A hermitian operator is one that satisfies < Hx|y > = < x|Hy >, where x and y are real or complex vectors or functions and H is a linear operator and < x|y > denotes the inner product of x and y.
Some properties: Suppose we know that Hx = λx where &lambda is an eigenvalue of the hermitian operator H. Then we can prove that &lambda must be real. Consider:
< Hx|x > = < x|Hx >, therefore < λx|x > = < x|λx > which by the linearity or the inner product reduces to: &lambda*< x|x > = &lambda< x|x > (where * denotes the complex conjugate). Since cannot be zero by the non-negativity of the inner product (here x represents a magnitude which is by definition non-negative), &lambda* = &lambda therefore &lambda must be real.
Given that x and y are eigenvectors of the hermitian operator H, and &lambda and ɸ are the eigenvalues of H to x and y respectively (note also that &lambda ≠ ɸ), then we can prove that x and y are orthogonal.
Consider that Hx = λx and Hy = ɸy. Now, < Hx|y > = < x|Hy > which means that < λx|y > = < x|ɸy > so by the linearity of the inner product, &lambda< x|y > = ɸ< x|y > . Since &lambda ≠ ɸ, < x|y > = 0, ergo x is orthogonal to y.