Let the product of two rectangular matrices be a square matrix.

This will happen if and only if not only the number of columns of the first matrix is ​​equal to the number of rows of the second, but also the number of rows of the first is equal to the number of columns of the second:

In this situation, the following theorem, called the Binet-Cauchy theorem, holds.

Theorem 3. The determinant of the matrix AB is equal to zero if , and is equal to the sum products of all minors of the order of matrix A by the corresponding minors of the mth order of matrix B, if .

The correspondence of minors is understood here in the following sense: the numbers of the columns of the matrix A that make up the minor coincide with the numbers of the rows of the matrix B that make up the corresponding minor.

In formula notation:

where - minor of matrix A, composed of columns with numbers - minor of matrix B, composed of rows with numbers

The Binet-Cauchy theorem can be proved in a similar way to the proof of the theorem on the determinant of the product of two square matrices (which, of course, is special case Binet-Cauchy theorems). However, in this case one would have to use Laplace's theorem in a general formulation.

We give a proof based on another idea. Let's write down in detail

Now apply the linearity property of the determinant to the first column. Get

where all determinants have columns, starting from the second one, the same as in the original form.

Let us now apply the linearity property to the second columns of the determinants that make up this sum. Get

where the indices run independently of the values ​​of . Here, all determinants have columns, starting from the third one, the same as in the original form.

In the same way, we continue the decomposition of the determinant into the sum of determinants, applying the linearity property to the third columns. We get as a result

where the indices take independently from each other all values ​​from 1 to . There are only terms here. Take out the common factor from each column. Get

If then the indices will be “so crowded” that among their values ​​there will be at least one pair of equals. But then all the determinants included in the terms will be equal to zero as having equal columns. Therefore, at .

Let now If among the values ​​of the indices there is at least one pair of equals, then the corresponding term is equal to zero. All such terms can be discarded and the sum extended to pairwise distinct values ​​of the indices remains. Sets of such values ​​may differ both in the composition of values ​​and in the order, if the composition is the same. Such sets are called placements. Let us denote by the set of values ​​of the indices it, arranged in ascending order: so that with the same composition, the values ​​of the indices will form permutations of the elements

We first carry out the summation over all possible sets of the same composition, i.e., over the permutations of the elements, and then add the resulting sums over the possible compositions.

where in the inner sum the summation is carried out over all sets of components of the permutation of numbers within the inner sum, the determinants differ only in the order of the columns. Putting the columns in ascending order of index values, we get:

All terms of the inner sum include the same determinant as a factor. It can be taken out of the sum sign

After taking the minor of matrix A out of the sign of the internal sum, a precious legacy in the form of a factor remains, the presence of which allows us to conclude that the internal sum is equal to the determinant

Indeed, it is the sum of all possible products of the elements of the matrix of this determinant, taken one from each row (after all, (axis ) runs through all possible permutations of numbers) and one from each column.

Theorem (Cauchy-Binet formula)

Let, - and -matrices, respectively, and

In other words, for , the determinant of a matrix is ​​the sum of the products of all possible minors of order in and the corresponding minors of a matrix of the same order.

Exercise 1. Let's show with an example

Let, and, then according to the Cauchy-Binet formula:

Proof of the theorem:

Since, it is possible to write

The determinant is an additive and homogeneous function of each of its columns. Using this fact for each of the columns in, we express as a sum of determinants:

Those terms in the summation that have two or more matching indices are zero, since in these cases the minors will have at least two matching columns. Thus, it is necessary to consider only those summation terms in which the indices are different. We distribute these remaining members into groups of members each in such a way that in each group the members differ only in the order of the indices. Note also that we can write

where. Therefore, the sum over the terms in which is a permutation of numbers is given by the expression:

Rearranging the elements so that the first indices are in ascending order, we bring this expression to the form:

where is a permutation of numbers, as is obvious. It now follows from the determinant of the determinant function that this expression is simply:

Consequence. Determinant of the product of two multiple matrices is equal to the product multiplier determinants.

This follows from the Theorem for

Federal Agency for Education

Murmansk State Pedagogical University

Faculty of Applied Mathematics, Programming and Economics

Department of Algebra, Geometry and Applied Mathematics

Course work

Determinant of product of rectangular matrices.

Cauchy-Binet theorem.

Completed by a student

II course groupPMI

Reshotkina Natalia Nikolaevna

Scientific adviser:

PhD in Physics and Mathematics

Sciences, Associate Professor of the Department of AG and PM

Mostovskoy Alexander Pavlovich

Murmansk

TOCo "1-3" h z u Introduction. PAGEREF _Toc169771091 h 4

Chapter I. PAGEREF _Toc169771092 h 5

§ 1 Definition, notation and types of matrices. PAGEREF _Toc169771093 h 5

Properties of addition and multiplication of matrices by scalars: PAGEREF _Toc169771094 h 7

Chapter II. PAGEREF _Toc169771095 h 7

§1 Matrix multiplication. PAGEREF _Toc169771096 h 7

§2 Properties of matrix multiplication. PAGEREF _Toc169771097 h 8

§3 Technique of matrix multiplication. PAGEREF _Toc169771098 h 9

§4 Transposition of the product of matrices. PAGEREF _Toc169771099 h 10

Chapter III. PAGEREF _Toc169771100 h 10

§1 Invertible matrices… PAGEREF _Toc169771101 h 10

§2 Elementary matrices… PAGEREF _Toc169771102 h 12

Chapter IV… PAGEREF _Toc169771103 h 13

§1 Determinants. PAGEREF _Toc169771104 h 13

§2 The simplest properties of determinants. PAGEREF _Toc169771105 h 14

§3 Basic properties of determinants. PAGEREF _Toc169771106 h 14

§4 Minors and algebraic additions. PAGEREF _Toc169771107 h 18

Theorems about determinants. PAGEREF _Toc169771108 h 18

§5 Determinant of product of matrices. PAGEREF _Toc169771109 h 21

Necessary and sufficient conditions for the determinant to be equal to zero ... PAGEREF _Toc169771110 h 22

§6 Matrix partitioning. PAGEREF _Toc169771111 h 23

§7 Theorem (Binet-Cauchy formula) PAGEREF _Toc169771112 h 25

Conclusion. PAGEREF _Toc169771113 h 28

Literature. PAGEREF _Toc169771114 h 30

Appendix. PAGEREF _Toc169771115 h 31

Introduction

When solving various problems of mathematics, one often has to deal with tables of numbers called matrices. With the help of matrices, it is convenient to solve systems of linear equations, perform many operations with vectors, solve various computer graphics problems and other engineering problems.

The purpose of this work: theoretical justification and the need for practical application of the Cauchy-Binet theorem:

Let , - And -matrices, respectively,

Then

In other words, when matrix determinant is the sum of products of all possible order minors in to the corresponding matrix minors the same order

The work consists of four chapters, contains a conclusion, a list of references and an application program for the Cauchy-Binet theorem. Chapter I discusses the elements linear algebra– matrices, operations on matrices and properties of matrix addition, and multiplication by a scalar. Chapter II is devoted to the multiplication of matrices and its properties, as well as the transposition of the product of two matrices. Chapter III deals with invertible and elementary matrices. Chapter IV introduces the concept of the determinant of a square matrix, discusses properties and theorems about determinants, and also provides a proof of the Cauchy-Binet theorem, which is the purpose of my work. In addition, a program is attached showing the mechanism for finding the determinant of the product of two matrices.

Chapter I

§ 1 Definition, notation and types of matrices

We define a matrix as a rectangular table of numbers:

Where the elements of the matrix aij(1≤i≤m, 1≤j≤n) are numbers from the field .For our field purposes is either the set of all real numbers or the set of all complex numbers. Matrix size , where m is the number of rows, n is the number of columns. If m=n, then the matrix is ​​said to be square, of order n. In general, a matrix is ​​called a rectangular matrix.

Each matrix with elements aij corresponds to an n×m matrix with elements aji. It is called transposed to and is denoted by =. Matrix rows become columns in and matrix columns become strings in

A matrix is ​​called null if all elements are 0:

A matrix is ​​called triangular if all its elements below the main diagonal are 0

A triangular matrix is ​​called diagonal if all elements outside the main diagonal are 0

A diagonal matrix is ​​said to be identity if all elements on the main diagonal are equal to 1

Matrix composed of elements located at the intersection of several selected rows of the matrix and several selected columns, is called a submatrix for the matrix

In particular, the rows and columns of a matrix can be considered as its submatrices.

§2Operations on matrices

We define the following operations:

I.

sum of two matrices with elements And matrix C with elements

II.

Matrix product per number

III.

Work matrices matrix C with elements

IV.

field of scalars, consider matrices over the field

Def. Two matrices are equal if they have the same dimension and have the same elements in the same places. In other words: is equal to the matrix

Def.Let And called column located element

Def.Let to matrix called in which column located element multiply by matrix need all the elements of the matrix multiply by a scalar

Definition.Opposite to matrix called matrix

Properties of addition and multiplication of matrices by scalars:

1) Matrix addition associative and commutative.

2)

3)

but)

b)

4)

Chapter II§1 Matrix Multiplication

Def.Product matrices on the matrix called matrix

They say that is the scalar product on the

§2 Properties of matrix multiplication

1.

Matrix multiplication is associative:

1) And

Proof:

Let and defined

We define matrices:

but)

b)

(1) matrices, then have the same dimension

2) Let us show that at the same places in the matrices identical elements are located

Conclusion:Matrices have the same dimensions and have the same elements in the same places.

2.

Matrix multiplication is distributive

Proof:

as defined and defined

dimensions

matrices have the same dimension, we show the location of the same elements:

Conclusion: The same elements are located in the same places.

3. matrices, then the proof is similar to property 2.

4.

Proof:

5. Matrix multiplication is generally not commutative. Let's look at this with an example:

§3 Matrix multiplication technique

scalar field,

Properties:

1)

Work can be considered as the result of multiplying the columns of the matrix on the left and as a result of multiplying the rows of the matrix on the on right.

2)

Let matrix

Let whose coefficients are the elements of the matrix

3)

Matrix columns §4 Transposition of the product of matrices

scalar field,

if

Proof:

1) Let

- dimensions

2)those

on the column

Chapter III§1 Invertible matrices

field of scalars, set

Definition.Square matrix order is called the identity matrix

Let

Theorem 1

performed

Proof:

Therefore is the identity matrix. It plays the role of a unit in matrix multiplication.

Definition. square matrix so that the conditions are met

Matrix called inverse to denoted inverse to

Theorem 2

If

Proof:

Let the matrix those.

Notation: The set of all invertible order matrices over the field denoted

Theorem 3

Fair statements:

1)algebra

2)Group

Proof:

a) Let

back to

Similarly: invertible matrix i.e.

b)

v) reversible i.e.

2) Let us prove the second assertion that Group. To do this, we check the group axioms:

1)

2)

3)

Group

Consequence:

1)

The product of invertible matrices is an invertible matrix

2)

If reversible, then

3)

4)

§2 Elementary matrices

Let field of scalars

Definition. An elementary matrix is ​​a matrix obtained from an identity matrix as a result of one of the following elementary transformations:

1)

Row (column) multiplication to a scalar

2)

Adding to any row (column) another row(column) multiplied by a scalar

Designation:

Example: Elementary matrices of order 2

Designation:

Chapter IV§1 Qualifiers

Matrix determinant multiplied by the sign of the corresponding substitution.

The second order determinant is equal to the product of the elements of the main diagonal subtract the product of the elements on the side diagonal.

For

We got the triangle rule:

SHAPE*MERGEFORMAT

§2 The simplest properties of determinants

1)

The determinant of a matrix with a zero row (column) is zero

2)

The determinant of a triangular matrix is ​​equal to the product of the elements located on the main diagonal

The determinant of a diagonal matrix is ​​equal to the product of the elements located on the main diagonal. Matrix diagonal if all elements located outside the main diagonal are equal to zero.

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Example

and the corresponding minors have the form

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The Binet-Cauchy formula in this case gives the equality