What are eigenvectors and eigenvalues. Eigenvalues (numbers) and eigenvectors. Examples of solutions

How to insert mathematical formulas on a website?

If you ever need to add one or two mathematical formulas to a web page, then the easiest way to do this is as described in the article: mathematical formulas are easily inserted onto the site in the form of pictures that are automatically generated by Wolfram Alpha. In addition to simplicity, this universal method will help improve the visibility of the site in search engines. It has been working for a long time (and, I think, will work forever), but is already morally outdated.

If you regularly use mathematical formulas on your site, then I recommend you use MathJax - a special JavaScript library that displays mathematical notation in web browsers using MathML, LaTeX or ASCIIMathML markup.

There are two ways to start using MathJax: (1) using a simple code, you can quickly connect a MathJax script to your website, which will be automatically loaded from a remote server at the right time (list of servers); (2) download the MathJax script from a remote server to your server and connect it to all pages of your site. The second method - more complex and time-consuming - will speed up the loading of your site's pages, and if the parent MathJax server becomes temporarily unavailable for some reason, this will not affect your own site in any way. Despite these advantages, I chose the first method as it is simpler, faster and does not require technical skills. Follow my example, and in just 5 minutes you will be able to use all the features of MathJax on your site.

You can connect the MathJax library script from a remote server using two code options taken from the main MathJax website or on the documentation page:

One of these code options needs to be copied and pasted into the code of your web page, preferably between tags and or immediately after the tag. According to the first option, MathJax loads faster and slows down the page less. But the second option automatically monitors and loads the latest versions of MathJax. If you insert the first code, it will need to be updated periodically. If you insert the second code, the pages will load more slowly, but you will not need to constantly monitor MathJax updates.

The easiest way to connect MathJax is in Blogger or WordPress: in the site control panel, add a widget designed to insert third-party JavaScript code, copy the first or second version of the download code presented above into it, and place the widget closer to the beginning of the template (by the way, this is not at all necessary , since the MathJax script is loaded asynchronously). That's all. Now learn the markup syntax of MathML, LaTeX, and ASCIIMathML, and you are ready to insert mathematical formulas into your site's web pages.

Any fractal is constructed according to a certain rule, which is consistently applied an unlimited number of times. Each such time is called an iteration.

The iterative algorithm for constructing a Menger sponge is quite simple: the original cube with side 1 is divided by planes parallel to its faces into 27 equal cubes. One central cube and 6 cubes adjacent to it along the faces are removed from it. The result is a set consisting of the remaining 20 smaller cubes. Doing the same with each of these cubes, we get a set consisting of 400 smaller cubes. Continuing this process endlessly, we get a Menger sponge.

SYSTEM OF HOMOGENEOUS LINEAR EQUATIONS

A system of homogeneous linear equations is a system of the form

It is clear that in this case , because all elements of one of the columns in these determinants are equal to zero.

Since the unknowns are found according to the formulas , then in the case when Δ ≠ 0, the system has a unique zero solution x = y = z= 0. However, in many problems the interesting question is whether a homogeneous system has solutions other than zero.

Theorem. In order for a system of linear homogeneous equations to have a non-zero solution, it is necessary and sufficient that Δ ≠ 0.

So, if the determinant Δ ≠ 0, then the system has a unique solution. If Δ ≠ 0, then the system of linear homogeneous equations has an infinite number of solutions.

Examples.

Eigenvectors and eigenvalues of a matrix

Let a square matrix be given , X– some matrix-column, the height of which coincides with the order of the matrix A. .

In many problems we have to consider the equation for X

where λ is a certain number. It is clear that for any λ this equation has a zero solution.

The number λ for which this equation has non-zero solutions is called eigenvalue matrices A, A X for such λ is called eigenvector matrices A.

Let's find the eigenvector of the matrix A. Because the E∙X = X, then the matrix equation can be rewritten as or . In expanded form, this equation can be rewritten as a system of linear equations. Really .

And therefore

So, we have obtained a system of homogeneous linear equations for determining the coordinates x 1, x 2, x 3 vector X. For a system to have non-zero solutions it is necessary and sufficient that the determinant of the system be equal to zero, i.e.

This is a 3rd degree equation for λ. It's called characteristic equation matrices A and serves to determine the eigenvalues of λ.

Each eigenvalue λ corresponds to an eigenvector X, whose coordinates are determined from the system at the corresponding value of λ.

Examples.

VECTOR ALGEBRA. THE CONCEPT OF VECTOR

When studying various branches of physics, there are quantities that are completely determined by specifying their numerical values, for example, length, area, mass, temperature, etc. Such quantities are called scalar. However, in addition to them, there are also quantities, to determine which, in addition to the numerical value, it is also necessary to know their direction in space, for example, the force acting on the body, the speed and acceleration of the body when it moves in space, the magnetic field strength at a given point in space and etc. Such quantities are called vector quantities.

Let us introduce a strict definition.

Directed segment Let's call a segment, relative to the ends of which it is known which of them is the first and which is the second.

Vector called a directed segment having a certain length, i.e. This is a segment of a certain length, in which one of the points limiting it is taken as the beginning, and the second as the end. If A– the beginning of the vector, B is its end, then the vector is denoted by the symbol; in addition, the vector is often denoted by a single letter. In the figure, the vector is indicated by a segment, and its direction by an arrow.

Module or length A vector is called the length of the directed segment that defines it. Denoted by || or ||.

We will also include the so-called zero vector, whose beginning and end coincide, as vectors. It is designated. The zero vector does not have a specific direction and its modulus is zero ||=0.

Vectors are called collinear, if they are located on the same line or on parallel lines. Moreover, if the vectors and are in the same direction, we will write , opposite.

Vectors located on straight lines parallel to the same plane are called coplanar.

The two vectors are called equal, if they are collinear, have the same direction and are equal in length. In this case they write .

From the definition of equality of vectors it follows that a vector can be transported parallel to itself, placing its origin at any point in space.

For example .

LINEAR OPERATIONS ON VECTORS

Multiplying a vector by a number.

The product of a vector and the number λ is a new vector such that:

The product of a vector and a number λ is denoted by .

For example, there is a vector directed in the same direction as the vector and having a length half as large as the vector.

The introduced operation has the following properties:

Vector addition.

Let and be two arbitrary vectors. Let's take an arbitrary point O and construct a vector. After that from the point A let's put aside the vector. The vector connecting the beginning of the first vector with the end of the second is called amount of these vectors and is denoted .

The formulated definition of vector addition is called parallelogram rule, since the same sum of vectors can be obtained as follows. Let's postpone from the point O vectors and . Let's construct a parallelogram on these vectors OABC. Since vectors, then vector, which is a diagonal of a parallelogram drawn from the vertex O, will obviously be a sum of vectors.

It is easy to check the following properties of vector addition.

Vector difference.

A vector collinear to a given vector, equal in length and oppositely directed, is called opposite vector for a vector and is denoted by . The opposite vector can be considered as the result of multiplying the vector by the number λ = –1: .

An eigenvector of a square matrix is one that, when multiplied by a given matrix, results in a collinear vector. In simple words, when a matrix is multiplied by an eigenvector, the latter remains the same, but multiplied by a certain number.

Definition

An eigenvector is a non-zero vector V, which, when multiplied by a square matrix M, becomes itself increased by some number λ. In algebraic notation it looks like:

M × V = λ × V,

where λ is the eigenvalue of the matrix M.

Let's look at a numerical example. For ease of recording, numbers in the matrix will be separated by a semicolon. Let us have a matrix:

M = 0; 4;
6; 10.

Let's multiply it by a column vector:

V = -2;

When we multiply a matrix by a column vector, we also get a column vector. In strict mathematical language, the formula for multiplying a 2 × 2 matrix by a column vector will look like this:

M × V = M11 × V11 + M12 × V21;
M21 × V11 + M22 × V21.

M11 means the element of matrix M located in the first row and first column, and M22 means the element located in the second row and second column. For our matrix, these elements are equal to M11 = 0, M12 = 4, M21 = 6, M22 10. For a column vector, these values are equal to V11 = –2, V21 = 1. According to this formula, we get the following result of the product of a square matrix by a vector:

M × V = 0 × (-2) + (4) × (1) = 4;
6 × (-2) + 10 × (1) = -2.

For convenience, let's write the column vector into a row. So, we multiplied the square matrix by the vector (-2; 1), resulting in the vector (4; -2). Obviously, this is the same vector multiplied by λ = -2. Lambda in this case denotes the eigenvalue of the matrix.

An eigenvector of a matrix is a collinear vector, that is, an object that does not change its position in space when multiplied by a matrix. The concept of collinearity in vector algebra is similar to the term of parallelism in geometry. In a geometric interpretation, collinear vectors are parallel directed segments of different lengths. Since the time of Euclid, we know that one line has an infinite number of lines parallel to it, so it is logical to assume that each matrix has an infinite number of eigenvectors.

From the previous example it is clear that eigenvectors can be (-8; 4), and (16; -8), and (32, -16). These are all collinear vectors corresponding to the eigenvalue λ = -2. When multiplying the original matrix by these vectors, we will still end up with a vector that differs from the original by 2 times. That is why, when solving problems of finding an eigenvector, it is necessary to find only linearly independent vector objects. Most often, for an n × n matrix, there are an n number of eigenvectors. Our calculator is designed for the analysis of second-order square matrices, so almost always the result will find two eigenvectors, except for cases when they coincide.

In the example above, we knew the eigenvector of the original matrix in advance and clearly determined the lambda number. However, in practice, everything happens the other way around: the eigenvalues are found first and only then the eigenvectors.

Solution algorithm

Let's look at the original matrix M again and try to find both of its eigenvectors. So the matrix looks like:

M = 0; 4;
6; 10.

First we need to determine the eigenvalue λ, which requires calculating the determinant of the following matrix:

(0 − λ); 4;
6; (10 − λ).

This matrix is obtained by subtracting the unknown λ from the elements on the main diagonal. The determinant is determined using the standard formula:

detA = M11 × M21 − M12 × M22
detA = (0 − λ) × (10 − λ) − 24

Since our vector must be non-zero, we accept the resulting equation as linearly dependent and equate our determinant detA to zero.

(0 − λ) × (10 − λ) − 24 = 0

Let's open the brackets and get the characteristic equation of the matrix:

λ 2 − 10λ − 24 = 0

This is a standard quadratic equation that needs to be solved using a discriminant.

D = b 2 − 4ac = (-10) × 2 − 4 × (-1) × 24 = 100 + 96 = 196

The root of the discriminant is sqrt(D) = 14, therefore λ1 = -2, λ2 = 12. Now for each lambda value we need to find the eigenvector. Let us express the system coefficients for λ = -2.

M − λ × E = 2; 4;
6; 12.

In this formula, E is the identity matrix. Based on the resulting matrix, we create a system of linear equations:

2x + 4y = 6x + 12y,

where x and y are the eigenvector elements.

Let's collect all the X's on the left and all the Y's on the right. Obviously - 4x = 8y. Divide the expression by - 4 and get x = –2y. Now we can determine the first eigenvector of the matrix, taking any values of the unknowns (remember the infinity of linearly dependent eigenvectors). Let's take y = 1, then x = –2. Therefore, the first eigenvector looks like V1 = (–2; 1). Return to the beginning of the article. It was this vector object that we multiplied the matrix by to demonstrate the concept of an eigenvector.

Now let's find the eigenvector for λ = 12.

M - λ × E = -12; 4
6; -2.

Let's create the same system of linear equations;

-12x + 4y = 6x − 2y
-18x = -6y
3x = y.

Now we take x = 1, therefore y = 3. Thus, the second eigenvector looks like V2 = (1; 3). When multiplying the original matrix by a given vector, the result will always be the same vector multiplied by 12. This is where the solution algorithm ends. Now you know how to manually determine the eigenvector of a matrix.

determinant;
trace, that is, the sum of the elements on the main diagonal;
rank, that is, the maximum number of linearly independent rows/columns.

The program operates according to the above algorithm, shortening the solution process as much as possible. It is important to point out that in the program lambda is designated by the letter “c”. Let's look at a numerical example.

Example of how the program works

Let's try to determine the eigenvectors for the following matrix:

M = 5; 13;
4; 14.

Let's enter these values into the cells of the calculator and get the answer in the following form:

Matrix rank: 2;
Matrix determinant: 18;
Matrix trace: 19;
Calculation of the eigenvector: c 2 − 19.00c + 18.00 (characteristic equation);
Eigenvector calculation: 18 (first lambda value);
Eigenvector calculation: 1 (second lambda value);
System of equations for vector 1: -13x1 + 13y1 = 4x1 − 4y1;
System of equations for vector 2: 4x1 + 13y1 = 4x1 + 13y1;
Eigenvector 1: (1; 1);
Eigenvector 2: (-3.25; 1).

Thus, we obtained two linearly independent eigenvectors.

Conclusion

Linear algebra and analytical geometry are standard subjects for any freshman engineering major. The large number of vectors and matrices is terrifying, and it is easy to make mistakes in such cumbersome calculations. Our program will allow students to check their calculations or automatically solve the problem of finding an eigenvector. There are other linear algebra calculators in our catalog; use them in your studies or work.

Definition 9.3. Vector X is called the eigenvector of the matrix A, if there is such a number λ, that the equality holds: Ах = λх, that is, the result of applying to X linear transformation specified by the matrix A, is the multiplication of this vector by the number λ . The number itself λ is called the eigenvalue of the matrix A.

Substituting into formulas (9.3) x` j = λx j , we obtain a system of equations for determining the coordinates of the eigenvector:

. (9.5)

This linear homogeneous system will have a nontrivial solution only if its main determinant is 0 (Cramer's rule). By writing this condition in the form:

we obtain an equation for determining the eigenvalues λ , called the characteristic equation. Briefly it can be represented as follows:

| A - λE | = 0, (9.6)

since its left side contains the determinant of the matrix A-λE. Polynomial relative λ | A - λE| is called the characteristic polynomial of matrix A.

Properties of the characteristic polynomial:

1) The characteristic polynomial of a linear transformation does not depend on the choice of basis. Proof. (see (9.4)), but hence, . Thus, it does not depend on the choice of basis. This means that | A-λE| does not change when moving to a new basis.

2) If the matrix A linear transformation is symmetric (i.e. and ij =a ji), then all roots of the characteristic equation (9.6) are real numbers.

Properties of eigenvalues and eigenvectors:

1) If you choose a basis from the eigenvectors x 1, x 2, x 3, corresponding to the eigenvalues λ 1, λ 2, λ 3 matrices A, then in this basis the linear transformation A has a matrix of diagonal form:

(9.7) The proof of this property follows from the definition of eigenvectors.

2) If the eigenvalues of the transformation A are different, then their corresponding eigenvectors are linearly independent.

3) If the characteristic polynomial of the matrix A has three different roots, then in some basis the matrix A has a diagonal appearance.

Let's find the eigenvalues and eigenvectors of the matrix Let's create a characteristic equation: (1- λ )(5 - λ )(1 - λ ) + 6 - 9(5 - λ ) - (1 - λ ) - (1 - λ ) = 0, λ ³ - 7 λ ² + 36 = 0, λ 1 = -2, λ 2 = 3, λ 3 = 6.

Let's find the coordinates of the eigenvectors corresponding to each found value λ. From (9.5) it follows that if X (1) ={x 1 ,x 2 ,x 3) – eigenvector corresponding λ 1 =-2, then

- a cooperative but uncertain system. Its solution can be written in the form X (1) ={a,0,-a), where a is any number. In particular, if we require that | x (1) |=1, X (1) =

Substituting into system (9.5) λ 2 =3, we obtain a system for determining the coordinates of the second eigenvector - x (2) ={y 1 ,y 2 ,y 3}:

, where X (2) ={b,-b,b) or, provided | x (2) |=1, x (2) =

For λ 3 = 6 find the eigenvector x (3) ={z 1 , z 2 , z 3}:

, x (3) ={c,2c,c) or in the normalized version

x(3) = It can be noticed that X (1) X (2) = ab–ab= 0, x (1) x (3) = ac-ac= 0, x (2) x (3) = bc- 2bc + bc= 0. Thus, the eigenvectors of this matrix are pairwise orthogonal.

Lecture 10.

Quadratic forms and their connection with symmetric matrices. Properties of eigenvectors and eigenvalues of a symmetric matrix. Reducing a quadratic form to canonical form.

Definition 10.1. Quadratic form of real variables x 1, x 2,…, x n is called a polynomial of the second degree in these variables that does not contain a free term and terms of the first degree.

Examples of quadratic forms:

(n = 2),

(n = 3). (10.1)

Let us recall the definition of a symmetric matrix given in the last lecture:

Definition 10.2. A square matrix is called symmetric if , that is, if the matrix elements that are symmetrical about the main diagonal are equal.

Properties of eigenvalues and eigenvectors of a symmetric matrix:

1) All eigenvalues of a symmetric matrix are real.

Proof (for n = 2).

Let the matrix A has the form: . Let's create a characteristic equation: