square root of negative one. As we learned back in first year algebra, multiplying (x + y) * (x - y) yields x² - y², and since bi squared is -b², (a + bi) * (a - bi) = a² + b², a real number. This makes the complex conjugate a useful number.

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.