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1、矩阵的初等变换及其应用（Elementary transformation of matrix and its application）Elementary transformation of matrix and its applicationWang DanElementary transformation of matrix and its applicationAbstractElementary transformation of matrix is an important method of studying matrix, and it is the core of appli
2、cation in linear algebra. This paper introduces some concepts and properties associated with the matrix, on the basis of matrix rank, the basis for judgment matrix is invertible, after inverse matrix equations, eigenvalues and eigenvectors, two types of standard form, and illustrate the application
3、of elementary transformation of matrix in the above is how to play the role of.Keywords: matrix, elementary transformation, applicationThe, elementary, transformation, of, matrix, and, its, applicationsAbstractElementary transformation matrix is an important means of Matrix is the core linear algebr
4、a applications. This article briefly describes some of the concepts and properties associated with the matrix as a basis, the rank of a matrix to determine whether a matrix is reversible after inverse matrix, seeking basic solutions line equations find eigenvalues, and eigenvectors, quadratic standa
5、rd Shape and so on. Illustrate the elementary transformation matrix in the above applications is how to play a role.Keywords:, matrix, elementary, transformation, applicationCatalog1. introduction 62. the related concepts of matrix 72.1 definition of matrix 72.2 transpose of matrix 72.3 elementary t
6、ransformation of matrix and elementary matrix 73. the application of elementary transformation of matrix 83.1, the rank of the matrix 83.2 the inverse matrix of the matrix 103.3 using elementary transformation to solve matrix equation 113.4 find the solution of linear equations 12The conditions for
7、the existence of nonzero solutions of 3.4.1 homogeneous linear equations are 13Conditions for the existence of solutions of 3.4.2 nonhomogeneous linear equations 143.5 find the eigenvalues and eigenvectors of the matrix 153.6, use elementary transformation, two times as standard type 17Summary 19Ref
8、erences 191. introductionIn the course of studying linear algebra, I find that the elementary transformation of matrix is very extensive and runs through the whole chapter. It is the key to solve the problem in linear algebra. Linear equations is the beginning of the elementary transformation matrix
9、, the matrix effect can also be said to be of linear algebra, each knowledge point of linear algebra and linear algebra and matrix are closely related, each in mathematics both can play a role. Biology, economics, physics, cryptography requires knowledge of mathematics, the significance of matrix el
10、ementary transformation of matrix, as can be imagined, is the complex matrix into a simple form is easy to calculate and understand.In real life, many aspects involve the knowledge of matrices,In studying the virtual aircraft model, we will find that the operation of the matrix plays a crucial role.
11、 The plane surface appears to be smooth, but the geometric structure is perplexing, the flow equation is more difficult, must also consider other external factors, but we use the matrix knowledge to be able to solve the problem very well. There are many other applications, for example, matrix eigenv
12、alues and eigenvectors is the key to solve many problems in physics, mechanics and engineering technology; now the game company and Bank Account confidential security, but also the use of matrix theory invented the matrix card; simulation in equipment monitoring system in engineering, radio and tele
13、vision; large screen display works, TV teaching, command and control center etc. mainly used matrix switcher and so on.2. concepts related to matrices2.1 definition of matrixTable is a rectangular matrix. Similar to the cross and the determinant is called a row, called vertical columns, with a line,
14、 the line and the line and the row element matrix for short note.Transpose of the 2.2 matrixLet a matrix be called a matrixFor the transpose of the matrix, rememberElementary transformation and elementary matrix of 2.3 matrices1, the following three transformations called matrices, called the matrix
15、 of the primary row (column) transform, collectively referred to as the elementary transformation of the matrix:(1) the two row (column) of the exchange matrix(2) the elements of a row (column) of a matrix are multiplied by a nonzero constant(3) a constant of the elements of a row (column) of a matr
16、ix added to the corresponding element of another row (column)Elementary row and column transformations are collectively referred to as elementary transformations2. The matrix obtained by elementary transformation of a unit matrix is called elementary matrix.Three types of elementary matrices:(1) ele
17、mentary commutative matrices: the second and the second lines of the commutative unit matrix(2) the elementary multiplied matrix: the row (column) of the unit matrix takes the nonzero constant, i.e.(3) elementary doubly matrix: the first row of a unit matrix is added to the first line, or the first
18、row is multiplied to the next columnIf the matrix is transformed into a matrix by a finite elementary transformation, it is said to be equivalent3, matrix equivalence has the following properties:(1) reflexivity, that is, the self equivalence of any matrix;(2) symmetry, that is, the equivalence of a
19、ny matrix, if and equivalence;(3) transitivity is equivalent to any matrix, and if and equivalence, equivalence, and equivalence;The application of elementary transformation of 3. matrices3.1, the rank of the matrixMany methods for matrix rank, general definition method, elementary transformation me
20、thod, formula method and comprehensive method, but when the specific element of the matrix is known, using elementary transformation method is for non zero row (column) number.The highest order of a nonzero divisor defined in a 3.1.1 matrix is called the rank of a matrix. That is, there is a rank or
21、der of no 0, and all orders of variables (if any) are 0, then the rank of the matrix is (or / or rank)(1)(2) the rank of the zero matrix is 0(3) the rank of a ladder matrix = the number of nonzero rows in a rowTheorem 3.1.1 the elementary transformation of a matrix does not change the rank of a matr
22、ixTheorem 3.1.2 row rank of a matrix = row rank of a matrixTheorem 3.1.3, the equivalent matrices have the same rank, but their inverse is not true, that is, the matrices with the same rank may not be equivalent, and the matrices of the same type and the same rank are equivalent to each otherFind th
23、e rank of a matrix, and give a brief introduction of the most common method:(1) definition method:If the matrix has a nonzero order, and all the sub orders (if any) are all 0, then.If there is a nonzero order in the matrix, and all of the order variables containing this order are 0.The usage of matr
24、ix rank can be calculated with simple formula omit a lot.(2) the number of zero rows in Central Africa is the rank of the matrix.This is because the elementary row transformation does not change the rank of the matrix, in addition, it can be transformed into a column ladder rank by the elementary co
25、lumn transformation, and the elementary transformation can be used as the standard form to obtain the rank.Example 1 find the rank of a matrix.Solution 1: take the 2 order of the upper left of the matrixHowever, there are only 3 lines in the matrix, so it is necessary to find the 3 order of the vari
26、ables contained in the matrix.Solution 2: to do elementary row transformationDue to non-zero behavior 2.It can be seen that the definition method is only suitable for the calculation of simple matrix, but if it is a higher order matrix, it is very inconvenient to calculate.3.2, the inverse matrix of
27、 the matrixThe definition of 3.2.1 is set as a square matrix, if the order matrix existsHere is the rank unit matrix, which is called the invertible matrix, and is called the inverse matrix.Note (1) if it is invertible, its inverse matrix is unique, and the inverse matrix is;(2) the invertible probl
28、em of the matrix is the case of the opponents matrix.Set the invertible matrix of order, and the inverse matrix is as follows:Example 2 is set up as a invertible square matrix, and the resulting matrix is denoted by the following line and column(1) proved to be reversible;(2) seekingProof: (1) since
29、 the left multiplication of the elementary matrix corresponds to the two rows of the interchange, so there isBecause, so the matrix is reversible(2)Example 3 uses the elementary transformation of the matrix to find the inverse matrix of the matrixSolution:soIn short, we in the inverse matrix with el
30、ementary transformation, we must first selected by elementary row transformation or elementary column transformation, note that if using elementary row transformation must be from first to last by elementary row transformation, using elementary column transformation must be from first to last by ele
31、mentary column transformation.But in the inverse does not need to check whether the reversible matrix, elementary transformation can be directly obtained, if the simplest form of a square matrix transform unit is not left after the show, the original matrix is irreversible.3.3 using elementary trans
32、formation to solve matrix equation(1) if it is reversible, then(2) if it is reversible, then(3) if both are reversible, thenFirst of allAgainThis can be obtainedThe matrix equations of type can only be elementary row transformations (on the left); the pair can only be elementary column transformatio
33、ns (on the right)Example 4 solving matrix equationSolution: let the original equation be.therefore3.4 solving the system of linear equationsSet a system of linear equations with unknown quantitiesIts matrix form is,Among them,The coefficient matrix called linear equation is called the augmented matr
34、ix.Conditions for nonzero solutions of 3.4.1 homogeneous linear equations(1) the necessary and sufficient condition for the existence of nonzero solutions of homogeneous linear equations is the rank of the coefficient matrix(2) when the number of equations of a homogeneous linear equation group is l
35、ess than the number of unknown quantities (mn), there must be nonzero solutions(3) if the order matrix is square, the system of equations has nonzero solution(4) if the order matrix is square, then the system of equations has only zero solutionFirst, the coefficient matrix is transformed into a ladd
36、er matrix by using elementary row transformation, and if there is only zero solution, if there is a nonzero solution, it continues to be calculated;The ladder? Matrix to the simplest form, a non zero row non zero element corresponding to the unknown quantity, the unknown amount of free unknown quant
37、ity, revenuer, after making one of a free variable is 1, the remaining 0, basic system of solutions can be obtained.The linear combination of the solutions of the parameters is the general solution of the equationExample 5 solving linear equationsSolution: the coefficient matrix is transformed into
38、the simplest form by elementary row transformationsoThat is, there are 2 free unknownsWith the same set of equationsFor the selection of free unknown, and transferred toThe general solution is()Represented as a vector matrixConditions for the existence of solutions of 3.4.2 nonhomogeneous linear equ
39、ations(1) if the set is a matrix, then the necessary and sufficient condition for the solution of the nonhomogeneous linear equation set is that the rank of the coefficient matrix is equal to the rank of the augmented matrix(2) if the set of nonhomogeneous linear equations is solvable, thenThe solut
40、ion is unique and the second set of equations has only zero solutions.(3) there are infinitely many solutions to the system of nonhomogeneous linear equations(4) the solution of a system of nonhomogeneous linear equations without elementExample 6 for solving nonhomogeneous linear equationsSolution:
41、an elementary row transformation of the augmented matrixThat wasTherefore, the general solution of the original equation set is any constant3.5 find the eigenvalues and eigenvectors of the matrixThe definition of 3.5.1 is a matrix of order, if there exists a number and a zero dimensional column vect
42、or, theThat isSet up is called an eigenvalue of a square matrix, and nonzero column vectors are called eigenvectors of the square corresponding to (or belong to) eigenvaluesThe characteristic polynomial of a 3.5.2 determinant (or) called a matrix (Note: the sub polynomial of a characteristic polynom
43、ial is) is a characteristic equation of a matrix:Let the order matrix be the unit matrix of the order, the eigenvalues of the matrix, and the matrixWith the elementary transformation, the upper triangular matrix can be obtained, and the product of the elements on the principal diagonal of the matrix
44、 is 0The value is the eigenvalue of the matrix.Example 7 uses the elementary transformation method of matrix to find the eigenvalues and eigenvectors of the matrixSolution:The product of the principal diagonal elements of the order is zero, i.e.EigenvalueThenTherefore, the corresponding eigenvectors
45、 areAll the corresponding eigenvectors are.WhenTherefore, the corresponding eigenvectors areThe entire feature vector at this time is.3.6, use the elementary transformation, and the two form is the standard typeTwo order homogeneous polynomials with variablesReferred to as the yuan two times, referr
46、ed to as the two times.Order, rememberThen the two type can be expressed asA matrix of symmetric matrices of two order.When a series of elementary column transformations are applied to a matrix, the same elementary row transformation is applied to the block,When the block diagonal matrixWhen the chi
47、ld blocks are reduced, the. At this point, if the order, then into a standard shapeExamples are 8 and two times as standard.Solution: the quadratic matrix is twoImplementing elementary transformationIn this way, by coordinate transformation, of whichThe two form is a standard shapeNote: two types can be standardized in a variety of ways, and their standard shapes are not unique.Sum upTo solve some problems in algebra when using the elementary matrix transform can simplify the problem, such as th