Quadratic form is a mathematical concept related to matrix, which refers to the form of a quadratic polynomial, which only contains quadratic term (square of independent variable) and linear term (independent variable). In matrices, quadratic forms usually appear in the form of symmetric matrices. In linear algebra, the quadratic form of matrix usually contains the eigenvalues and eigenvectors of symmetric matrix, and uses them to calculate the properties of quadratic form.

Quadratic form is widely used in mathematics, physics and engineering. Mathematically, it is used to define the length and angle of a vector, and provide concepts such as vertical, parallel and orthogonal. In physics, quadratic form is used to describe the energy and inertia of physical systems, which plays an important role in the fields of machinery, electronics, thermodynamics and so on. In engineering, quadratic form is used to solve the problems of least square fitting, image processing and data mining. In a word, quadratic form is an important mathematical tool and widely used in many fields.

Quadratic forms have many interesting characteristics and properties. For example, quadratic forms can always be classified and described by eigenvalues and eigenvectors of matrices. In addition, the positive and negative characteristics of quadratic form can be judged by positive definite, semi-positive definite and negative definite of matrix. In addition, any quadratic form can be transformed into the simplest and most manageable form through diagonalization and canonical form of matrix. Finally, the quadratic form can be transformed and simplified by linear transformation and orthogonal transformation, which makes the calculation and processing easier.