Gram-Schmidt process

The Gram-Schmidt process is an algorithm for producing an orthonormal basis from a set of vectors that are not orthogonal. These vectors are orthogonal in the sense that their inner product is for any two basis vectors ${\vec {v}}_{i}$ and ${\vec {v}}_{j}$ is:

$\langle {\vec {v}}_{i}|{\vec {v}}_{j}\rangle =\delta _{ij}$

where $\delta _{ij}$ is the Kronecker delta, equal to 1 if i=j and 0 otherwise. They are then said to be orthonormal as the norm of any vector is 1: $||{\vec {v}}_{i}||=1$. It is used in quantum mechanics to create a basis of eigenfunctions to represent wavefunctions.

See also

• Basis
• Norm