Understanding matrix (2)

 

Then we understand the matrix.

In the previous article, "the matrix is the description of motion." So far, it seems that everyone has no opinions yet. But I believe that sooner or later there will be a netizen from the mathematics department. Because the concept of motion is linked to calculus in mathematics and physics. When we study calculus, some people will follow the instructions to tell you that elementary mathematics is a constant mathematics, a static mathematics, a variable mathematics, and a motion mathematics. It is widely said that almost everyone knows this sentence. But there are not many people who really know what this sentence means. In short, in our human experience, motion is a continuous process. from point A to point B, even if the fastest light goes, it takes timePoint by pointThrough the path between AB, this brings about the concept of continuity. However, if we do not define the limit, we can't explain it. The ancient Greek mathematics was very strong, but they lacked the Limit Concept, so they could not explain the movements. The famous paradox of zhino (four paradox s, such as flying arrows and flying legs, Achilles, and turtles) let's get it done. This article is not about calculus, so I will not talk about it much. If you are interested, you can go to Professor Qi minyou's "review calculus". I just read the beginning of this book to understand the truth of the sentence "Advanced Mathematics is a mathematical study of motion.

However, in my article "Understanding matrix", the concept of "motion" is not a continuous movement in calculus, but an instantaneous change. For example, at, after a "movement","Transition"To B, which does not need to go through any point between A and B. Such "motion" or "transition" is against our daily experience. However, those who have some knowledge about quantum physics will immediately point out that quantum (such as electronics) jumps on different levels of energy in an instant and has such a kind of transition behavior. Therefore, this kind of movement does not exist in nature, but we cannot observe it at a macro level. However, in any case, the word "motion" is often used here to produce ambiguity. To be more precise, it should be "transition ". Therefore, this sentence can be changed:

"Matrix is the description of transition in Linear Space ".

This is too physical, that is, too specific, not mathematical, that is, not abstract. So we finally use a correct math term --Transform. In this case, everyone should understand,The so-called transformation is actually the transition from one point (element/object) to another point (element/object) in the space.. For example, a topological transformation is a transition from one point to another in a topological space. For example, an affine transformation is the transition from one point to another in an affined space. Additionally, this affine space and vector space are siblings. All the friends who do computer graphics know that, although a 3D object only needs 3D vectors to be described, all the computer graphics transformation matrices are 4x4. For the reason, many books have written "for convenience", which seems to me to be an attempt to pass through. The real reason is that the graphic transformation applied in computer graphics is actually carried out in the affine space rather than the vector space. Think about it. In vector space, when a vector is moved in parallel, it is still the same vector. However, the real world and other long parallel line segments cannot be considered the same thing, therefore, the living space of computer graphics is actually an affine space. The Matrix Representation of the affine transformation is 4x4 at all. If you are interested, you can go to computer graphics- ry tool algorithm details.

Once we understand the concept of "transformation", the definition of the matrix becomes:

"A matrix is the description of the transformation in a linear space ."

So far, we finally got a definition that looks more mathematical. But I have to say a few more words. This is generally the case in teaching materials. A linear transformation T in a linear space V can be expressed as a matrix after being elected as a group of bases. Therefore, we need to clarify what linear transformation is, what is base, and what is a group of base. The definition of linear transformation is very simple. It has a transformation T, so that for any two different objects x and y in the linear space V, as well as any real numbers A and B, there are:
T (AX + by) = at (x) + Bt (Y ),
T is called linear transformation.

The definition is written in this way, but the definition cannot be intuitively understood. What kind of transformation is linear transformation? As we have just said, transformation is a point transition from space to another point, while linear transformation, it is the movement from one point of a linear space V to another point of another linear space w. This sentence contains a layer of meaning, that is, a point can be transformed not only to another point in the same linear space, but also to another point in another linear space. No matter how you change it, as long as the objects in the linear space are before and after the transformation, this transformation must be a linear transformation, and it must be described using a non-singular matrix. A transformation you describe using a non-singular matrix must be a linear transformation. Some may ask, why do we emphasize non-singular matrices? The so-called non-singular, only meaningful to the square matrix, so what is the situation of non-square matrix? This will be lengthy. At last, we need to take linear transformation as a ing and discuss its ing nature and the concepts of kernel and image of linear transformation. I don't think this is important. If you have time, write it later.Below we will only discuss the most common and useful transformation, that is, linear transformation within the same linear space. That is to say, if the matrix mentioned below is not described, it is a square matrix, and it is a non-singular square matrix. To learn a subject, the most important thing is to grasp the main content and quickly establish the overall concept of this subject. You don't have to consider all the details and special circumstances from the very beginning.

Next, what is the base? This question will be further discussed later.Consider the base as the coordinate system in a linear space.Note that it is a coordinate system, not a coordinate value, which is a "unity of opposites ". In this way, "selecting a group of bases" means selecting a coordinate system in a linear space. This means.

Well, the definition of the matrix is improved as follows:

"A matrix is a description of linear transformation in a linear space. In a linear space, as long as we select a group of bases, any linear transformation can be described using a definite matrix ."

The key to understanding this sentence is to distinguish "linear transformation" from "A description of linear transformation.One is the object, and the other is the expression of the object. It is like in the familiar object-oriented programming, an object can have multiple references. Each reference can be called by a different name, but it is the same object. If you do not have an image, you can simply make a vulgar analogy.

For example, if you want to take a picture of a pig, you can take a picture of the pig as long as you select a camera position. This picture can be seen as a description of this pig, but it is only a one-sided description. Because I can take a picture of this pig with another camera, I can get a different picture, this is another one-sided description of this pig. All the pictures taken in this way are described by the same pig, but they are not the same pig.

Similarly, for a linear transformation, as long as you select a group of bases, you can find a matrix to describe this linear transformation. If you change the base group, a different matrix is obtained. All these matrices are described in the same linear transformation, but they are not linear transformations themselves.

However, the problem arises. If you give me two pictures of pig, how do I know that the two pictures contain the same pig? Similarly, how do I know that these two matrices describe the same linear transformation? If it is a different matrix description of the same linear transformation, it is the brother of the family. If you don't know each other, it won't be a joke.

Fortunately, we can find a property of the matrix brothers of the same linear transformation, that is:

If matrix A and matrix B are two different descriptions of the same linear transformation (the reason is that different bases are selected, that is, different coordinate systems are selected ), then, a non-singular matrix P can be found to satisfy the following relationship between A and B:

A = P-1BP

If linear algebra is a bit familiar, you can see that this is the definition of the similarity matrix. Yes,Similar matrices are different descriptors of the same linear transformation.According to this definition, photos from different angles of the same pig can also be similar photos. It is a little vulgar, but it is understandable.

In the formula above, the matrix P is actually a transformation relationship between the base based on matrix A and the Base Based on matrix B. This conclusion can be proved in a very intuitive way (rather than the formal proof in general textbooks). If I have time, I will add this proof to my blog later.

This discovery is too important.The original family of similarity matrices are described in the same linear transformation!No wonder this is so important! Graduate courses in engineering include matrix theory, matrix analysis, and other courses, which talk about a variety of similar transformations, such as what is similar standard type, and what is the content of the right corner, the matrix obtained after transformation is required to be similar to the previous matrix. Why is this requirement? Only in this way can we ensure that the two matrices before and after the transformation describe the same linear transformation. Of course, the descriptions of different matrices of the same linear transformation do not distinguish the ring from the actual computing nature. Some descriptive matrices are much better than other matrices. It is easy to understand that photos of the same pig are also ugly. Therefore, the similarity transformation of a matrix can convert an ugly matrix into a more beautiful matrix, ensuring that both matrices describe the same linear transformation.

In this way, the matrix is basically clearly described as a side of linear transformation. However, it is not that simple, or linear algebra has a more amazing nature than that, that is,A matrix can be used not only as a description of linear transformation, but also as a set of base descriptions. As a matrix of transformation, we can not only transform one point in a linear space to another, but also change one coordinate system (base) table in a linear space to another coordinate system (base) go. In addition, the transformation point and the transformation coordinate system have the same effect. The most interesting mysteries of linear algebra are contained in them. With this understanding, many theorems and rules in linear algebra become clearer and more intuitive.

Leave this in the next article.

Because there are other things to do, it may take a few days to write the next article.