Formation of a solid image matrix with a separate perception of elements of an integrated object. The formation of a solid image matrix with a separate perception of elements of an integrated object What is the image of the matrix

Definition 1. The linear operator A is called the set of all elements representing in the form where.

The image of the linear operator A is a linear subspace of space. Its dimension is called rank operator BUT.

Definition 2.The kernel of the linear operator A is called the set of all vectors for which.

The core is a linear subspace of the space X. Its dimension is called defect operator BUT.

If the operator A acts in-dimensional space x, then the following ratio is + \u003d.

Operator A called non-degenerateif his kernel. The rank of a non-degenerate operator is equal to the dimension of the space X.

Let the matrix of the linear transformation and the space X in some basis, then the coordinates of the image and the pre-edge are associated with the relationship

Therefore, the coordinates of any vector satisfy the system of equations

It follows that the kernel of the linear operator is a linear shell of the fundamental solution system of this system.

Tasks

1. Prove that the rank of the operator is equal to the rank of its matrix in an arbitrary basis.

Calculate the kernels of linear operators specified in some basis of the space x the next matrices:

5. Prove that.

Calculate the rank and defect of the operators specified by the following matrices:

6. . 7. . 8. .

3. Own vectors and own values \u200b\u200bof a linear operator

Consider a linear operator A, acting in a measuring space X.

Definition. The number L is called its own value of the operator A, if, such that. In this case, the vector is called its own vector of operator A.

The most important property of its own vectors of a linear operator is that its own vectors corresponding to pairs of various eigenvalues linearly independent.

If it is a matrix of a linear operator A in the basis of the space x, then eigenvalues \u200b\u200bL and its own vectors of the operator A are determined as follows:

1. Own values \u200b\u200bare found as the roots of the characteristic equation (algebraic equation):

2. The coordinates of all linearly independent eigenvectors corresponding to each individual eigenvalue are obtained by solving a system of homogeneous linear equations:

the matrix of which has rank. The fundamental solutions of this system are the vector - columns from the coordinates of their own vectors.

The roots of the characteristic equation are also called eigenvalues \u200b\u200bof the matrix, and system solutions - the matrix's own vectors.

Example.Find your own self-implementing values \u200b\u200bof the operator A specified in some base matrix

1. To determine eigenvalues, and solve the characteristic equation:

Hence your own value, its multiplicity.

2. To determine your own vectors and solve the system of equations:

Equivalent system of basic equations has the form

Therefore, any own vector is a column vector, where C is an arbitrary constant.

3.1. The operator of a simple structure.

Definition. The linear operator A, acting in N-dimensional space, is called a simple structure operator if exactly N linearly independent eigenvectors corresponds to it. In this case, you can construct the basis of space from the operator's own vectors, in which the operator's matrix has the simplest diagonal view.

where - the eigenvalues \u200b\u200bof the operator. Obviously, it is true: if in a certain base of the space X, the operator's matrix has a diagonal view, the basis consists of its own vectors of the operator.

The linear operator A is the operator of a simple structure if and only if each eigenvalue of multiplicity corresponds to exactly linearly independent eigenvectors. Since its own vectors have solutions of a system of equations that, therefore, each root of the characteristic equation of multiplicity must match the grade matrix.

Any sizes matrix corresponding to the operator of a simple structure, similar to a diagonal matrix

where the matrix of the transition T from the original basis for the base from its own vectors has its own columns of the vector column from the coordinates of the eigenvectric vectors of the matrix (operator A).

Example.Lead a matrix of a linear operator to a diagonal view

We comprise a characteristic equation and find its roots.

Where does your own values \u200b\u200bof multiplicity and multiplicity.

The first one's own meaning. It corresponds to its own vectors whose coordinates are

system solution

The rank of this system is 3, so there is only one independent solution, for example, vector.

Own vectors, corresponding, are determined by the system of equations

the rank of which is 1 and, therefore, there are three linearly independent solutions, for example,

Thus, each own value of multiplicity corresponds to exactly linearly independent eigenvectors and, therefore, the operator is a simple structure operator. The transition matrix T has a view

and the relationship between similar matrices is determined by the relation

Tasks

Find your own vectors and eigenvalues

linear operators specified in some base matrices:

Determine which of the following linear operators can be brought to a diagonal type by switching to a new basis. Find this basis and the corresponding matrix:

10. Prove that the eigenvectors of the linear operator corresponding to different eigenvalues \u200b\u200bare linearly independent.

11. Prove that if the linear operator A acting in, has ns of different values, then any linear operator in the permutation with A, has the basis of eigenvectors, and any own vector A will be own and for V.

Invariant subspaces

Definition 1.. Subspace L. linear space X is called invariant relative to the operator A acting in x, if its image also belongs for each vector.

The main properties of invariant subspaces are determined by the following ratios:

1. If the invariant subspaces relative to the operator A, then their sum and intersection are also invariant with respect to the operator A.

2. If the space x is decomposed into the direct sum of subspaces and () and invariantly relative to A, then the operator's matrix in the base, which is the combination of bases and is a block matrix

where - square matrices0 - zero matrix.

3. In any invariant relative to the operator, a subspace operator has at least one own vector.

Example 1.Consider the kernel of some operator A acting in X. by definition. Let be . Then, since the zero vector is contained in every linear subspace. Consequently, the kernel is invariant relative to the subspace.

Example 2.Let in some framework of the space X operator A is defined by the matrix are determined by the equation and

5. Prove that any subspace invariant with a relatively non-degenerate operator A will be invariant and relative to the reverse operator.

6. Let the linear conversion of A-dimensional space in the base has a diagonal matrix with various elements on the diagonal. Find all subspaces invariant with respect to A, and determine their number.

IN vector space V. over an arbitrary field P. set linear operator 

Definition9.8. Nucleus Linear operator  called many space vectors V. The way of which is zero vector. Adopted designation for this set: Ker., i.e.

Ker. = {x. | (h.) = o.}.

Theorem 9.7. The kernel of the linear operator is a subspace of space V..

Definition 9.9. Dimension the core of the linear operator is called defect linear operator. dim Ker. = d..

Definition 9.10.Waylinear operator  called many images space vectors V. . Designation for this set IM., i.e. IM. = {(h.) | h. V.}.

Theorem 9.8. Form linear operator is a subspace of space V..

Definition 9.11. Dimension the image of the linear operator is called rank linear operator. dim IM. = r..

Theorem 9.9. Space V. It is the direct amount of the kernel and the image of the linear operator specified in it. The sum of rank and defect linear operator is equal to the dimension of space V..

Example 9.3. 1) In space R.[x.] ( 3) find the rank and defect operator differentiation. We find those polynomials, the derivative of which is zero. These are polynomials of zero degree, therefore, Ker. = {f. | f. = c.) I. d.\u003d 1. Derivatives of polynomials, the degree of which does not exceed three, form a plurality of polynomials, the degree of which does not exceed two, therefore, IM. = R.[x.] ( 2) and r. = 3.

2) If linear the operator is set by the matrix M.(), then to find his kernel should be solved equation  ( h.) = aboutwhich in the matrix form looks like this: M.()[x.] = [about]. Of this follows that the basis of the kernel of the linear operator is the fundamental set of solutions of a homogeneous system of linear equations with the main matrix M.(). The system of forming a linear operator make up vectors  ( e. 1), (e. 2), …, (e. n.). The basis of this system of vectors gives the basis of the image of a linear operator.

9.6. Reversible linear operators

Definition9.12. Linear operator  called reversibleif exist linear operator ψ such what is running equality ψ \u003d ψ \u003d , where  - the identity operator.

Theorem 9.10. If linear operator  Reverse that operator ψ is determined singly and called inverse for operator .

In this case, the operator is reverse for the operator , denotes  -1.

Theorem 9.11. Linear operator  Reversely and only when reversible the matrix M.(), while M.( –1) = (M.()) –1 .

From this theorem it follows that the rank of the reversible linear operator is equal dimension spaces, and the defect is zero.

Example 9.4. 1) Determine whether the linear operator  if  ( x.) = (2h. 1 – h. 2 , –4h. 1 + 2h. 2).

Decision. Make a matrix of this linear operator: M.() \u003d. As
\u003d 0 then matrix M.() irreversible, and therefore irreversible and linear operator .

2) To find linear operator, back operator  if (x.) = (2h. 1 + h. 2 , 3h. 1 + 2h. 2).

Decision.The matrix of this linear operator, equal M.() =
, reversible, because | M.()| ≠ 0. (M.()) –1 =
, so  -1 = (2h. 1 – h. 2 , –3h. 1 + 2h. 2).

1

Finding out the principles of integration of discrete information at a separate perception of elements of a complex object is an urgent interdisciplinary problem. The article discusses the process of constructing an object of an object, which is a complex of blocks, each of which combines a set of small elements. A conflict situation was chosen as the object under study, since it was stably in the field of attention with a constant strategy for analyzing information. The circumstances of the situation were component parts The object and separately perceived as the pre-conflict. The task of this work was in mathematical expression of the matrix, reflecting the image of the problem behavioral situation. The solution of the problem was based on the data of the visual analysis of the design of the graphic composition, the elements of which corresponded to situational circumstances. The size and graphic features of the elements selected, as well as their distribution in the compositions served as a guide to highlight the rows and columns in the image matrix. The study showed that the design of the matrix is \u200b\u200bdetermined, in first, behavioral motivation, secondly, causal relations of situational elements and a sequence of obtaining information, as well as, thirdly, the allocation of information fragments in accordance with their weight parameters. It can be assumed that the marked matrix vector principles of the formation of the behavioral situation are characteristic of building images and other objects to which attention is directed.

visualization

perception

discreteness of information

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The results of studies of the perception of incomplete images have expanded the prospect of studying the principles that determine the integration of discrete information and the installation of solid images. Analysis of the characteristics of the identification of fragmented images upon presentation of a changing number of fragments made it possible to trace three strategies for constructing a solid image in the conditions of the deficit of information. Strategies differed in assessing the importance of cash portions of information for the formation of a solid image. In other words, each strategy was characterized by manipulation by weight parameters of cash portions of information. The first strategy provided for the equivalent image fragments - its identification was accomplished after the accumulation of information to a level sufficient for a complete presentation with a relatively presented object. The second strategy was based on a differentiated approach to the assessment of the weight of the fragments of cash information. The estimate was given in accordance with the hypothesis extended relative to the essence of the object. The third strategy was determined by the motivation of the maximum use of cash, which was given high weight and was considered a sign or prototype real object. An important point In the previously done work was the consideration of the brain mechanisms, which provided a change of strategies depending on the dominant emotion and behavioral motivation. There are in mind nonspecific brain systems and heterogeneity of neural modules operating under the control of the central control. Studies conducted, as well as those known from literary sources, left an open question about the principles of information distribution in a solid image. To answer the question, it was necessary to observe the formation of the image of the object on which the attention was concentrated for a long time and the selected strategy for building the image remains unchanged. A conflict situation could serve as such an object, since it was stably in the field of attention with a constant second strategy for analyzing circumstances. The controversial sides rejected the first strategy due to an increase in the duration of the conflict and did not apply the third strategy, avoiding erroneous solutions.

purpose This work was to clarify the principles of constructing an image matrix based on elements of information obtained at a separate perception of components. comprehensive objectwhich was focused on. The following tasks were solved: first, the object was chosen on which a consistently long time was focused on, and secondly, the image visualization method was used to trace the fragmentation of the information obtained during the perception of the object, and then, thirdly, to formulate the principles of solid distribution Fragments in the matrix.

Materials and research methods

As a multicomponent object that has been stably in the field of attention with a constant cash analysis strategy, has served a problem behavioral situation. The problem was caused by the conflict in the relationship of family members, as well as employees of production and educational institutions. Experiments in which an analysis of the image of a situation was preceded by mediation aimed at resolving contradictions between the controversial parties. Before the beginning of media negotiations, representatives of the controversial parties received an offer to participate as subjects in experiments using a technique that helps analyze the situation. The visualization technique provided for the construction of a graphic composition reflecting the design of the image that occurred at a separate perception of components of the complex object. The technique served as a tool for studying the process of forming a solid image from a set of elements corresponding to the items of the object. The subject group consisted of 19 women and 8 men aged 28 to 65 years. To obtain a solid visual image of a situation, the subjects were offered to perform the following actions: 1) Restore the circumstances of the conflict situation - events, relations with people, the motives of their own behavior and the surrounding people; 2) appreciate the circumstances to understand the essence of the situation; 3) divide the circumstances on favorable and unfavorable to resolve the conflict and try to trace their relationship; 4) pick up suitable, in your opinion, a graphic element (circle, square, triangle, line or point) for each of the circumstances that characterize the situation; 5) form a composition of graphic elements, taking into account the importance and relationship of the circumstances transmitted by these elements, and draw the resulting composition on a paper sheet. Graphic compositions were analyzed - the ordering and the ratio of the size of the image elements was evaluated. Random disordered compositions were rejected, and the subjects were proposed to again consider the relationship of situational circumstances. The results of generalized analysis The composition served as a guideline for the formulation of the mathematical expression of the matrix of the image.

Results of research and discussion

Each graphic composition, through which the subject represented the design of the behavioral image, was original. Examples of compositions are illustrated in the figure.

Graphic compositions reflecting images of problem behavioral situations in which there were subjects (each element of the composition corresponds to situational circumstances)

The uniqueness of the compositions testified to the responsible approach of the tests for the analysis of situations, taking into account their distinctive features. The number of elements in the composition and dimension of the elements, as well as the design of the composition reflected the estimate of the circumstance complex.

After the originality of the compositions was noted, the study turned to identifying the principal features of the image design. In an effort to build a solid composition reflecting the image of the situation, the subjects distributed elements in accordance with their individual preferences, as well as taking into account the causal relationship of the circumstances and the combination of circumstances over time. Seven subjects preferred to mount the composition in the form of a picture, the construction of which was determined in advance drawn plan. In fig. 1 (A, B, D) are examples of such compositions. Two tests before the preparation of the composition chose the idea based on the plan, consciously, and five intuitively, not giving a logical explanation, why they stopped at the selected version. The remaining twenty subjects created a schematic composition, paying attention only to causal relationships of circumstances and a combination of circumstances over time (Fig. 1, B, D, E). Related and coincidences in the time circumstances were combined in the composition. The experiments did not interpret the essence of the conflict using the graphic composition data. Such interpretation was subsequently carried out within the framework of mediation, when it turned out the readiness of the parties to negotiations.

The analysis of the compositions allowed to trace not only the difference, but also the versatility of the principles of the formation of a situation. First, the compositions consisted of graphic elements, each of which reflected the circumstances with commonplace. The community of circumstances was due to causal and temporary relations. Secondly, the circumstances had unequal significance to understand the essence of the problem situation. That is, circumstances differed in the weight parameters. Highly significant circumstances were depicted by graphic elements in an enlarged amount, compared with less significant. The noted features of the image were taken into account in the preparation of the matrix of the image. It is understood that the size and graphic features of the selected elements, as well as their spatial position in the graphics composition, served as a guideline for building an information matrix, reflecting the image of the situation and being it mathematical model. The rectangular matrix presented in the form of a table is divided into strings and columns. With reference to the formable image of the problem situation in the matrix, lines were allocated in which weighted elements of the samples were located combined with causal and temporal relations, and columns containing elemental data differing in weight parameters.

(1)

Each individual line reflected the formation of part of the image or, in other words, the prototype of the object. The larger the rows and the greater M, the more totally the object was perceived, since the structural and functional properties that served in its protrases were fully taken into account. The number of columns N was determined by the number of parts noted when constructing the Preparation. It can be assumed that the larger the information fragments of high and low weight were accumulated, the more complete the prototype corresponded to reality. The matrix (1) was characterized by dynamism, since its dimension varied in accordance with the complete image of the perceived object.

It appropriate to note here that completeness is not the only figure of the image. The images presented on the artists can often lose photos of detail and in accordance with reality, but at the same time can exceed the association with other images, on the initiation of imagination and on provocating emotions. The remark made helps to understand the importance of AMN parameters denoting the weight of information fragments. Increased weight leveled a lack of cash data. As the study showed strategies to overcome uncertainty, the recognition of the high significance of cash fragments accelerated decision-making in a problem situation.

So, the process of the formation of a solid image is amenable to interpretation, given it with manipulation of information within the matrix. Manipulation is expressed by an arbitrary or involuntary (conscious targeted or intuitive unconscious) change in the weight parameters of information fragments, that is, a change in the value of AMN. In this case, the value of Bm increases or decreases, which characterizes the importance of the sample, and the resulting image of Br is simultaneously changing. If you refer to the matrix model of forming an image that covers the set of data relative to the object, the image organization is described as follows. Denote by the vector of the samples containing M component, through

where t is a transposition sign, and each element of the versions of the examples is:

Then the choice of the resulting image can be carried out according to the rule of Laplace:

where Br is the final result of the formation of a solid image, which has its components of the value BM, AMN - a set of values \u200b\u200bthat determine the position and weight parameters of the variable in a string corresponding to the sobrase. Under conditions of limited information, the final result may increase by increasing the weight values \u200b\u200bof the cash data.

At the end of the discussion of the presented material regarding the principles of the formation of the image, attention is paid to the need to concretize the term "image", since there is no generally accepted interpretation in the literature. The term, first of all, denotes the formation of a solid system of information fragments that correspond to the details of the object in the field of attention. Moreover, large details of the object are reflected by the subsystems of information fragments constituting the samples. As an object, an object, a phenomenon, process, as well as a behavioral situation may appear. The formation of the image is provided by associations of the received information and the one that is contained in memory and is associated with a perceived object. Consolidation of information fragments and associations When creating an image is implemented within the framework of the matrix, the design and vector of which is selected consciously or intuitively. The selection depends on the preferences specified by the motivation of behavior. It is especially paid attention to the fundamental moment - the discreteness of the information used for the installation of a solid matrix of the image. Otherism, as shown, is provided by non-specific brain systems that control the processes of analyzing the information received and its integration in memory. Some may occur with minimum values \u200b\u200bof N and M equal to one. The image acquires a high value due to an increase in the weight parameters of the cash information, and the phillet of the image increases as the values \u200b\u200bof N and M (1) increase.

Conclusion

The visualization of the elements of the image made it possible to trace the principles of its design in the conditions of separate perception of the circumstances of the problem behavioral situation. As a result of the work performed, it was shown that the construction of a solid image can be considered as the distribution of information fragments in the structure of the matrix. Its design and vector are determined, firstly, behavioral motivation, secondly, causal relationships of the circumstances and the temporary sequence of obtaining information, as well as, thirdly, the allocation of information fragments in accordance with their weight parameters. The integrity of the image matrix is \u200b\u200bensured by the integration of discrete information reflecting the perceived object. Nonspecific brain systems make up the mechanism responsible for integrating information in a solid image. Filming up the matrix principles of forming a complex object expands the prospect of understanding the nature of not only wholeness, but also other properties of the image. This refers to the integrity and preservation of the figurative system, as well as the value and subjectivity due to the disadvantage full information Regarding the object.

Bibliographic reference

Lavrov V.V., Rudinsky A.V. The formation of a solid image matrix with a separate perception of elements of a complex object // International Journal of Applied and fundamental studies. - 2016. - № 7-1. - P. 91-95;
URL: https://applied-research.ru/ru/article/view?id\u003d9764 (Date of handling: 15.01.2020). We bring to your attention the magazines publishing in the publishing house "Academy of Natural Science"