The main branch of numerical analysis is devoted to the development of effective algorithms for matrix calculation, which has been a topic for a century and is an expanding research field.
The matrix decomposition method simplifies the theoretical and practical calculation.
The customized algorithm for specific matrix structures (such as sparse matrix and near-angle matrix) speeds up the operation speed in finite element method and other calculations.
Infinite matrix appears in planetary theory and atomic theory. A simple example of infinite matrix is the matrix representing the derivative operator of Taylor series of functions.
Application of matrix:
When Heisenberg put forward the first quantum mechanical model in 1925, he theoretically expressed the operators acting on quantum states with infinite-dimensional matrices. This practice can also be seen in matrix mechanics. For example, a density matrix is a "mixed" quantum state represented by a linear combination of "pure" quantum states in a quantum system.
Another matrix is an important tool to describe the scattering experiment that constitutes the cornerstone of experimental particle physics. When particles collide in the accelerator, the particles without interaction enter the action zone of other particles moving at high speed, and their momentum changes to form a series of new particles. This collision can be explained as the scalar product of the linear combination of the result particle state and the incident particle state.
Refer to the above content: Baidu Encyclopedia-Matrix