Algebraic and Geometric Significance of Matrix Operations

This week, I completed the knowledge points of linear algebra. Some knowledge points are remembered, some are not remembered, and some even if they are remembered, it feels like rote memorization and has not been integrated into my own knowledge system. Thinking again and again, I think the fundamental reason is that I don’t know what these formulas are used for, what specific or figurative meanings they have, so I plan to sort out the actual meaning of these concepts or formulas as as a whole. Many basic terms and Definitions are not summarized here.

In general, matrices and vectors, in the algebraic sense, represent the problem of solving equations, while in the geometric sense, they are the problem of coordinate transformation.

But in fact, the operation we perform from one n-dimensional coordinate space to another n-dimensional coordinate space is still a system of equations. Therefore, matrices, systems of equations and coordinate transformations have strong commonalities.

Algebraic meaning

We slowly sort out the relationship between determinants, vectors, matrices, systems of equations, and coordinate space.

First of all, not every matrix has its own determinant, only square matrices have determinants.

A matrix corresponding to an n-order determinant is a coefficient matrix of a system of equations consisting of n equations and n unknowns.

The key significance of the value of the determinant is whether it is equal to 0, which is related to whether the system of equations has a solution and whether there is a unique solution.

First of all, when does the determinant value equal 0? In one case, there is at least one behavior of 0, and the corresponding system of equations appears to be that the coefficients of n unknowns in that row are all equal to 0, and in the other case, there are two rows that correspond to proportional, and the system of equations appears to be that There are two rows of coefficients that correspond to proportional.

If it is a system of homogeneous linear equations, that is, the right side of the equal sign of all equations in the system of equations is 0. In these two cases, it is equivalent to having n unknowns, but the number of valid equations is less than n, then we are bound to be unable to find the only solution to the equation, which is Cramer’s rule. When | A | = 0, homogeneous linear equations There are non-zero solutions.

If it is a system of non-homogeneous linear equations, in the first case, if the right side of the equation with coefficients of 0 is not equal to 0, then it is obvious that the system of equations has no solution. If it is equal to 0, it is equivalent to that this is not a valid equation; and In the second case, if the two coefficients correspond to the disproportionate values on the right side of the proportional equation, there is no solution. If they are proportional, it is equivalent to that the two equations are actually one, that is, there is an invalid equation.

The number of effective equations is actually the rank of the coefficient matrix, that is, the rank of the determinant. Therefore, there is no unique solution to the equation system corresponding to the coefficient matrix whose rank of the matrix is less than n. In fact, the number of effective equations is not enough, and no unique solution can be found.

The process of obtaining a matrix rank is to do elementary row transformation and transform it into a step type. For the equations, this process is a process of continuous simplification and transformation into a system of equations with the same solution. Note that only elementary row transformation is a system of equations with the same solution., can not do column transformation.

Although only the elementary row transformation is a system of equations with the same solution, all elementary transformations do not change the rank of the matrix.

At the same time, if the rank of the determinant is n, we convert it to a stepped determinant (if you continue to do column changes, you can become a diagonal matrix). If each row is a vector, then this vector group is linearly Independent, so the coefficient matrix composed of linearly independent vector groups has a unique solution to the equation system. If it is linearly related, it means that the number of effective equations is less than n.

Geometric meaning

The system of equations mentioned above is actually a process of space coordinate transformation (that is, the process of projecting vector x to the coordinate space corresponding to the A matrix), that is, the coefficient matrix of the equation system is obtained by column partitioning. The n-dimensional vectors linearly represent the n-dimensional vectors composed of the values on the right side of the equation system. The n column vectors of the coefficient matrix are equivalent to n coordinate axes. If the rank is equal to n, it means that the n coordinate axes are linearly independent, and the equations There is only one solution, that is, there is only one way to express it.

The left multiplication of a vector by a matrix can have two geometric meanings. One is to project the vector into a new coordinate system (this projection matrix is actually the representation of the coordinate axis of the new coordinate system in the original coordinate system), that is, not to change the vector, but to represent the vector in another coordinate system, one is to represent scaling rotation,

The matrix of the former left multiplication must be an invertible matrix, because you can transform from the A coordinate system to the B coordinate system, you need to change it back. The process of changing it back is left multiplication Inverse Matrix, which is also why the determinant value of the invertible matrix cannot be One reason for 0, if it is zero, it means that a dimension is lost, so it will not come back.

Eigenvalues and eigenvectors

This paragraph can be seen directly http://haoeric.github.io/matrix-eigenvectors/

This blog describes in detail what eigenvalues and eigenvectors are.

I will only extract the most useful paragraph for me here:

If an n × n square matrix A is multiplied by a non-zero vector x with the following conditions:

Ax = λx
Then the vector x is an eigenvector of the matrix A, and λ (non-zero) is the eigenvalue corresponding to the eigenvector x.

The left side of the equation can be seen as the projection of a non-zero vector x into a coordinate system based on each behavior of the square matrix A, and the right side of the equation can be seen as the expansion and contraction transformation of the vector x, and the expansion and contraction ratio is λ. Therefore, if a vector is projected into a space defined by a square matrix with only scaling changes and no rotation changes, then the vector is an eigenvector of the square matrix, and the scaling ratio is the eigenvalue.

Similarity of matrices

Let A and B be matrices of order n, if there is an invertible matrix P, such that

$$P^{-1}AP = B$$

Then B is said to be the similarity matrix of A
If A is similar to a diagonal matrix, it is said that A can be similarly diagonal, and the diagonal matrix is called the similar standard type of A.

If two matrices are similar, the eigenvalues and eigenpolynomials of the two matrices are equal.

A sufficient and necessary condition for the similarity diagonalization of an n-order square matrix is that there are n linearly independent eigenvectors

Linear independence is not the same as orthogonality. Orthogonality must be linearly independent, but not necessarily the other way around. If linear correlation is understood as two vectors in parallel, then linear correlation is intersection, and orthogonality is vertical intersection.

Real symmetric matrices must be similar to diagonalized, and the transition matrix is an orthogonal matrix, and the eigenvectors of different eigenvalues of real symmetric matrices are not only linearly independent, but also orthogonal.

The geometric significance of matrix similarity: Suppose there is a vector v in space, and we can represent its rotation and scaling as a left multiplication of a matrix, which is similar to the matrix represented by the same transformation in different coordinate systems.

• First, convert v from the coordinate system corresponding to B to the coordinate system corresponding to A, which is Pv
• Then in the A coordinate system, do operations such as rotation and scaling, that is APv
• Then put the vector back into the coordinate system corresponding to B, that is, P ^ {-1} APv. The result of this set of operations is equivalent to rotating and scaling directly in the coordinate system of B, that is, Bv = P ^ {-1} AP

Then the meaning of similar diagonalization is that in the B coordinate system, we can express it as a separate scaling for each coordinate axis, which coincides with the definition of eigenvalues and eigenvectors (if a vector is projected into a square The space defined by the matrix only changes in expansion and contraction without rotation, then the vector is an eigenvector of the square matrix, and the scaling ratio is the eigenvalue).

There is a conjecture here, does it mean that the feature space of the coordinate space corresponding to A is the coordinate space corresponding to B?

When understanding the eigenvalues, the geometric meaning of the left-multiplied matrix is a projection transformation, and the vector is projected into the vector space represented by the matrix, while in the process of understanding similarity, the geometric meaning of the left-multiplied P matrix is a projection transformation, and the left-multiplied A matrix The geometric meaning is to perform rotation and scaling.

Matrix contract

You can also read this blog: https://www.cnblogs.com/yanghh/p/13492614.html

I only excerpt the paragraph that is most useful to me:

The polynomial or quadratic matrices obtained in different reference frames are different, but we have to know whether they are the same graph, so there is the concept of contract.

The matrix of two contracts is the representation of the same figure under different bases.

It should be noted that under the premise of considering the quadratic type of matrix contract, it is a real symmetric matrix.

The similarity of a matrix represents the representation of the transformation of the same vector in different coordinate spaces, and the contract of a matrix represents the representation of the same figure in different coordinate spaces.

From an algebraic point of view, the contract is for the real symmetric matrix, and the real symmetric matrix must be similar to a diagonal matrix, and the transition matrix is an orthogonal matrix, that is to say, the real symmetric matrix must be both similar and contract and diagonal matrix. In fact, the process of finding the diagonal matrix of the real symmetric matrix contract is the process of formulating the quadratic type into a standard type, such as

$$f(x_1, x_2, x_3) = 2x_1^2 + 3x_2^2 + 5x_3^2 + 4x_1x_2 - 8x_2x_3 - 4x_3x_1$$

This is a quadratic form, the coefficient matrix is a real symmetric matrix, and we can match it with perfect squares

$$f = 2(x_1 + x_2 - x_3)^2 + (x_2 - 2x_3)^2 - x_3^2$$

Then introduce new variables

$$\begin{cases} y_1 = x_ 1 + x_2 - x_3 \\ y_2 = x_2 - 2x_3 \\ y_3 = x_3 \end{cases} That is \begin{cases} x_1 = y_ 1 - y_2 - y_3 \\ x_2 = y_2 + 2y_3 \\ x_3 = y_3 \end{cases}$$

The standard form of f is

$$f = 2y_1^2 + y_2^2 - y_3^2$$

Transition matrix is

$$\begin{bmatrix} 1 & -1 & -1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{bmatrix}$$

Reference link:
http://haoeric.github.io/matrix-eigenvectors/
https://www.cnblogs.com/yanghh/p/13492614.html