Finding the kernel of the range of a linear operator. Formation of a matrix of an integral image with separate perception of the elements of a complex object. Solution example in MATLAB environment
1Elucidation of the principles of integration of discrete information with separate perception of the elements of a complex object is an urgent interdisciplinary problem. The article deals with 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 under study, since it was steadily in the field of attention with an unchanged information analysis strategy. The circumstances of the situation were the constituent parts of the object and were separately perceived as prototypes of the conflict. The aim of this work was to express mathematically a matrix that reflected the image of a problematic behavioral situation. The solution to 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 guide for the selection of rows and columns in the image matrix. The study showed that the construction of the matrix is \u200b\u200bdetermined, firstly, by behavioral motivation, secondly, by the cause-and-effect relationships of situational elements and the sequence of obtaining information, and also, thirdly, by highlighting fragments of information in accordance with their weight parameters. It can be assumed that the noted matrix vector principles of forming the image of a behavioral situation are characteristic 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 prospect of studying the principles that determine the integration of discrete information and the assembly of whole images. An analysis of the features of recognition of fragmented images upon presentation of a varying number of fragments made it possible to trace three strategies for constructing an integral image in conditions of a lack of information. The strategies differed in assessing the significance of the 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 the image fragments - its identification was performed after the accumulation of information to a level sufficient for a full-fledged 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 the motivation for maximum use of 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 the brain mechanisms that ensured a change in strategies depending on the dominant emotion and behavioral motivation. This refers to the nonspecific systems of the brain and the heterogeneity of neural modules operating under the control of central control. The conducted studies, as well as those that are known from literary sources, left open the question of the principles of the distribution of information in an integral manner. 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 of building the image remains unchanged. A conflict situation could serve as such an object, since it was consistently in the field of attention with the unchanged second strategy of analyzing the circumstances. The disputable parties rejected the first strategy because of the increased duration of the conflict and did not apply the third strategy, avoiding erroneous decisions.
goal This work consisted in elucidating the principles of constructing an image matrix on the basis of elements of information obtained during the separate perception of the components of a complex object 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 received during the perception of the object, and then, thirdly, 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 a conflict in the relations of family members, as well as employees of industrial and educational institutions. The experiments in which the analysis of the image of the situation was carried out preceded mediation aimed at resolving contradictions between the disputed parties. Before the beginning of mediation negotiations, representatives of the disputable parties received an offer to participate as subjects in experiments using a technique that facilitates situation analysis. The visualization technique provided for the construction of a graphical composition that reflected the construction of the image that arose when the components of a complex object were perceived separately. 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 an integral 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 of their own behavior and those around them; 2) evaluate the circumstances in terms of significance for understanding the essence of the situation; 3) divide the circumstances into favorable and unfavorable for the resolution of the conflict and try to trace their relationship; 4) select the graphic element that is suitable, in your opinion, (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 significance and interconnection of the circumstances conveyed by these elements, and draw the resulting composition on a piece of paper. Graphic compositions were analyzed - the ordering and the ratio of the sizes of the elements of the image were assessed. Random disordered compositions were rejected, and the subjects were asked to re-examine the interconnection 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 represented the construction of the image of the behavioral situation was original. Examples of compositions are illustrated in the figure.
Graphic compositions reflecting the images of problematic behavioral situations in which the subjects were (each element of the composition corresponds to the 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 a set 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 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 in time. Seven subjects preferred to mount the composition in the form of a drawing, the construction of which was determined by a previously drawn up figurative plan. In fig. 1 (a, b, d) gives examples of such compositions. Before composing the composition, two subjects chose the idea underlying the plan deliberately, and five subjects 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 causal relationships of circumstances and the combination of circumstances in time (Fig. 1, c, e, f). Connected and coincidental circumstances were combined in the composition. The experiments did not interpret the essence of the conflict using the data of the graphic composition. This interpretation was carried out later in the framework of mediation, when the readiness of the parties to negotiate was ascertained.
The 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 the situation. First, the compositions consisted of graphic elements, each of which reflected circumstances that had a commonality. The generality of circumstances was due to the cause-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 terms of weight. Highly significant circumstances were depicted by graphic elements in an enlarged size, compared with less significant ones. The noted features of the image were taken into account when compiling the image matrix. It means that the size and graphic features of the selected elements, as well as their spatial position in the graphic composition, served as a reference point for building an information matrix that reflected the image of the situation and was its mathematical model. A rectangular matrix, presented in a table, is divided into rows and columns. With regard to the image of the problem situation being formed, the rows were distinguished in the matrix, which contained the weighted elements of the prototypes, united by cause-and-effect and time relations, and the columns containing the element data differing 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 taken into account more fully. The number of columns n was determined by the number of details marked when constructing the preimage. 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 dimensionality changed in accordance with the completeness of the image of the perceived object.
It is pertinent to note here that completeness is not the only indicator of the quality of an image. The images presented on the canvases of artists often play photographs in detail and in accordance with reality, but at the same time they can surpass in association with other images, in stimulating 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. The study of strategies for overcoming uncertainty has shown that the recognition of the high importance of available pieces of information accelerated decision-making in a problem situation.
So, the process of forming an integral image lends itself to interpretation if we relate it to 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 of bm, which characterizes the significance of the preimage, increases or decreases, and at the same time the resulting image br changes. If we turn to the matrix model of the formation of an image, covering a set of data on an object, then the organization of the image is described as follows. We denote the vector of inverse images 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 the Laplace rule:
where br is the final result of the formation of an integral image, which has bm as its components, and amn is a complex of values \u200b\u200bthat determine the position and weight parameters of the variable in the line corresponding to the preimage. With 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 concretize 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 the subsystems of information fragments that make up prototypes. The object can be an object, phenomenon, process, as well as a behavioral situation. The formation of an image is provided by associations of the received information and that which is contained in memory and is associated with the perceived object. Consolidation of information fragments and associations when creating an image is implemented within the framework of a matrix, the design and vector of which are chosen consciously or intuitively. The choice depends on the preferences given by the motivations for the behavior. Here, special attention is paid to the fundamental point - the discreteness of the information used to mount the integral matrix of the image. Integrity, as shown, is provided by non-specific brain systems that control the processes of analyzing the information received and its integration in memory. Integrity can occur when the minimum values \u200b\u200bof n and m are 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 \u200b\u200bof n and m increase (1).
Conclusion
The 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 the 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 time sequence of obtaining information, and also, thirdly, by highlighting fragments of information in accordance with their weight parameters. The integrity of the image matrix is \u200b\u200bensured by the integration of discrete information that reflects the perceived object. Nonspecific brain systems constitute the mechanism responsible for integrating information into a coherent image. Elucidation of the matrix principles of the 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 figurative system, as well as the value and subjectivity due to the lack of complete information about the object.
Bibliographic reference
V.V. Lavrov, A.V. Rudinsky FORMATION OF THE MATRIX OF THE WHOLE IMAGE WITH SEPARATE PERCEPTION OF THE ELEMENTS OF A COMPLEX OBJECT // International Journal of Applied and Fundamental Research. - 2016. - No. 7-1. - S. 91-95;URL: https://applied-research.ru/ru/article/view?id\u003d9764 (date accessed: 01/15/2020). We bring to your attention the journals published by the "Academy of Natural Sciences"
AT vector space V over an arbitrary field P given linear operator .
Definition 9.8. Core linear operator is the set of vectors of the 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.The wayof a linear operator is the set of images space vectors V ... The notation for this set Im, i.e. Im = {(x) | x V}.
Theorem 9.8. Form 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 range of the linear operator given 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 us find those polynomials whose derivative is equal to zero. These are polynomials of degree zero, therefore, Ker = {f | f = c) and d\u003d 1. 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 operator defined by matrix M(), then to find its kernel it is necessary to solve the equation ( x) = aboutwhich looks like this in matrix form: M()[x] = [about]. Of this implies that the basis of the kernel of a linear operator is a fundamental set of solutions of a homogeneous system of linear equations with the basic matrix M(). Generator system of the image of a linear operator constitute vectors ( e 1), (e 2), …, (e n). The basis of this system of vectors gives the basis for the range of the linear operator.
9.6. Invertible linear operators
Definition9.12. Linear operator is called reversibleif exists linear operator ψ such what is being done the equality ψ \u003d ψ \u003d , where is the identity operator.
Theorem 9.10. If linear operator is reversible, then operator ψ uniquely defined and called reverse for operator .
In this case, the operator inverse for the operator is denoted by –1.
Theorem 9.11. Linear operator is invertible if and only if its matrix is \u200b\u200binvertible M(), while M( –1) = (M()) –1 .
This theorem implies that the rank of an invertible linear operator is dimensions space, and the defect is zero.
Example 9.4 1) Determine if linear operator if ( x) = (2x 1 – x 2 , –4x 1 + 2x 2).
Decision... Let's compose the matrix of this linear operator: M() \u003d. Because 
\u003d 0 then the matrix M() is irreversible, which means that the linear
operator
.
2) To find linear operator, back operator if (x) = (2x 1 + x 2 , 3x 1 + 2x 2).
Decision.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 a linear operator A is a linear subspace of the space. Its dimension is called operator rank AND.
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 AND.
If the operator A acts in the -dimensional space X, then the following relation is true + \u003d.
Operator A is called non-degenerateif its core. The rank of a non-degenerate operator is equal to the dimension of the space X.
Let be the matrix of the linear transformation A of the space X in some basis, then the coordinates of the image and the 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 the linear span 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 the space X by the following matrices:
5. Prove that.
Calculate the rank and defect of operators given by the following matrices:
6. . 7. . 8. .
3. Eigenvectors and eigenvalues \u200b\u200bof a linear operator
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 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 \u200b\u200bl and the eigenvectors of the operator A are defined as follows:
1. The eigenvalues \u200b\u200bare 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 from the coordinates of the eigenvectors.
The roots of the characteristic equation are also called the eigenvalues \u200b\u200bof the matrix, and the solutions of the system are called the eigenvectors of the matrix.
Example.Find the eigenvectors and eigenvalues \u200b\u200bof the operator A defined in some basis by the matrix
1. To determine the eigenvalues, we compose and solve the characteristic equation:
Hence its own meaning, its multiplicity.
2. To determine the eigenvectors, we compose and solve the system of equations:
An equivalent system of basic equations has the form
Therefore, every eigenvector is a column vector, where c is an arbitrary constant.
3.1 Simple structure operator.
Definition. A linear operator A acting 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 basis of the space from the eigenvectors of the operator, in which the matrix of the operator has the simplest diagonal form
where are the eigenvalues \u200b\u200bof 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 the eigenvectors are solutions of the system of equations, therefore, a matrix of rank must correspond to each root of the characteristic equation of multiplicity.
Any matrix of size corresponding to an operator of simple structure is similar to the diagonal matrix
where the transition matrix T from the original basis to the basis of eigenvectors has as its columns column vectors from the coordinates of the eigenvectors of the matrix (operator A).
Example.Bring the matrix of a linear operator to a diagonal form
Let's compose the characteristic equation and find its roots.
Whence the eigenvalues \u200b\u200bof 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, a vector.
The eigenvectors corresponding 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 relationship between similar matrices and is determined by the relation
Tasks
Find eigenvectors and eigenvalues
linear operators given 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 \u200b\u200bare 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 under an operator A acting in X if its image also belongs to each vector.
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 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 the union of the bases and is the block matrix
where are square matrices, 0 is a zero matrix.
3. In every subspace invariant under 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 be . Then, since the zero vector is contained in any linear subspace. Consequently, the kernel is an A-invariant subspace.
Example 2.Let in some basis of the space X the operator A is given by the matrix determined by the equation and
5. Prove that any subspace 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 with respect to A and determine their number.
Changing the coordinates of the vector and the operator matrix when passing 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 decompose the “new” basis vectors into linear combinations of the “old” basis vectors:
The matrix standing here m column of which 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
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 the transition matrix from the “old” basis to the “new” one.
Let us now see how matrices and the same operator are related to each other in different bases and spaces of the Matrix and are determined by equalities Let This equality in the basis is equivalent to the matrix equality
and in the basis of matrix equality (here the same notation is used as in (1)). Using theorem (1), we have
since the column is arbitrary, we obtain the equality
The following result is proved.
Theorem 2.If the matrix of the operator in the basis and the matrix of the same operator in the basis then
Remark 1.Two arbitrary matrices and related by the relation where is some nondegenerate 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
Decision. 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. This result can be generalized to operators acting from one linear space to another. Let the operator act from a linear space into 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 a 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 there be given a linear operator acting from a linear space into a linear space The following concepts are useful in solving linear equations.
Definition 1. Kernel operatorcalled the set
Operator imagecalled the set
It is easy to prove the following statement.
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 spaces and, accordingly) and solve the corresponding algebraic system of equations. Let us now explain how the image of an operator can be calculated.
Let the matrix of the operator in in the bases and Denote by the th column of the matrix The belonging of a vector to an image means that there are numbers such that the vector column is represented in the form i.e. is an element of the space of linear combinations of matrix columns Having chosen a basis in this space (for example, the maximum collection of linearly independent matrix columns), we first calculate the image matrix operator : and then build the image of the operator:
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).
Decision.Let us choose some basis in space (for example, a standard basis). In this basis, the matrix of the projection operator is found from the equality Let us find the images of the basis vectors. Since the plane passes through the axis, then
In this way,
Hence, 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).
