Find the basis of the kernel. Formation of a matrix of an integral image with separate perception of the elements of a complex object. Elementary theory of linear operators
1Clarification of the principles of integration of discrete information during separate perception of elements of a complex object is an urgent interdisciplinary problem. The article discusses the process of constructing an image 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 of study, since it was consistently in the field of attention with a constant strategy for analyzing information. The circumstances of the situation were components of the object and were separately perceived as prototypes of the conflict. The task of this work was to mathematically express a matrix that reflected the image of a problematic behavioral situation. The solution to the problem was based on data from a visual analysis of the design of a graphic composition, the elements of which corresponded to situational circumstances. The size and graphic features of the selected elements, as well as their distribution in the composition, served as a guide for identifying rows and columns in the image matrix. The study showed that the design of the matrix is determined, firstly, by behavioral motivation, secondly, by the cause-and-effect relationships of situational elements and the sequence of obtaining information, and thirdly, by the selection of pieces of information in accordance with their weight parameters. It can be assumed that the noted matrix vector principles of forming an image of a behavioral situation are characteristic of constructing 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 perspective of studying the principles that determine the integration of discrete information and the montage of complete images. Analysis of the features of recognition of fragmented images when presented with a changing number of fragments made it possible to trace three strategies for constructing a complete image in conditions of information deficiency. The strategies differed in their assessment of the significance of available pieces of information for the formation of a coherent image. In other words, each strategy was characterized by manipulation of the weight parameters of available pieces of information. The first strategy provided for the equivalence of fragments of the image - its identification was carried out after the accumulation of information to a level sufficient for a complete understanding of the presented object. The second strategy was based on a differentiated approach to assessing the weight of pieces of available information. The assessment was given in accordance with the hypothesis put forward regarding the essence of the object. The third strategy was determined by the motivation to make maximum use of available information, which was given high weight and was considered a sign or prototype of a real object. An important point in previous work was the consideration of brain mechanisms that ensure a change in strategies depending on the dominant emotion and behavioral motivation. This refers to nonspecific brain systems and the heterogeneity of neural modules operating under the control of central control. The conducted studies, like those known from literary sources, left open the question of the principles of information distribution in a complete image. To answer the question, it was necessary to observe the formation of the image of the object on which attention has been focused for a long time and the chosen strategy for constructing the image remains unchanged. A conflict situation could serve as such an object, since it was consistently in the field of attention with the second strategy of analyzing the circumstances remaining constant. The disputing parties rejected the first strategy due to the increase in the duration of the conflict and did not apply the third strategy, avoiding erroneous decisions.
Target This work was to clarify the principles of constructing an image matrix based on elements of information obtained through separate perception of the components of a complex object to which attention was directed. We solved the following problems: firstly, we chose an object on which attention was focused for a stable long time, secondly, we used the image visualization method to trace the fragmentation of information obtained during the perception of the object, and then, thirdly, to formulate the principles of integral distribution fragments in the matrix.
Materials and research methods
A problematic behavioral situation served as a multicomponent object that was stably in the field of attention with an unchanged strategy for analyzing available information. The problem was caused by conflict in relations between family members, as well as employees of industrial and educational institutions. Experiments in which the image of the situation was analyzed preceded mediation aimed at resolving contradictions between the disputing parties. Before the start of mediation negotiations, representatives of the disputing parties received an offer to participate as subjects in experiments using a technique that facilitates the analysis of the situation. The visualization technique involved the construction of a graphic composition that reflected the construction of the image that arose during the separate perception of the components of a complex object. The technique served as a tool for studying the processes of forming an integral image from a set of elements corresponding to the details of the object. The group of subjects consisted of 19 women and 8 men aged from 28 to 65 years. To obtain a complete visual image of the situation, subjects were asked to perform the following actions: 1) restore in their memory the circumstances of the conflict situation - events, relationships with people, motives for their own behavior and those around them; 2) evaluate the circumstances according to their significance for understanding the essence of the situation; 3) divide the circumstances into favorable and unfavorable for resolving the conflict and try to trace their relationship; 4) select, in your opinion, a suitable graphic element (circle, square, triangle, line or point) for each of the circumstances that characterize the situation; 5) form a composition from graphic elements, taking into account the significance and relationship of the circumstances conveyed by these elements, and draw the resulting composition on a piece of paper. Graphic compositions were analyzed - the orderliness and size ratio of the image elements were assessed. Random, disordered compositions were rejected, and subjects were asked to reconsider the interrelationship of situational circumstances. The results of the generalized composition analysis served as a guide for formulating the mathematical expression of the image matrix.
Research results and discussion
Each graphic composition through which the subject represented the construction of the image of a behavioral situation was original. Examples of compositions are illustrated in the figure.
Graphic compositions reflecting images of problematic behavioral situations in which the subjects were located (each element of the composition corresponds to situational circumstances)
The uniqueness of the compositions testified to the responsible approach of the subjects to the analysis of situations, taking into account their distinctive features. The number of elements in the composition and the dimension of the elements, as well as the design of the composition, reflected the assessment of the complex of circumstances.
After the originality of the compositions was noted, the study turned to identifying the fundamental features of the image design. In an effort to build an integral composition reflecting the image of the situation, the subjects distributed elements in accordance with their individual preferences, as well as taking into account the cause-and-effect relationships of circumstances and the combination of circumstances over time. Seven subjects preferred to mount the composition in the form of a drawing, the construction of which was determined by a pre-drawn figurative plan. In Fig. 1 (a, b, d) gives examples of such compositions. Before drawing up the composition, two subjects chose the idea that formed the basis of the plan consciously, and five intuitively, without giving a logical explanation of why they settled on the chosen option. The remaining twenty subjects created a schematic composition, paying attention only to the cause-and-effect relationships of circumstances and the combination of circumstances over time (Fig. 1, c, e, f). Related and coincidental circumstances were combined in the composition. The experiments did not interpret the essence of the conflict using graphic composition data. This interpretation was subsequently carried out within the framework of mediation, when the readiness of the parties to negotiate was determined.
Analysis of the compositions made it possible to trace not only the difference, but also the universality of the principles of forming the image of a situation. Firstly, the compositions consisted of graphic elements, each of which reflected circumstances that had a commonality. The commonality of circumstances was due to cause-and-effect and temporal relationships. Secondly, the circumstances were of unequal importance for understanding the essence of the problem situation. That is, the circumstances differed in weight parameters. Highly significant circumstances were depicted with graphic elements in an increased size compared to less significant ones. The noted features of the image were taken into account when compiling the image matrix. This means that the size and graphic features of the selected elements, as well as their spatial position in the graphic composition, served as a guide for constructing an information matrix that reflected the image of the situation and was its mathematical model. A rectangular matrix, represented as a table, is divided into rows and columns. In relation to the image of the problem situation being formed, rows were identified in the matrix, which contained weighted elements of the prototypes, united by cause-and-effect and temporal relationships, and columns containing elemental data that differed in weight parameters.
(1)
Each individual line reflected the formation of a part of the image or, in other words, a prototype of the object. The more lines and the larger m, the more totally the object was perceived, since the structural and functional properties that served as its prototypes were more fully taken into account. The number of columns n was determined by the number of details noted when constructing the prototype. It can be assumed that the more information fragments of high and low weight were accumulated, the more fully the prototype corresponded to reality. Matrix (1) was characterized by dynamism, since its dimension changed in accordance with the completeness of the image of the perceived object.
It is appropriate to note here that completeness is not the only indicator of image quality. Images presented on artists’ canvases are often inferior to photographs in terms of detail and correspondence to reality, but at the same time they can be superior in association with other images, in stimulating the imagination and in provoking emotions. The remark made helps to understand the significance of the parameters amn, which indicate the weight of information fragments. Weight gain offset the lack of available data. As a study of strategies for overcoming uncertainty has shown, recognizing the high significance of available pieces of information accelerated decision-making in a problem situation.
So, the process of forming an integral image can be interpreted if we correlate it with the manipulation of information within the matrix. Manipulation is expressed by a voluntary or involuntary (conscious, purposeful 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 bm, which characterizes the significance of the prototype, increases or decreases, and at the same time the resulting image br changes. If we turn to the matrix model of image formation, covering a set of data regarding an object, then the organization of the image is described as follows. Let us denote the vector of preimages containing m components by
where T is the transposition sign, and each element of the preimage vector has the form:
Then the choice of the resulting image can be carried out according to Laplace's rule:
where br is the final result of the formation of a solid image, which has the values bm as its components, amn is a set of values that determine the position and weight parameters of the variable in the line corresponding to the preimage. In conditions of limited information, the final result can be increased by increasing the weights of the available data.
At the end of the discussion of the presented material regarding the principles of image formation, attention is drawn to the need to specify the term “image”, since there is no generally accepted interpretation in the literature. The term, first of all, means the formation of an integral 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 subsystems of information fragments that make up the prototypes. The object can be an object, phenomenon, process, as well as a behavioral situation. The formation of an image is ensured by associations of the information received and that which is contained in memory and associated with the perceived object. The consolidation of information fragments and associations when creating an image is realized within the framework of a matrix, the design and vector of which are chosen consciously or intuitively. The choice depends on the preferences set by the motivations of behavior. Here, special attention is paid to the fundamental point - the discreteness of information used to assemble the integral image matrix. Integrity, as shown, is ensured by nonspecific brain systems that control the processes of analysis of received information and its integration in memory. Integrity can occur at minimum values of n and m equal to one. The image acquires high value due to an increase in the weight parameters of the available information, and the completeness of the image increases as the values of n and m (1) increase.
Conclusion
Visualization of the elements of the image made it possible to trace the principles of its design in conditions of separate perception of the circumstances of a problematic behavioral situation. As a result of the work carried out, it was shown that the construction of a complete image can be considered as the distribution of information fragments in the structure of the matrix. Its design and vector are determined, firstly, by behavioral motivation, secondly, by the cause-and-effect relationships of circumstances and the temporal sequence of obtaining information, and thirdly by the selection of pieces of information in accordance with their weight parameters. The integrity of the image matrix is ensured by the integration of discrete information reflecting the perceived object. Nonspecific brain systems constitute the mechanism responsible for integrating information into a coherent image. Clarification of the matrix principles of formation of the image of a complex object expands the perspective of understanding the nature of not only integrity, but also other properties of the image. This refers to the integrity and safety of the image system, as well as the value and subjectivity caused by the lack of complete information regarding the object.
Bibliographic link
Lavrov V.V., Rudinsky A.V. FORMATION OF A MATRIX OF AN INTEGRATED IMAGE DURING SEPARATE PERCEPTION OF ELEMENTS OF A COMPLEX OBJECT // International Journal of Applied and Fundamental Research. – 2016. – No. 7-1. – P. 91-95;URL: https://applied-research.ru/ru/article/view?id=9764 (date of access: 01/15/2020). We bring to your attention magazines published by the publishing house "Academy of Natural Sciences"
IN vector space V over an arbitrary field P set to linear operator .
Definition9.8. Core linear operator is the set of vectors in space V, whose image is the zero vector. Accepted notation for this set: Ker, i.e.
Ker = {x | (X) = o}.
Theorem 9.7. The kernel of a linear operator is a subspace of the space V.
Definition 9.9. Dimension the kernel of a linear operator is called defect linear operator. dim Ker = d.
Definition 9.10.In a manner linear operator is the set of images space vectors V. Notation for this set Im, i.e. Im = {(X) | X V}.
Theorem 9.8. Image linear operator is a subspace of the space V.
Definition 9.11. Dimension the image of a linear operator is called rank linear operator. dim Im = r.
Theorem 9.9. Space V is the direct sum of the kernel and the image of the linear operator specified in it. The sum of the rank and defect of a linear operator is equal to the dimension of the space V.
Example 9.3. 1) In space R[x] ( 3) find rank and defect operator differentiation. Let's find those polynomials whose derivative is equal to zero. These are polynomials of degree zero, therefore Ker = {f | f = c) And d= 1. The derivatives of polynomials whose degree does not exceed three form a set of polynomials whose degree does not exceed two, therefore, Im =R[x] ( 2) and r = 3.
2) If linear the operator is given by a matrix M(), then to find its kernel one must solve equation ( X) = O, which in matrix form looks like this: M()[x] = [O]. From It follows that the basis of the kernel of a linear operator is the fundamental set of solutions of a homogeneous system of linear equations with the underlying matrix M(). System of generators of the image of a linear operator make up the vectors ( e 1), (e 2), …, (e n). The basis of this system of vectors gives the basis of the image of the linear operator.
9.6. Invertible linear operators
Definition9.12. Linear operator is called reversible, if exists linear operator ψ such what is being done equality ψ = ψ = , where is the identity operator.
Theorem 9.10. If linear operator reversible, That operator ψ is uniquely defined and is called reverse For operator .
In this case the operator, the inverse of the operator , denoted –1.
Theorem 9.11. Linear operator is invertible if and only if its matrix is invertible M(), while M( –1) = (M()) –1 .
From this theorem it follows that the rank of an invertible linear operator is equal to dimensions space, and the defect is zero.
Example 9.4 1) Determine whether linear is invertible operator , if ( x) = (2X 1 – X 2 , –4X 1 + 2X 2).
Solution. Let's create a matrix for this linear operator: M() = . Because 
= 0 then the matrix M() is irreversible, which means it is irreversible and linear
operator
.
2) Find linear operator, back operator , if (x) = (2X 1 + X 2 , 3X 1 + 2X 2).
Solution. The matrix of this linear
operator equal to
M() = 
, is reversible, since | M()| ≠ 0.
(M()) –1 = 
, therefore –1
= (2X 1 – X 2 ,
–3X 1 + 2X 2).
Definition 1. The image of a linear operator A is the set of all elements representable in the form , where .
The image of the linear operator A is a linear subspace of the space. Its dimension is called operator rank A.
Definition 2. The kernel of a linear operator A is the set of all vectors for which .
The kernel is a linear subspace of the space X. Its dimension is called operator defect A.
If the operator A acts in the -dimensional space X, then the following relation + = is true.
Operator A is called non-degenerate, if its core . The rank of a non-degenerate operator is equal to the dimension of the space X.
Let be the matrix of linear transformation A of space X in some basis, then the coordinates of the image and inverse image are related by the relation
Therefore, the coordinates of any vector satisfy the system of equations
It follows that the kernel of a linear operator is a linear shell of the fundamental system of solutions of a given system.
Tasks
1. Prove that the rank of an operator is equal to the rank of its matrix in an arbitrary basis.
Calculate the kernels of linear operators defined in a certain basis of space X by the following matrices:
5. Prove that .
Calculate the rank and defect of the operators given by the following matrices:
6. . 7. . 8. .
3. Eigenvectors and eigenvalues of the linear operator
Let us consider a linear operator A acting in the -dimensional space X.
Definition. The number l is called the eigenvalue of the operator A if , such that . In this case, the vector is called the eigenvector of operator A.
The most important property of eigenvectors of a linear operator is that the eigenvectors corresponding to pairwise different eigenvalues linearly independent.
If is the matrix of the linear operator A in the basis of the space X, then the eigenvalues l and eigenvectors of the operator A are determined as follows:
1. Eigenvalues are found as the roots of the characteristic equation (algebraic equation of the th degree):
2. The coordinates of all linearly independent eigenvectors corresponding to each individual eigenvalue are obtained by solving a system of homogeneous linear equations:
whose matrix has rank . The fundamental solutions of this system are column vectors of the coordinates of the eigenvectors.
The roots of the characteristic equation are also called the eigenvalues of the matrix, and the solutions of the system are called the eigenvectors of the matrix.
Example. Find the eigenvectors and eigenvalues of the operator A, specified in a certain basis by the matrix
1. To determine the eigenvalues, we compose and solve the characteristic equation:
Hence the eigenvalue, its multiplicity.
2. To determine the eigenvectors, we compose and solve a system of equations:
The equivalent system of basic equations has the form
Therefore, every eigenvector is a column vector, where c is an arbitrary constant.
3.1.Operator of a simple structure.
Definition. A linear operator A operating in an n-dimensional space is called an operator of simple structure if it corresponds to exactly n linearly independent eigenvectors. In this case, it is possible to construct a space basis from the eigenvectors of the operator, in which the operator matrix has the simplest diagonal form
where are the eigenvalues of the operator. Obviously, the converse is also true: if in some basis of the space X the matrix of the operator has a diagonal form, then the basis consists of the eigenvectors of the operator.
A linear operator A is an operator of simple structure if and only if each eigenvalue of multiplicity corresponds to exactly linearly independent eigenvectors. Since eigenvectors are solutions to a system of equations, therefore, each root of the characteristic equation of multiplicity must correspond to a rank matrix.
Any matrix of size corresponding to a simple structure operator is similar to a diagonal matrix
where the transition matrix T from the original basis to the basis of eigenvectors has as its columns the column vectors from the coordinates of the eigenvectors of the matrix (operator A).
Example. Reduce the linear operator matrix to diagonal form
Let's create a characteristic equation and find its roots.
Where do the eigenvalues of multiplicity and multiplicity come from?
First eigenvalue. It corresponds to eigenvectors whose coordinates are
system solution
The rank of this system is 3, so there is only one independent solution, for example, the vector .
The eigenvectors corresponding to are determined by the system of equations
whose rank is 1 and therefore there are three linearly independent solutions, for example,
Thus, each eigenvalue of multiplicity corresponds to exactly linearly independent eigenvectors and, therefore, the operator is an operator of simple structure. The transition matrix T has the form
and the connection between similar matrices is determined by the relation
Tasks
Find eigenvectors and eigenvalues
linear operators defined in a certain basis by matrices:
Determine which of the following linear operators can be reduced to diagonal form by passing to a new basis. Find this basis and its corresponding matrix:
10. Prove that the eigenvectors of a linear operator corresponding to different eigenvalues are linearly independent.
11. Prove that if a linear operator A acting in , has n different values, then any linear operator B commutes with A, has a basis of eigenvectors, and any eigenvector of A will be an eigenvector of B.
INVARIANT SUBSPACES
Definition 1.. A subspace L of a linear space X is said to be invariant under the operator A acting in X if for each vector its image also belongs to .
The main properties of invariant subspaces are determined by the following relations:
1. If and are invariant subspaces with respect 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 a direct sum of subspaces and () and is invariant with respect to A, then the matrix of the operator in the basis, which is a union of bases, is a block matrix
where are square matrices, 0 is a zero matrix.
3. In any subspace invariant with respect to operator A, the operator has at least one eigenvector.
Example 1. Let us consider the kernel of some operator A acting in X. By definition. Let . Then , since the zero vector is contained in every linear subspace. Consequently, the kernel is a subspace invariant under A.
Example 2. Let in some basis of the space X the operator A be given by a matrix defined by the equation and
5. Prove that any subspace that is invariant under a non-degenerate operator A will also be invariant under the inverse operator.
6. Let a linear transformation of an A -dimensional space in the basis have a diagonal matrix with different elements on the diagonal. Find all subspaces invariant under A and determine their number.
