Schmidt orthogonalization is an important mathematical method, which is used to transform a group of linearly independent vectors into an orthogonal vector group. The formula is (α, β) = α β = α. Schmidt orthogonalization is widely used in signal processing, image processing and machine learning.
In the process of Schmidt orthogonalization, different orthogonalization methods can be used, such as QR decomposition and Gram-Schmidt decomposition. Among them, QR decomposition is a common method By decomposing the matrix into orthogonal matrix and upper triangular matrix, the vector group can be orthogonalized. Gram-Schmidt decomposition is another commonly used method. A set of orthogonal vectors can be obtained by orthogonalizing the vector groups one by one.
Geometric significance of Schmidt orthogonalization
1, orthogonal basis vector: The result of Schmidt orthogonalization is a set of mutually orthogonal vectors, which are called orthogonal basis vectors. In geometry, orthogonal basis vectors are very important. They can be used to represent all directions in space. Orthogonal basis vectors have the following characteristics: they are perpendicular to each other and the length is 1. Any vector can be uniquely expressed as a linear combination of orthogonal basis vectors.
2. Basis transformation: Schmidt orthogonalization is essentially a basis transformation, that is, a given set of vectors is transformed into another set of orthogonal basis vectors. Basis transformation has great application in geometry. It can transform one coordinate system into another, thus describing different geometric properties. Schmidt orthogonalization can be regarded as a method of basis transformation. Through orthogonalization, a new set of basis vectors can be obtained to describe the geometric characteristics of the space where the original vector is located.
3. Orthogonalization error: Schmidt orthogonalization will introduce an orthogonalization error, that is, the error between the orthogonalized vector and the original vector. The error can be expressed as the product of orthogonal matrix and vector. The orthogonalization error can be used to evaluate the accuracy and stability of Schmidt orthogonalization. In practical application, it is usually necessary to control the orthogonalization error in a small range to ensure the accuracy and reliability of orthogonalization.
Refer to the above content: Baidu Encyclopedia-Schmidt orthogonalization