Let A and B be N-order symmetric matrices over the complex number field, then the contraction of A and B over the complex number field is equivalent to the same rank of A and B..

Let A and B be N-order symmetric matrices in real number field, then the contract of A and B in real number field is equivalent to that A and B have the same positive and negative inertia index (that is, the number of positive and negative is equal).

The contractual relationship is an equivalent relationship, that is, it satisfies:

1, reflexivity: any matrix shrinks with itself;

2. Symmetry: If A and B contract, the contraction of B and A can be deduced.

Extended data:

Linearity refers to the proportion and linear relationship between quantities, which can be mathematically understood as a function with a constant first derivative.

Nonlinear refers to the non-proportional nonlinear relationship, and the first derivative is not constant.

Linear algebra originated from the study of two-dimensional and three-dimensional rectangular coordinate systems. Here, the vector is a directional line segment, which is expressed by length and direction. Such a vector can be used to represent a physical quantity, such as a force, or it can be added and multiplied by a scalar. This is the first example of real vector space.

Modern linear algebra has been extended to study arbitrary or infinite dimensional space. What is the dimension? n？ The vector space of is called n-dimensional space. Most useful conclusions in 2D and 3D spaces can be extended to these high-dimensional spaces. Although many people can't easily imagine n? A vector in dimensional space, such a vector (that is, n? Tuple) is used to indicate that the data is very effective.

And the result? n？ Tuple, the vector is n? An "ordered" list of elements in which most people can effectively summarize and manipulate data.

For example, eight-dimensional vectors can be used in economics to represent the gross national product (GNP) of eight countries. When arranging the order of all countries, such as (China, America, Britain, France, Germany, Spain, India and Australia), you can use vectors (V 1, V2, V3, V4, V5, V6, V7 and V8) to show their respective GNP in a certain year. Every country's GNP here is in its own position.

Vector space (linear space), as a purely abstract concept used to prove theorems, is a part of abstract algebra and has been well integrated into this field.

Some famous examples are: irreversible linear mapping or matrix group, and linear mapping ring in vector space. Linear algebra also plays an important role in mathematical analysis, especially in? In vector analysis, high-order derivatives are described, tensor products and commutative mappings are studied.

Vector space is defined on a field, such as a real number field or a complex number field. Linear operators map the elements of one linear space to another linear space (or the same linear space), and keep the consistency of addition and scalar multiplication in vector space. The set of all these transformations is itself a vector space.

If the basis of a linear space is determined, all linear transformations can be expressed as a table called a matrix. The in-depth study of matrix properties and matrix algorithms (including determinant and eigenvector) is also considered as a part of linear algebra.

Baidu encyclopedia-linear algebra

Baidu Encyclopedia-Matrix