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Eigenvalues and eigenvectors and their application cases

Author：Wang Rong, Liao Xiaolian

Nationality：U.S.A

Source：Eigenvalues and eigenvectors and their applications [J]. Education Modernization, 2018,5 (27): 258-261.

Absrtact: Eigenvalues and eigenvectors are one of the important problems in Higher Algebra research, and also an important tool in mathematical research and application. They are also widely used in other academic fields and life. In this paper, we will use the knowledge of eigenvalues and eigenvectors to discuss their applications in physical mechanics, biology, the relationship between economic development and the growth of environmental pollution, and the stability of weather. Key words: eigenvalues; eigenvectors; application case citation format: Wang Rong, et al.

Eigenvalues and eigenvectors are basic concepts in Higher Algebra courses: let A be a linear transformation of linear space V over the number field P. If there exists a non-zero vector_in the number field P so that A_=_0_, then a eigenvalue of A is called a eigenvalue of A, and a eigenvector of A belongs to the eigenvalue of_0 [1]. For any eigenvalue lambda 0 of linear transformation A, the set of all suitable vectors a of condition A=lambda 0a, that is, the set of all eigenvectors of A belonging to lambda 0 and adding zero vectors, is a subspace of V, called a characteristic subspace of A, denoted as Vlambda 0. Eigenvalues and eigenvectors have many special properties, such as: eigenvectors belonging to different eigenvalues are linearly independent; similar matrices have the same eigenvalue polynomial [1]. A N application case of eigenvalues and eigenvectors (1) Application case of eigenvalues and eigenvectors in mathematics 1. Use eigenvalues and eigenvectors to judge whether linear transformation can be diagonalized lemma 2.1 [2] Let_be a linear transformation on n-dimensional linear space V, where A is a matrix of linear transformation_under a set of bases in V, all of which are not_1, _2,... And_n. The same eigenvalue. The necessary and sufficient conditions for the diagonalization of_are as follows: (1) A has n linearly independent eigenvectors; (2) There exists a set of bases composed of eigenvectors of_in V; (3) the sum of dimensions of eigenspace of_belonging to different eigenvalues is equal to n; (4) V = Vlambda 1 Vlambda 2 * Vlambda n. _diagonalization means that matrix A can be diagonalized, or there exists an invertible matrix P, so that P-1AP = D, where D = diag {lambda 1, lambda 2,... Lambda n}. Example 1 sets up a matrix.

Try to determine whether matrix A can be diagonalized; if A can be diagonalized, please diagonalize A.

Solution: as a result of

So the eigenvalues of A are 1,1,2. Thus the eigenvectors corresponding to eigenvalue-2 are, and the two linearly independent eigenvectors corresponding to eigenvalue-1 are,. So A can be diagonalized.

order

Then there are

1. Using eigenvalues and eigenvectors to find the general terms of sequence is a particularly important content in the problem of sequence. Exploring the method of finding the general terms of sequence is also a hot issue in middle school mathematics. Next, we use eigenvalues and eigenvectors to give an example of a recursive method for solving general terms of sequence. Example 2 assumes that the sequence {xn} satisfies the conditions: xn = 2xn-1 xn-2-2xn-3, where n gt; 5, and X1 = 1, x2=-2, X3 = 3. Try to find the general term xn of the sequence.

Solution: Construct a system of linear equations:

Written in matrix form as follows:

，

Be

Next, the eigenvalues and eigenvectors of matrix A are obtained. From

So the eigenvalue of A is:the eigenvectors corresponding to the three eigenvalues are:

order

be

thus

By calculating and substituting the values ，the general term of the sequence is

(2) Application of Eigenvalues and Eigenvectors in Other Disciplines 1. Leslie Population Model in Applied Biology of Eigenvalues and Eigenvectors [3], often used to study dynamics

The relationship between the age distribution of females in a population and the increase of the total number of females is another practical application of eigenvalues and eigenvectors. Case 3: The maximum survival age of females in an animal population is 15 years, and the females are divided into three age groups [0, 5], [5, 10], [10, 15]. Statistical data show that the fertility rates of females in three age groups are 0, 4, 3, the survival rates are 0.5, 0.25 and 0, and the number of females in three age groups at the initial time is 500, 1000, respectively. 500. The Leslie population model was used to analyze the age distribution and quantity growth of female animals in this animal population. Solution: The Max survival age of animals is L = 15, A1 is the fertility rate of age group I, B1 is the survival rate (that is, the ratio of the number of females in age group I 1 to the total number of females in age group I), Xi (k) is the number of females in age group I of the animal population at time TK [4]. By the meaning of the question, there are A1 = 0, A2 = 4, A3 = 3, B1 = 0.5, B2 = 0.25, B3 = 0, n = 3.

Thus, in the initial state, the age distribution of females in the population tends to be stable after a certain period of time. The ratio of the number of females in the three age groups is the same. At this time, the number of females in the three age groups of the population is respectively.

Level will affect the quality of the environment, on

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Eigenvalues and eigenvectors and their application cases

Author：Wang Rong, Liao Xiaolian

Nationality：U.S.A

Source：Eigenvalues and eigenvectors and their applications [J]. Education Modernization, 2018,5 (27): 258-261.

Absrtact: Eigenvalues and eigenvectors are one of the important problems in Higher Algebra research, and also an important tool in mathematical research and application. They are also widely used in other academic fields and life. In this paper, we will use the knowledge of eigenvalues and eigenvectors to discuss their applications in physical mechanics, biology, the relationship between economic development and the growth of environmental pollution, and the stability of weather. Key words: eigenvalues; eigenvectors; application case citation format: Wang Rong, et al.

Eigenvalues and eigenvectors are basic concepts in Higher Algebra courses: let A be a linear transformation of linear space V over the number field P. If there exists a non-zero vector_in the number field P so that A_=_0_, then a eigenvalue of A is called a eigenvalue of A, and a eigenvector of A belongs to the eigenvalue of_0 [1]. For any eigenvalue lambda 0 of linear transformation A, the set of all suitable vectors a of condition A=lambda 0a, that is, the set of all eigenvectors of A belonging to lambda 0 and adding zero vectors, is a subspace of V, called a characteristic subspace of A, denoted as Vlambda 0. Eigenvalues and eigenvectors have many special properties, such as: eigenvectors belonging to different eigenvalues are linearly independent; similar matrices have the same eigenvalue polynomial [1]. A N application case of eigenvalues and eigenvectors (1) Application case of eigenvalues and eigenvectors in mathematics 1. Use eigenvalues and eigenvectors to judge whether linear transformation can be diagonalized lemma 2.1 [2] Let_be a linear transformation on n-dimensional linear space V, where A is a matrix of linear transformation_under a set of bases in V, all of which are not_1, _2,... And_n. The same eigenvalue. The necessary and sufficient conditions for the diagonalization of_are as follows: (1) A has n linearly independent eigenvectors; (2) There exists a set of bases composed of eigenvectors of_in V; (3) the sum of dimensions of eigenspace of_belonging to different eigenvalues is equal to n; (4) V = Vlambda 1 Vlambda 2 * Vlambda n. _diagonalization means that matrix A can be diagonalized, or there exists an invertible matrix P, so that P-1AP = D, where D = diag {lambda 1, lambda 2,... Lambda n}. Example 1 sets up a matrix.

Try to determine whether matrix A can be diagonalized; if A can be diagonalized, please diagonalize A.

Solution: as a result of

So the eigenvalues of A are 1,1,2. Thus the eigenvectors corresponding to eigenvalue-2 are, and the two linearly independent eigenvectors corresponding to eigenvalue-1 are,. So A can be diagonalized.

order

Then there are

1. Using eigenvalues and eigenvectors to find the general terms of sequence is a particularly important content in the problem of sequence. Exploring the method of finding the general terms of sequence is also a hot issue in middle school mathematics. Next, we use eigenvalues and eigenvectors to give an example of a recursive method for solving general terms of sequence. Example 2 assumes that the sequence {xn} satisfies the conditions: xn = 2xn-1 xn-2-2xn-3, where n gt; 5, and X1 = 1, x2=-2, X3 = 3. Try to find the general term xn of the sequence.

Solution: Construct a system of linear equations:

Written in matrix form as follows:

，

Be

Next, the eigenvalues and eigenvectors of matrix A are obtained. From

So the eigenvalue of A is:the eigenvectors corresponding to the three eigenvalues are:

order

be

thus

By calculating and substituting the values ，the general term of the sequence is

(2) Application of Eigenvalues and Eigenvectors in Other Disciplines 1. Leslie Population Model in Applied Biology of Eigenvalues and Eigenvectors [3], often used to study dynamics

The relationship between the age distribution of females in a population and the increase of the total number of females is another practical application of eigenvalues and eigenvectors. Case 3: The maximum survival age of females in an animal population is 15 years, and the females are divided into three age groups [0, 5], [5, 10], [10, 15]. Statistical data show that the fertility rates of females in three age groups are 0, 4, 3, the survival rates are 0.5, 0.25 and 0, and the number of females in three age groups at the initial time is 500, 1000, respectively. 500. The Leslie population model was used to analyze the age distribution and quantity growth of female animals in this animal population. Solution: The Max survival age of animals is L = 15, A1 is the fertility rate of age group I, B1 is the survival rate (that is, the ratio of the number of females in age group I 1 to the total number of females in age group I), Xi (k) is the number of females in age group I of the animal population at time TK [4]. By the meaning of the question, there are A1 = 0, A2 = 4, A3 = 3, B1 = 0.5, B2 = 0.25, B3 = 0, n = 3.

Thus, in the initial state, the age distribution of females in the population tends to be stable after a certain period of time. The ratio of the number of females in the three age groups is the same. At this time, the number of females in the three age groups of the population is respectively.

Level will affect the quality of the environment, on the other hand, the environment will also restrict economic development. Generally speaking, if xt.yt is the level of environmental pollution and economic development in the region after t years, then the growth model of economic

令

development and environmental pollution is established.

Then the matrix form of t

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