# How to calculate the formula for Schmidt’s orthogonalization

### How is the “schmidt orthogonalization factor” calculated?

Calculation formula: (α, β) = α – β = αT – β = βT – α = ∑XiYi

1, schmidt orthogonalization: Schmidt orthogonalization (Schmidt orthogonalization) is the method of turning a set of linearly unrelated vectors into a set of unit orthogonal vectors. Starting from any linearly unrelated vector group α1, α2, ……, αm in Euclidean space, the orthogonal vector group β1, β2, ……, βm is obtained such that the set of vectors consisting of α1, α2, …… , αm is equivalent to the vector group β1, β2, ……, βm, and then each vector in the orthogonal vector group undergoes unitization, a standard orthogonal vector group is obtained, this method is called Schmidt orthogonalization.

2, Theorem:

Generally, with mathematical induction can be proved:

Set is a linear irrelevant vector group, if you make

is an orthogonal vector group, if you make

get a standard orthogonal vector group , and the vector group and equivalent.

The method illustrated above of constructing a standard orthogonal set of vectors using a linearly independent set of vectors is the Schmidt orthogonalization method.

### How to calculate schmidt orthogonalization coefficient is (α2,β1)/(β1,β1)

(α,β)=α-β=αT-β=βT-α=∑XiYi

Using the above formula, you can find it.

For example, (α2,β1)=0*1+1*1+2*1+1*0=3

Similarly, (β1,β1)=1*1+1*1+1*1+0*0=3

So, (α2,β1)/(β1,β1)=3/3=1

### What is the Schmidt orthogonalization formula?

The Schmidt orthogonalization formula is as follows:

Schmidt orthogonalization (Schmidt orthogonalization) is a method for finding orthogonal bases in Euclidean space. Starting from any linearly independent vector group α1, α2, ……, αm in Euclidean space, the orthogonal vector group β1, β2, ……, βm is found such that the vector group consisting of α1, α2, …… , αm is equivalent to the vector group β1, β2, ……, βm, and then each vector in the orthogonal vector group undergoes unitization, a standard orthogonal vector group is obtained, and this method is called Schmidt orthogonalization.

Related information:

Schmidt’s orthogonalization first requires the vector sets b1,b2,b3… Must be linearly independent. The general solution is eigenvectors, eigenvectors of the same eigenvalue are not necessarily linearly irrelevant, but eigenvectors of different eigenvalues must be linearly correlated.

Select the vector b1 as the reference vector c1, then c2 is equal to b2 minus the inner product of b2 and c1 divided by c1 and the inner product of c1 and multiplied by c1, remember that the plurality must be in the form of a matrix. Including c3 is equal to b3 minus the inner product of b3 and c1 multiplied by b1 minus the inner product of c3 and b2 divided by the inner product of c2 and c2 multiplied by c2.