Find the basis of the kernel. Formation of a matrix of an integral image with separate perception of elements of a complex object

1

Elucidation of the principles of integration of discrete information in the case of separate perception of the elements of a complex object is an actual interdisciplinary problem. The article discusses the process of building 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 an unchanged information analysis strategy. The circumstances of the situation were constituent parts object and were separately perceived as prototypes of the conflict. The task of this work was to mathematically express the matrix that reflected the image of a problematic behavioral situation. The solution of the problem was based on the data of visual analysis of the design of the graphic composition, the elements of which corresponded to the situational circumstances. The size and graphic features of the selected elements, as well as their distribution in the composition, served as a guideline for highlighting 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 allocation of information fragments 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 typical for the construction of images and other objects to which attention is directed.

visualization

perception

discreteness of information

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The results of studies on the perception of incomplete images have expanded the perspective of studying the principles that determine the integration of discrete information and the montage of integral images. An analysis of the features of the recognition of fragmented images upon presentation of a changing number of fragments made it possible to trace three strategies for constructing an integral image in conditions of information deficiency. The strategies differed in assessing the significance of available portions of information for the formation of an integral image. In other words, each strategy was characterized by the manipulation of the weight parameters of the available pieces of information. The first strategy provided for the equivalence of image fragments - its identification was carried out after the accumulation of information to a level sufficient for a complete representation 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 motivation maximum use available information, which was endowed with high weight and was considered a sign or prototype of a real object. An important point in the work done earlier was the consideration of brain mechanisms that ensured the change of strategies depending on the dominant emotion and behavioral motivation. This refers to non-specific systems of the brain and the heterogeneity of neuronal modules operating under the control of the central control. The conducted studies, as well as those that are known from literary sources, left open the question of the principles of distribution of information in a whole image. To answer the question, it was necessary to observe the formation of the image of the object on which attention is 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 unchanged. 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 of this work was to clarify the principles of constructing an image matrix based on the elements of information obtained during the separate perception of the components complex facility to which attention was directed. We solved the following tasks: firstly, we chose an object on which attention was focused for a 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 methods of research

A problematic behavioral situation served as a multicomponent object that was consistently in the field of attention with an unchanged strategy for analyzing available information. The problem was caused by a conflict in the relations of family members, as well as employees of production and educational institutions. The experiments, in which the analysis of the image of the situation was carried out, preceded the mediation aimed at resolving the contradictions between the disputable parties. Before the start of mediation negotiations, representatives of the disputable parties received an offer to participate as subjects in experiments using a methodology that facilitates the analysis of the situation. The visualization technique provided for 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 28 to 65 years. To obtain a complete visual image of the situation, the subjects were asked to perform the following actions: 1) restore in memory the circumstances of the conflict situation - events, relationships with people, motives for their own behavior and those around them; 2) evaluate the circumstances by their significance for understanding the essence of the situation; 3) divide the circumstances into favorable and unfavorable for conflict resolution and try to trace their relationship; 4) select a suitable, in your opinion, graphic element (circle, square, triangle, line or dot) for each of the circumstances that characterize the situation; 5) form a composition from graphic elements, taking into account the significance and interrelation of the circumstances transmitted by these elements, and draw the resulting composition on a paper sheet. The graphic compositions were analyzed - the ordering and the ratio of the sizes of the elements of the image were evaluated. Random disordered compositions were rejected, and the subjects were asked to reconsider the relationship of situational circumstances. The results of the generalized analysis of the composition served as a guideline for formulating the mathematical expression of the image matrix.

Research results and discussion

Each graphic composition, through which the subject presented 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 (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 an assessment of a set of circumstances.

After the originality of the compositions was noted, the study turned to identifying the fundamental features of the design of the image. In an effort to build an integral composition that reflects the image of the situation, the subjects distributed the 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 assemble the composition in the form of a picture, the construction of which was determined by a pre-compiled figurative plan. On fig. 1 (a, b, d) examples of such compositions are given. Before compiling the composition, two subjects consciously chose the idea underlying the plan, and five intuitively, without giving a logical explanation why they stopped at 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). Circumstances connected and coinciding in time were combined in the composition. In the experiments, the interpretation of the essence of the conflict was not carried out using the data of the graphic composition. Such an interpretation was subsequently carried out within the framework of mediation, when the readiness of the parties to negotiate was ascertained.

Analysis of the compositions made it possible to trace not only the difference, but also the universality of the principles of formation of the image of the situation. Firstly, the compositions consisted of graphic elements, each of which reflected circumstances that had a commonality. The generality of the circumstances was due to causal and temporal relationships. Secondly, the circumstances were of unequal significance for understanding the essence of the problem situation. That is, the circumstances differed in weight parameters. Highly significant circumstances were depicted by graphic elements in an enlarged 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 building an information matrix that reflected the image of the situation and was its mathematical model. A rectangular matrix presented as a table is divided into rows and columns. In relation to the formed image of the problem situation in the matrix, rows were distinguished, in which there were weighted elements of preimages, united by causal and temporal relationships, and columns containing elemental data that differed in weight parameters.

(1)

Each separate line reflected the formation of a part of the image or, in other words, the prototype of the object. The more lines and the more 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 during the construction of 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.

Here it is appropriate to note that completeness is not the only indicator of the quality of an image. The images presented on the canvases of artists often lose photos in terms of detail and correspondence to reality, but at the same time they can surpass in association with other images, in exciting the imagination and in provoking emotions. This remark helps to understand the significance of the amn parameters, which denote the weight of information fragments. The increase in weight offset the lack of available data. As the study of strategies for overcoming uncertainty has shown, the recognition of the high significance of available pieces of information accelerated decision-making in a problem situation.

So, the process of forming a whole image can be interpreted if we correlate it with the manipulation of information within the framework of the matrix. Manipulation is expressed by an arbitrary 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 the resulting image br simultaneously changes. If we turn to the matrix model of forming an image that covers a set of data on an object, then the organization of the image is described as follows. Denote the vector of preimages containing m components by

where T is the transposition sign, and each element of the vector of preimages has the form:

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

where br is the final result of the formation of an integral 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 end 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. An object, a phenomenon, a process, as well as a behavioral situation can act as an object. The formation of the image is provided by the associations of the received information and that which is contained in the memory and is associated with the perceived object. Consolidation of information fragments and associations when creating an image is implemented within the matrix, the design and vector of which are chosen consciously or intuitively. The choice depends on the preferences given by the motivations of behavior. Here, special attention is drawn to the fundamental point - the discreteness of the information used to mount the entire matrix of the image. Integrity, as shown, is provided by non-specific brain systems that control the processes of analyzing the information received and integrating it into memory. Integrity can arise at the minimum values ​​of n and m equal to one. The image acquires a high value due to the 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 construction in the 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 an integral image can be considered as the distribution of information fragments in the matrix structure. 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 allocation of information fragments in accordance with their weight parameters. The integrity of the image matrix is ​​ensured by the integration of discrete information that reflects the perceived object. The non-specific systems of the brain constitute the mechanism responsible for integrating information into a coherent image. Elucidation of the matrix principles of forming 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 preservation of the figurative system, as well as the value and subjectivity, due to the lack of complete information regarding the object.

Bibliographic link

Lavrov V.V., Rudinsky A.V. FORMING A MATRIX OF A WHOLE IMAGE WITH 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 the journals published by the publishing house "Academy of Natural History"

Changing the coordinates of the vector and matrix of the operator during the transition to a new basis

Let a linear operator act from space into itself, and let two bases be chosen in the linear space: and Let us expand the “new” basis vectors into linear combinations of the “old” basis vectors:

Matrix standing here whose column is the coordinate column of the th basis vector in the “old” basis is called the transition matrix from the “old” basis to the “new”“. If now the coordinates of the vector are in the “old” basis and the coordinates of the same vector are in the “new” basis, then the equality takes place

Since the expansion in terms of the basis is unique, it follows that

The following result is obtained.

Theorem 1.The coordinates of the vector in the basis and the coordinates of the same vector in the basis are related by relations (2), where is the transition matrix from the “old” basis to the “new” .

Let us now see how the matrices and the same operator are related in different bases and spaces Matrices and are defined by the equalities Let This equality in the basis is equivalent to the matrix equality

and in the basis to matrix equality (here the same notation is used as in (1)). Using theorem (1), we will have

since the column is arbitrary, from here we obtain the equality

The following result is proved.

Theorem 2.If the operator matrix in the basis and the matrix of the same operator in the basis then

Remark 1. Two arbitrary matrices and related by where is some nonsingular matrix are called similar matrices. Thus, two matrices of the same operator in different bases are similar.

Example 1 The operator matrix in the basis has the form

Find the matrix of this operator in the basis Calculate the coordinates of the vector in the basis

Solution. The transition matrix from the old basis to the new one and its inverse matrix have the form

therefore, by Theorem 2, the matrix of the operator and the new basis will be as follows:

Remark 2. One can generalize this result to operators acting from one linear space to another. Let the operator act from a linear space to another linear space and let two bases be chosen in the space: and and in the space - two bases and Then we can compose two matrices and of the linear operator

and two matrices and transitions from “old” bases to “new” ones:

It is easy to show that in this case the equality

Let a linear operator acting from a linear space to a linear space be given. The following concepts are useful in solving linear equations.

Definition 1. operator core is called a set

Operator way is called a set

It is easy to prove the following assertion.

Theorem 3.The kernel and the image of a linear operator are linear subspaces of the spaces and, respectively, and the equality

To calculate the kernel of the operator, it is necessary to write the equation in matrix form (choosing bases in the spaces and respectively) and solve the corresponding algebraic system of equations. Let us now explain how the image of an operator can be computed.

Let the matrix of the operator in in bases and Denote by the -th column of the matrix. is an element of the space of linear combinations of matrix columns. Having chosen a basis in this space (for example, the maximum set of linearly independent matrix columns), we first calculate the image matrix operator: and then build the operator image:

Let us give an example of calculating the kernel and the image of an operator acting from space into itself. In this case, the bases and coincide.

Example 2 Find the matrix, kernel, and image of the projection operator onto the plane (three-dimensional space of geometric vectors).

Solution. We choose some basis in space (for example, a standard basis). In this basis, the matrix of the projection operator is found from the equality Find the images of the basis vectors. Since the plane passes through the axis, then

In this way,

So the operator matrix has the form

The kernel of the matrix operator is calculated from the equation

In this way,

(arbitrary constant).

The image of the matrix operator is spanned over all linearly independent columns of the matrix, i.e.

(arbitrary constants).

Definition 1. The image of a linear operator A is the set of all elements that can be represented as , where .

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

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 BUT.

If the operator A acts in -dimensional space X, then the following relation is valid + = .

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

Let - the matrix of linear transformation A of the space X in some basis, then the coordinates of the image and preimage are related by the relation

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

Hence it follows that the kernel of a linear operator is a linear envelope of the fundamental system of solutions of this 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 given in some 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 A LINEAR OPERATOR

Let us consider a linear operator A acting in - dimensional space X.

Definition. The number l is called an eigenvalue of the operator A if , such that . In this case, the vector is called the eigenvector of the operator A.

The most important property of the eigenvectors of a linear operator is that the eigenvectors corresponding to pairwise different eigenvalues are linearly independent.

If is the matrix of the linear operator A in the basis of the space X, then the eigenvalues ​​l and the eigenvectors of the operator A are defined 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 vector-columns of eigenvector coordinates.

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 eigenvectors and eigenvalues ​​of the operator A given in some 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 the 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 acting in 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 operator eigenvectors, in which the operator matrix has the simplest diagonal form

where are the eigenvalues ​​of the operator. It is obvious that 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 multiplicity eigenvalue corresponds to exactly linearly independent eigenvectors. Since the eigenvectors are solutions to the system of equations, therefore, each root of the characteristic multiplicity equation 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 its columns of column vectors from the coordinates of the eigenvectors of the matrix (operator A).

Example. Reduce the matrix of a linear operator to a diagonal form

We compose the characteristic equation and find its roots.

Whence the eigenvalues ​​of multiplicity and multiplicity.

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 and is determined by the relation

Tasks

Find eigenvectors and eigenvalues

linear operators defined in some basis by matrices:

Determine which of the following linear operators can be reduced to a 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 commuting with A has a basis of eigenvectors, and any eigenvector A will be eigenvector for B.

INVARIANT SUBSPACES

Definition 1.. A subspace L of a linear space X is called invariant with respect to 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 under the operator A, then their sum and intersection are also invariant under the operator A.

2. If the space X decomposes into a direct sum of subspaces and () and is invariant under A, then the matrix of the operator in the basis, which is the union of the bases, is the block matrix

where are square matrices, 0 is the zero matrix.

3. In any subspace invariant with respect to the operator A, the operator has at least one eigenvector.

Example 1 Consider the kernel of some operator A acting in X. By definition, . Let . Then , since the zero vector is contained in any linear subspace. Therefore, the kernel is a subspace invariant with respect to A.

Example 2 Let, in some basis of the space X, the operator A is given by the matrix defined by the equation and

5. Prove that any subspace , which is invariant with respect to the non-degenerate operator A, will also be invariant with respect to the inverse operator .

6. Let linear transformation An A-dimensional space in the basis has a diagonal matrix with distinct entries on the diagonal. Find all subspaces invariant under A and determine their number.

AT vector space V over custom field P linear operator .

Definition 9.8. core linear operator  is the set of space vectors 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 the linear operator is called defect linear operator. dim Ker = d.

Definition 9.10.way linear operator  is called 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 defined 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. Find those polynomials whose derivative is equal to zero. These are polynomials of degree zero, so Ker = {f | f = c) and d= 1. The derivatives of polynomials whose degree does not exceed three form the set of polynomials whose degree does not exceed two, therefore, Im =R[x] ( 2) and r = 3.

2) If linear operator given by matrix M(), then to find its kernel it is necessary to solve equation ( X) = about, which in matrix form looks like this: M()[x] = [about]. From This implies that the basis of the kernel of a linear operator is the fundamental set of solutions of the homogeneous system of linear equations with the main matrix M(). The 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. Reversible 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  turn, then operator ψ uniquely defined and called reverse for operator .

In this case, the operator inverse to 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 .

It follows from this theorem that the rank of an invertible linear operator is equal to dimensions space, and the defect is zero.

Example 9.4 1) Determine if a linear operator  if ( x) = (2X 1 – X 2 , –4X 1 + 2X 2).

Solution. Let us compose the matrix of this linear operator: M() = . Because
= 0 then matrix M() is irreversible, which means that the 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 =
, so  –1 = (2X 1 – X 2 , –3X 1 + 2X 2).