Linear Algebra: Chapter 2 determining factor 1

Chapter 2 determining factors

§ 1 Introduction

Solving equations is a basic problem in algebra. Especially in the algebra learned by Middle School, solving equations plays an important role. This chapter and the next chapter mainly discuss the General Multivariate equations, that is, linear equations.

The theory of linear equations is basic and important in mathematics.

For Binary Linear Equations

At that time, the equations had a unique solution:

We call it a second-level determinant, represented by symbols

.

Therefore, the above solution can be described as follows using the second-level determinant:

When the secondary Determinant

The system has a unique solution, that is

.

There is a similar conclusion for the ternary linear equations.

It is called an algebraic form as a three-level determinant, expressed:

.

When the third-level Determinant

The above three-element linear equations have a unique solution.

Where

.

In this chapter, we will extend this result to the linear equations.

For this reason, we first give the definition of the level-level determinant and discuss its nature. This is the main content of this chapter.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 Arrangement

I. Definition of Arrangement

Definition 1An ordered array is called a hierarchical arrangement.

It is also a level arrangement, which is arranged in a natural order in ascending order; other arrangements are more or less broken by the natural order.

Definition 2In an arrangement, if the front and back positions of a pair of numbers are in the opposite order of size, that is, the front number is greater than the back number, they are called a reverse order, the total number of reverse orders in an arrangement is called the number of reverse orders in this arrangement.

Sort the numbers in reverse order

Definition 3An even-number sort is called an even-number sort; an odd-number sort is called an odd sort.

It should be pointed out that we can also consider the arrangement composed of any different natural numbers, which is also called the level arrangement. For such a general level arrangement, we can also define the above concepts.

Ii. parity of Arrangement

Swaps the positions of one or two numbers in one arrangement, while the remaining numbers do not move. such a transformation is called a swap. obviously, if we perform the same Swap again consecutively, the order will be restored. from this we know that one swap will arrange all levels in pairs, so that each of the two pairs is arranged in the order of this pair.

Theorem 1The parity of changing the arrangement.

That is to say, after a swap, the odd arrangement becomes an even arrangement, and the even arrangement becomes an odd arrangement.

InferenceIn an all-level arrangement, the numbers of odd and even arrays are equal and each has its own number.

Theorem 2Any level of arrangement and arrangement can go through a series of exchange changes, and the number of exchanges has the same parity with this arrangement.

 

 

 

 

§ Level 3 Determinant

I. Concept of level determining

Before defining the level-3 determinant, let's take a look at the definition of level-2 and level-3 determinant.

, (1)

(2)

It can be seen from the definition of level 2 and level 3 determine that they are all product algebra and each product is composed of elements in the determinant located in different rows and different columns, and the expansion is exactly the product of all such possibilities. on the other hand, each product has a symbol. what principles does this symbol determine? In the expansion (2) of the third-level determinant, the general form of the item can be written

, (3)

It is an arrangement of 1, 2, 3. It can be seen that when it is an even arrangement, the corresponding items carry a positive number in (2), and when it is an odd arrangement, it carries a negative number.

Definition 4
Level determining factor

(4)

Equal to the product of all elements taken from different columns in different rows.

(5)

Algebra and, here is an arrangement, each item (5) is signed according to the following rules; when it is an even arrangement, (5) with a positive number, when it is an odd arrangement, (5) with a negative number. this definition can be written

, (6)

Sum of all levels.

The definition indicates that, in order to calculate the level determining factor, first make all the products that may be composed of different columns in different rows. the elements that constitute these products are arranged in a natural order by row indicators, and then determined by the parity of the column indicators.

It can be seen from the definition that the level-level determinant is composed of items.

Example 1Calculate the Determinant

.

Example 2Calculate the top triangle Determinant

. (7)

. (8)

This determinant is equal to the product of elements on the main diagonal (from the upper left to the lower right corner. in particular, all elements except the main diagonal line are zero. the value of the diagonal determine is equal to the product of the elements on the main diagonal.

It is easy to see that when the element of the determining factor is the number in the number field, its value is also a number in the number field.

Ii. Nature of the determinant

In the definition of the determinant, to determine the plus and minus signs of each item, sort the elements by row. in fact, the multiplication of numbers is an exchange, so the order of this element can be written at will. Generally, the items in the Level determine can be written

, (11)

There are two levels of arrangement. It is not difficult to prove that (11) is equal

. (12)

The advantage of determining the symbol of each item by (12) is that the row index and column indicator are symmetric, so in order to determine the symbol of each item, you can also sort each item by column, so the definition can be written

. (15)

Thus, the following attributes of the determinant are obtained:

Nature 1Transpose rows and columns.

. (16)

Property 1 indicates that the row and column in the determinant are symmetric, because of the nature of the relevant row, the column is also true.
For example, the determinant of the bottom triangle is obtained from (8 ).

.

 

 

 

 

 

 

 

 

 

§ Properties of Level 4 deciding factors

The Calculation of determining factors is an important and complex issue. there are a total of items in the level determining factor, and a multiplication is required to calculate it. when it is large, it is a considerable number. it is almost impossible to calculate the determinant directly from the definition. therefore, it is necessary to further discuss the nature of the determinant. using these properties, we can simplify the calculation of the determinant.

In the definition of the determinant, although each item is the product of an element, this element is taken from different rows and columns, so for a certain Row Element (for example) each item contains one and only one element. as a result, the items of the level-level determinant can be divided into groups, the items in the first group contain, the items in the second group contain, and so on. put forward the elements of the rows separately.

(1)

It represents the algebraic sum of the items contained after the proposed public factor. as to which items are irrelevant for the time being, we will discuss them in section 6. from the above discussion, we can know that there is no longer any element of the first row, that is, it is completely irrelevant to the element of the first row in the determinant. this is the result.

Nature 2

That is to say, the common factor of a row can be raised, or a row of a single number multiplied by the determinant is equivalent to using this number to multiply the determinant.

In this case, if one behavior in the determinant is zero, then the determinant is zero.

Nature 3

.

That is to say, if a row is the sum of two sets of numbers, then this determinant is equal to the sum of the two determine, and the two determine, in addition to this row, are all the same as the corresponding row of the original determine.

Nature 3 obviously can be promoted to a multi-group sum of a row.

Nature 4If two rows are the same in the determinant, the determinant is zero. The two rows are the same, that is, the corresponding elements of the two rows are equal.

Nature 5If the two rows in the determinant are proportional, the determinant is zero.

Nature 6Add a multiple of a row to another row without changing the determinant.

Nature 7Transpose the location of the two rows in the determinant.

Example 1Computing level determining factor

Example 2Calculate the Determinant

.

As the row and column rings make it easy to calculate the top and bottom triangles, a basic method to calculate the deciding factor is to convert the deciding factor into the top and bottom triangles for calculation.

Example 3A level determining factor, assuming that its elements meet

, (4)

It is proved that when the value is an odd number, This determinant is zero.

 

 

 

§ 5 Calculation of the determining factor

The following uses the nature of the determinant to give a method to calculate the determinant.

In section 3, we can see that a top triangle has a determining factor.

It is equal to the product of elements on the main diagonal line.

This computation is very simple. Next we will find a way to convert any level-based deciding factor into the top-triangle deciding factor for computation.

Definition 5Table of rows (horizontal) columns (vertical) arranged by number

(1)

It is called a matrix.

Number, called the element of matrix (1), is called the row indicator of the element, and is called the column indicator. when the elements of a matrix are all numbers in a certain number field, it is called a matrix in this number field.

A matrix is also called a matrix.

Define a level determining factor

It is called the determining factor of the matrix and is recorded.

Definition 6In the so-called number field, the primary Row Transformation of the matrix refers to the following three transformations:

1) a row of a non-zero number multiplication matrix;

2) Add the double of a row in the Matrix to another row, which is any number in it;

3) Position of the two rows in the interchange matrix.

In general, after a matrix is transformed by primary rows, it becomes another matrix. When the matrix is transformed into a matrix by primary rows, we write

If any row of a matrix starts from the first element to the bottom of the first non-zero element of the row, this matrix is called a step matrix.

It can be proved that any matrix can be converted into a step matrix after a series of Elementary Line transformations.

Now I will discuss the calculation of the determinant. A level determining factor can be viewed as determined by a level matrix. The matrix can be transformed by primary rows, and the nature of the determining factor is 2, 6, 7 illustrates the influence of the primary Row Transformation of the square matrix on the value of the determinant. each square matrix can be transformed into a step-shaped square matrix through a series of elementary rows. each Elementary Line Transformation of the matrix is performed for the, 7 by the nature of the determining factor. Correspondingly, the determining factor or constant, or a non-zero multiple of the difference, that is

Obviously, the determining factors of the trapezoid matrix are all of the top triangles, so they are easy to calculate.

ExampleComputing

It is not difficult to calculate. to use this method to calculate a level of numerical determinant, we only need to perform the multiplication and division. especially when it is relatively large, the superiority of this method is even more obvious. at the same time, we should also see that this method is completely mechanical, so we can use the electronic computer to calculate the determinant according to this method.

For a matrix, you can also define an elementary column transformation, that is

1) a column in a non-zero multiplication matrix;

2) Add the times of a column in the Matrix to another column, which is any number in the column;

3) Position of the two columns in the interchange matrix.

In order to calculate the deciding factor, we can also perform primary column transformation on the matrix. Sometimes, with primary Row Transformation and column transformation, the calculation of the determining factor can be simpler.

The primary Row Transformation and primary column transformation of the matrix are collectively referred to as primary transformations.