Gram-schmidt decomposition

Any set of linearly independent vectors can be converted into a set of orthogonal vectors by the Gram-Schmidt process. In three dimensions v₁ determines a line; the vectors v₁ and v₂ determine a plane. The vector q₁ is the unit vector in the direction v₁. The (unit) vector q₂ lies in the plane of v₁, v₂, and is normal to v₁ (on the same side as v₂). The (unit) vector q₃ is normal to the plane of v₁, v₂, on the same side as v₃.

In general, first set u₁= v₁, and then each u_i is made orthogonal to the preceding by subtraction of the projections of v_i in the directions of  :

The i vectors u_i span the same subspace as the v_i. The vectors are orthonormal. This leads to the following theorem:

Any (m,n) matrix A with linearly independent columns can be factorized into a product, A = QR. The columns of Q are orthonormal and R is upper triangular and invertible.

This ``classical Gram-Schmidt method is often numerically unstable, see Golub89 for a ``modified Gram-Schmidt method.