向量的施密特正交化

先看一个例子：
设三个向量分别为：

α1α2α3=(1,−1,0)T=(1,0,−1)T=(1,1,1)T

那么对 α1,α2 正交化：

β1β2=α1=(1,−1,0)T=α2−(α2,β1)(β1,β1)β1=(1,0,−1)T−1×1+0×(−1)+(−1)×01×1+(−1)×(−1)+0×0(1,−1,0)T=(1,0,−1)T−12(1,−1,0)T=12(1,1,−2)T

再对 β1,β2,β3 单位化：

γ1γ2γ3=12‾‾√(1,−1,0)T=16‾‾√(1,1,−2)T=13‾‾√(1,1,1)T

以下施密特正交化的标准定义引自Wiki

We define the projection operator by

proju(v)=⟨v,u⟩⟨u,u⟩u

where ⟨v,u⟩ denotes the inner product of the vectors v and u. This operator projects the vector v orthogonally onto the line spanned by vector u. If u=0, we define proj0(v):=0. i.e., the procjection map proj0 is the zero map, sending every vector to the zero vector.
The Gram-Schmidt process then works as follows:

u1u2u3u4uk=v1=v2−proju1(v2)=v3−proju1(v3)−proju2(v3),=v4−proju1(v4)−proju2(v4)−proju3(v4),⋮=vk−∑j=1k−1projuj(vk),e1=u1∥u1∥e2=u2∥u2∥e3=u3∥u3∥e3=u4∥u4∥ek=uk∥uk∥

The sequence u1,...,uk is the required system of orthogonal vectors, and the normalized vectors e1,...,ek from an orthonormal set. The calculation of the sequence u1,...,uk is known as Gram-Schmidt rothogonalization, while the calculation of the sequence e1,...,ek is known as Gram-Schmidt orthonormalization as the vectors are normalized.

部分符号和国内教材上有差异。