Tensor defined in geometric algebra is based on the generalization of vector and matrix. If you understand it in a popular way, you can regard a scalar as a zero-order tensor and a vector as a first-order tensor, then a matrix is a second-order tensor.
The strict definition of tensor is described by linear mapping. Similar to vectors, when several coordinate systems change, the set of ordered numbers satisfying a certain coordinate transformation relationship is defined as tensor. Geometrically speaking, it is a real geometric quantity, that is, it is something that does not change with the coordinate transformation of the reference system (in fact, it is the change of the base vector). The final result is that the combination of the basis vector and the components on the corresponding basis vector (i.e. tensor) remains unchanged. For example, the first-order tensor (vector) A can be expressed as a = x * i+y * j, and the tensor can represent very rich physical quantities because of the rich combinations of basis vectors.
Put it another way.
Tensor of type (p, q) is a mapping:
Give me a long talk
If a physical quantity is only a single value in a certain position of an object, it is an ordinary scalar, such as density. If you look at the same position from different directions, there will be different values. This number can be calculated by multiplying the observation direction by the matrix, which is the tensor.
There are many ways to understand tensor product, and there are different views in different contexts. But if compared with matrix product, I think tensor product is the product of everything, while matrix multiplication is concrete.
We have many matrices in our hands now, and then we want to multiply the two matrices. I can't figure out how to multiply at first, but I can guess some basic properties of the product, such as multiplying and matching by numbers and matching by addition, which is the distribution law. No matter what this product is, it should have these basic attributes. Then tensor product appears at this time, which represents the widest product and the weakest product, and only satisfies the basic properties mentioned above. Because it is the weakest product, all concrete products can be regarded as the concretization of the result of tensor product, that is, they can be regarded as the product of everything or an envelope.
In mathematics, tensor product writing
What can a vector represent?
For example, we can use the normal vector of a plane to represent this plane; Physically, forces can be represented by vectors and so on. It seems that vectors can represent many things, but when you think about it, vectors only represent two elements: size and direction.
There are many ways to represent vectors. We can use [0, 1] to represent two-dimensional vectors, or we can use lines with arrows in plane, three-dimensional or higher-dimensional space to represent vectors. We all know that (0,0)-> (1, 1) can represent a directed line segment (vector) from (0,0) to (1,1), so why can [0, 1] be used?
According to the previous explanation, we know that a vector is a directed line segment in space, which can be expressed by the product combination of the base of a set of coordinate systems and the corresponding components of the vector. Because there are many ways to define the coordinate system, there are many bases and corresponding components, but if you use the same set of base vectors by default, then the base vectors are unnecessary. At this time, if you want to represent a vector, you only need to give these three components, such as 0, 1 to represent a vector. If you add two brackets, this is the column representation of vectors that we often see in books (0 post a very loving picture.