The Gram-Schmidt process is a mathematical technique for orthogonalising a set of nonorthogonal vectors.

Consider two linearly independent nonorthogonal vectors and of the Hermitian operator :

where is the degenerate eigenvalue corresponding to and .

Let . With reference to the diagram above, , where we have used the fact that the magnitude of the unit vector is 1 in the second equality. Furthermore, ; that is, the vector is a multiple of the unit vector , with the multiple being . Therefore,

is a component of , which is orthogonal to . The eigenvalue of is unchanged versus that of because

In the presence of a third linearly independent vector that is nonorthogonal to and (where ), the vector that is orthogonal to is (c.f. eq111). To determine the component of that is orthogonal to both and , let be the component of that is orthogonal to :

We can immediately see that is orthogonal to both and because it is sum of two vectors and , each of which is orthogonal to (and hence the dot product is zero). Substituting in the above equation, noting that and are scalars and that , we have

Therefore, for a set of three linearly independent nonorthogonal vectors , the transformed set of vectors, which are orthogonal to one another, is , where

For a set of linearly independent nonorthogonal vectors , the transformed set of vectors, which are orthogonal to one another, is with the *k*-th transformed vector as

The corresponding orthonormal vectors are , , … , .

An example of the application of the Gram-Schmidt process is the orthogonalisation of nonorthogonal Slater-type orbitals in the Hartree-Fock method.