Schmidt’s simple algorithm for orthogonalization

What is the formula for Schmidt orthogonalization?

It is as follows:

Schmidt orthogonalization is a method for finding orthogonal bases in Euclidean space. Starting from any linearly independent vector group α1, α2, αm in the Euclidean space, the orthogonal vector group β1, β2, βm is obtained, so that from α1, α2, αm is equivalent to the vector group β1, β2, βm, and then each vector in the orthogonal vector group is unitarized to obtain a standard orthogonal vector group, this method is called Schmidt orthogonalization.

In linear algebra, if a set of vectors on an inner product space can be tensored into a subspace, then this set of vectors is called a basis of this subspace.Gram-Schmidt orthogonalization provides a way to derive an orthogonal basis of the subspace from a basis on this subspace, and can furthermore be used to derive the corresponding standard orthogonal basis.

This method of orthogonalization is named after Jrgen PedersenGram and Erhard Schmidt, however it was discovered earlier than them by Laplace and Cauchy. In Lie group decomposition, this method was popularized as Iwasawadecomposition.

In numerical calculations, the Gram-Schmidt orthogonalization is numerically unstable, and the rounding errors accumulated in the calculations can make the final results poorly orthogonal. Therefore, in practice, orthogonalization is usually performed using the Haushold transform or Givens rotation.

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 times b1 minus the inner product of c3 and b2 divided by the inner product of c2 and c2 times c2.