What Is a Matricectomy?

In mathematics, a matrix is a set of complex or real numbers arranged in a rectangular array ^[1] , which first came from a square matrix formed by the coefficients and constants of the system of equations. This concept was first proposed by Kelly, a British mathematician in the 19th century.

Research on matrices has a long history,

A number table of m rows and n columns arranged by m × n numbers a _{ij is} referred to as a matrix of m rows and n columns, referred to as an m × n matrix. Referred to as:

These m × n numbers are called the elements of matrix A , which are called elements for short. The number a _ij is located in the i-th row and j-th column of matrix A. It is called the (i, j) element of matrix A. The matrix of (j) elements can be described as (a _ij ) or (a _ij ) _{m × n} , and the m × n matrix A is also described as A _mn .

Element is

Matrix operations in

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n × n real symmetric matrix A if satisfied for all non-zero vectors

,corresponding

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Matrix image processing

The affine transformation of an image in image processing can generally be expressed as a form of an affine matrix multiplied by an original image ^[27] . For example,

This shows a linear transformation followed by a translation.

Matrix linear transformation and symmetry

Linear transformation and its corresponding symmetry play an important role in modern physics. For example, in quantum field theory, elementary particles are represented by the Lorenz group of the special theory of relativity, specifically, their performance under the spinor group. The concrete representation of the contained Pauli matrices and the more general Dirac matrix is an indispensable component in the physical description of fermions, and the performance of fermions can be expressed by spins. When describing the lightest three kinds of quarks, a group theory representation containing a special unitary group SU (3) is needed; physicists use a simpler matrix representation in calculations, called the Gailman matrix. This matrix is also used as the SU (3) gauge group, and the modern description of strong nuclear forcesthe basis of quantum chromodynamicsis exactly SU (3). There is also the Kabibo-Kobayashi-Yichuan matrix (CKM matrix): the basic quark states that are important in weak interactions are not the same as the quark states with different masses between the specified particles, but the two are in a linear relationship. This is what is expressed.

Linear combination of matrix quantum states

When Heisenberg proposed the first quantum mechanical model in 1925, he used an infinite-dimensional matrix to represent the operators in the theory that act on quantum states. This approach is also seen in matrix mechanics. For example, the density matrix is used to describe the "mixed" quantum states represented by linear combinations of "pure" quantum states in a quantum system ^[28] .

Another matrix is an important tool for describing the scattering experiments that form the cornerstone of experimental particle physics. When particles collide in the accelerator, particles that did not interact with each other enter the action area of other particles in high-speed motion, and the momentum changes to form a series of new particles. This collision can be interpreted as a scalar product of the linear combination of the resulting particle state and the incident particle state. The linear combination can be expressed as a matrix, called the S matrix, in which all possible interparticle interactions are recorded ^[29] .

Matrix normal mode

Another general application of matrices in physics is to describe linearly coupled harmonic systems. The equation of motion of such a system can be expressed in the form of a matrix, that is, a mass matrix multiplied by a generalized velocity to give a motion term, and a force matrix multiplied by a displacement vector to characterize the interaction. The best way to find the solution of the system is to find the eigenvectors of the matrix (by diagonalization, etc.), which is called the normal mode of the system. This kind of solution is very important when studying the dynamic mode of the molecule: the vibration of the atoms bonded by chemical bonds in the system can be expressed as the superposition of the normal vibration mode ^[30] . When describing mechanical vibrations or circuit oscillations, the normal mode solution is also needed ^[31] .

Matrix geometric optics

In geometric optics, you can find many places where matrices are needed. Geometric optics is an approximation theory that ignores the fluctuation of light waves. The model of this theory treats light as geometric rays. With paraxial approximation, if the angle between the light and the optical axis is small, the effect of the lens or reflective element on the light can be expressed as the product of a 2 × 2 matrix and a vector. The two components of this vector are the geometric properties of the ray (the slope of the ray, the vertical distance between the ray and the optical axis in the principal plane). This matrix is called the ray transfer matrix, and the internal elements encode the properties of the optical element. For refraction, this matrix is subdivided into two types: "refraction matrix" and "translation matrix". The refraction matrix describes the refraction behavior of light when it encounters a lens. The translation matrix describes the translational behavior of light rays traveling from one principal plane to another.

An optical system composed of a series of lenses or reflective elements can be simply described by its corresponding matrix combination ^[32] .