Understanding Matrix (ii)

Then understand the matrix.

"The matrix is the description of the motion", and so far, it seems that no one has any opinion about it. But I believe that sooner or later there will be friends from the mathematics department to make the decision. Because the concept of movement is associated with calculus in mathematics and physics. When we learn calculus, there will always be someone to tell you that elementary mathematics is the study of constant mathematics, the study of static mathematics, higher mathematics is variable mathematics, is the study of sports mathematics. People word of mouth, almost everyone knows this sentence. But it seems that there are not many people who really know what the meaning of this saying is. In short, in our human experience, motion is a continuous process, from point A to point B, even the fastest light, and it takes a time to pass through the path of AB, and this brings the concept of continuity. And the continuity of this thing, if not defined the concept of limits, simply can not explain. The Ancient Greek mathematics is very strong, but is lack of limit concept, so can not explain the movement, by the famous paradox of Zeno (flying Arrows, Scud Achilles, such as the tortoise, etc. four paradoxes) have to die. Because this article is not about calculus, so I will not say more. Interested readers can go to the "relive calculus" written by Professor Qi-min's friend. I read this book at the beginning of the section, only to understand that "advanced mathematics is the study of the Mathematics of movement," the truth of the phrase.

But in my "Understanding Matrix" article, the concept of "movement" is not a continuous movement in calculus, but an instantaneous change. For example, this moment at point a, after a "movement", all of a sudden " jump " to point B, which does not need to go through a point and B point between any point. Such "movement", or "jump", is a violation of our daily experience. But people who know a little bit of quantum physics will immediately point out that quantum (such as electrons) jumps in different energy-level orbits, which happen instantaneously, with such a transition behavior. So, there is no such phenomenon in nature, but we cannot observe it on the macroscopic. However, in any case, the word "sport" is used here, or it is prone to ambiguity, to say more precisely, should be "jump". So this sentence can be changed to:

The matrix is the description of the transitions in the linear space.

But this is too physical, that is, too specific, not math, which means not enough abstract. So we end up with a genuine mathematical term--a transformation --to describe this thing. In this way, we should understand that the so-called transformation, in fact, is the space from one point (Element/object) to another point (element/object) of the jump . For example, a topological transformation is a transition from one point to another in a topological space. Affine transformations, for example, are transitions from one point to another in affine space. Incidentally, this affine space and vector space are brothers. Computer graphics friends know that although the description of a three-dimensional object only need three-dimensional vector, but all the computer graphics transformation matrix is 4 x 4. The reason, many books are written "in order to use convenience," which in my view is simply an attempt to muddle through. The real reason is that graphic transformations applied in computer graphics are actually done in affine space rather than vector space. Think of it, in vector space, a vector parallel to move after the same vector, and the real world equal to two parallel segments of course can not be considered the same thing, so the living space of computer graphics is actually affine space. and the matrix representation of affine transformation is 4 x 4. Also pull far, interested readers can go to see "computer graphics--geometric tool algorithm detailed."

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

The matrix is a description of the transformations in the linear space. "

So far, we've finally got a definition that looks more mathematical. But I have to say a few more words. It is generally said in the textbook that a linear transformation t in a linear space V, when a group of bases is selected, can be represented as a matrix. So we have to make clear exactly what is linear transformation, what is base, what is called a group of bases. The definition of a linear transformation is very simple, with a transform t, so that for any of the two different objects x and Y in the middle of the linear space V, as well as arbitrary real numbers A and B, there are:
T (ax + by) = at (x) + BT (y),
Then you call t the linear transformation.

Definitions are written in this way, but it is not intuitive to see the definition of light. What kind of transformation is a linear transformation. As we have just said, the transformation moves from one point of space to another, and the linear transformation is the movement from one point of the linear space V to the other of the other of the linear space W. This sentence contains a layer of meaning, that is, a point can not only be transformed to another point in the same linear space, and can be transformed to another point in a linear space. No matter how you change, as long as the transformation is in the linear space before and after the object, this transformation must be a linear transformation, it must be a nonsingular matrix to describe. And you use a nonsingular matrix to describe a transformation that must be a linear transformation. Some people may ask why we should emphasize nonsingular matrices here. The so-called nonsingular, only to the square of significance, then the situation of the non-square. This will be more verbose, and finally, the linear transformation as a mapping, and the discussion of its mapping properties, as well as the linear transformation of the core and image concepts can be fully explained. I think this is not the point, if you do have the time, write a little later. here we only explore the most common and useful transformations, which are linear transformations within the same linear space. In other words, the following matrix, without explanation, is a phalanx, and is a nonsingular phalanx. Learning a knowledge, the most important thing is to grasp the backbone of the content, and quickly establish the whole concept of this knowledge, do not have to consider all the details and special circumstances at the outset, from the chaos.

Then say, what is the base. This question is a big one in the back, so just think of the base as a coordinate system in a linear space. Note that the coordinate system, not the coordinate value, is a "contradictory unity." Thus, "Select a group of bases" means to select a coordinate system in the linear space. That's what it means.

Well, finally, we'll refine the definition of the matrix as follows:

The matrix is a description of the linear transformation in the linear space. In a linear space, as long as we select a set of bases, then for any linear transformation, we can use a definite matrix to describe. "

the key to understanding this sentence is to distinguish between "linear transformation" and "a description of linear transformations". One is the object, the other is the expression of that object. Just as we are familiar with object-oriented programming, an object can have multiple references, each of which can be called a different name, but is the same object. If the image is not, then simply a very vulgar analogy.

For example, if you have a pig and you want to take a picture of it, you can take a picture of the pig as long as you have a shot position selected for the camera. This picture can be seen as a description of the pig, but only a one-sided description, because a camera position for the pig to take a picture, can get a different photo, but also the pig's another one-sided description. All these pictures are the same pig's description, but not the pig itself.

Similarly, for a linear transformation, as long as you select a group of bases, you can find a matrix to describe the linear transformation. To change a group of bases, we get a different matrix. All of these matrices are descriptions of this same linear transformation, but they are not linear transformations themselves.

But that's the problem. If you give me two pictures of pigs, how do I know they are the same pig in these two pictures? Similarly, you give me two matrices, how do I know that the two matrices are described by the same linear transformation? If it is the same linear transformation of the different matrix description, that is the Clan brothers, meet do not know, not become a joke.

Fortunately, we can find one of the properties of the Matrix Brothers of the same linear transformation, that is:

If the matrix A and B are two different descriptions of the same linear transformation (the reason for the difference is that a different base is selected, a different coordinate system is selected), then a nonsingular matrix P must be found to satisfy the relationship between A and B:

A = p-1bp

Linear algebra A slightly more familiar reader can see that this is the definition of a similar matrix. Yes, the so-called similarity matrix is a different description matrix of the same linear transformation. According to this definition, photographs of different angles of the same pig can also be similar photographs. A bit vulgar, but it can make people understand.

The matrix p in the above equation is actually a transformation relationship between the bases on which a matrix is based and the base of the B matrix. This conclusion can be proved in a very intuitive way (rather than the formal proof of the general textbook), and if there is time, I will add it later in my blog.

The discovery is too important. The original family of similar matrices is a description of the same linear transformation. no wonder it's so important. In the course of engineering postgraduate courses, such as Matrix theory, matrix analysis and other courses, including a variety of similar transformations, such as what similar standard type, diagonalization, and so on, all require the transformation after the matrix and the previous matrix similar, why so required. Because only in this way can we guarantee that the two matrices before and after the transformation describe the same linear transformation. Of course, the different matrix description of the same linear transformation is not a good ring in terms of actual operational properties. Some of the descriptive matrices are much better than the other matrix properties. This is easy to understand, the same pig photos also have beautiful and ugly points. So the similarity transformation of matrices can transform an ugly matrix into a more beautiful matrix, and ensure that both matrices describe the same linear transformation.

Thus, the matrix as a linear transformation of the description of the side, is basically clear. But the thing is not so simple, or that linear algebra has a more wonderful property than this, that the matrix can be used not only as a description of linear transformations, but also as a set of bases. As a matrix of transformation, not only one point in the linear space can be transformed to another point, but also a coordinate system (base) table in the linear space could be changed to another coordinate system (base). Moreover, the transformation point and the transformation coordinate system have the similar effect. The most interesting mysteries in linear algebra are contained therein. Understanding these elements, many theorems and rules in linear algebra become clearer and more intuitive.

Let's write this in the next article.

Because there are other things to do, the next one may be written in a few days.