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 the 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. Since 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, tension magnetic field at a given point in space, 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 equal to 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

1. 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 that of the vector.

   The introduced operation has the following properties:

2. 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's easy to check the following properties of vector addition.

3. 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: .

"The first part sets out the provisions that are minimally necessary for understanding chemometrics, and the second part contains the facts that you need to know for a deeper understanding of the methods of multivariate analysis. The presentation is illustrated with examples made in the Excel workbook Matrix.xls, which accompanies this document.

Links to examples are placed in the text as Excel objects. These examples are of an abstract nature; they are in no way tied to the problems of analytical chemistry. Real-life examples of the use of matrix algebra in chemometrics are discussed in other texts covering a variety of chemometric applications.

Most measurements made in analytical chemistry are not direct, but indirect. This means that in the experiment, instead of the value of the desired analyte C (concentration), another value is obtained x(signal), related but not equal to C, i.e. x(C) ≠ C. As a rule, the type of dependence x(C) is unknown, but fortunately in analytical chemistry most measurements are proportional. This means that with increasing concentration of C in a times, signal X will increase by the same amount, i.e. x(a C) = a x(C). In addition, the signals are also additive, so the signal from a sample that contains two substances with concentrations C 1 and C 2 will be equal to the sum of the signals from each component, i.e. x(C 1 + C 2) = x(C 1)+ x(C 2). Proportionality and additivity together give linearity. Many examples can be given to illustrate the principle of linearity, but it is enough to mention the two most striking examples - chromatography and spectroscopy. The second feature inherent in an experiment in analytical chemistry is multichannel. Modern analytical equipment simultaneously measures signals for many channels. For example, the intensity of light transmission is measured for several wavelengths at once, i.e. spectrum. Therefore, in the experiment we deal with many signals x 1 , x 2 ,...., x n, characterizing the set of concentrations C 1 , C 2 , ..., C m of substances present in the system under study.

Rice. 1 Spectra

So, an analytical experiment is characterized by linearity and multidimensionality. Therefore, it is convenient to consider experimental data as vectors and matrices and manipulate them using the apparatus of matrix algebra. The fruitfulness of this approach is illustrated by the example shown in, which presents three spectra taken at 200 wavelengths from 4000 to 4796 cm −1. First ( x 1) and second ( x 2) the spectra were obtained for standard samples in which the concentrations of two substances A and B are known: in the first sample [A] = 0.5, [B] = 0.1, and in the second sample [A] = 0.2, [B] = 0.6. What can be said about a new, unknown sample, the spectrum of which is indicated x 3 ?

Let us consider three experimental spectra x 1 , x 2 and x 3 as three vectors of dimension 200. Using linear algebra, one can easily show that x 3 = 0.1 x 1 +0.3 x 2, so the third sample obviously contains only substances A and B in concentrations [A] = 0.5×0.1 + 0.2×0.3 = 0.11 and [B] = 0.1×0.1 + 0.6×0.3 = 0.19.

1. Basic information

1.1 Matrices

Matrix called a rectangular table of numbers, for example

Rice. 2 Matrix

Matrices are denoted by capital bold letters ( A), and their elements - by corresponding lowercase letters with indices, i.e. a ij. The first index numbers the rows, and the second - the columns. In chemometrics, it is customary to denote the maximum value of an index with the same letter as the index itself, but in capital letters. Therefore the matrix A can also be written as ( a ij , i = 1,..., I; j = 1,..., J). For the example matrix I = 4, J= 3 and a 23 = −7.5.

Pair of numbers I And J is called the dimension of the matrix and is denoted as I× J. An example of a matrix in chemometrics is a set of spectra obtained for I samples for J wavelengths.

1.2. The simplest operations with matrices

Matrices can be multiply by numbers. In this case, each element is multiplied by this number. For example -

Rice. 3 Multiplying a matrix by a number

Two matrices of the same dimension can be element by element fold And subtract. For example,

Rice. 4 Matrix addition

As a result of multiplication by a number and addition, a matrix of the same dimension is obtained.

A zero matrix is ​​a matrix consisting of zeros. It is designated O. It's obvious that A+O = A, A−A = O and 0 A = O.

The matrix can be transpose. During this operation, the matrix is ​​flipped, i.e. rows and columns are swapped. Transposition is indicated by a prime, A" or index A t. Thus, if A = {a ij , i = 1,..., I; j = 1,...,J), That A t = ( a ji , j = 1,...,J; i = 1,..., I). For example

Rice. 5 Matrix transposition

It is obvious that ( A t) t = A, (A+B) t = A t+ B t.

1.3. Matrix multiplication

Matrices can be multiply, but only if they have the appropriate dimensions. Why this is so will be clear from the definition. Matrix product A, dimension I× K, and matrices B, dimension K× J, is called a matrix C, dimension I× J, whose elements are numbers

Thus for the product AB it is necessary that the number of columns in the left matrix A was equal to the number of rows in the right matrix B. An example of a matrix product -

Fig.6 Product of matrices

The rule for matrix multiplication can be formulated as follows. To find a matrix element C, standing at the intersection i-th line and j th column ( c ij) must be multiplied element by element i-th row of the first matrix A on j th column of the second matrix B and add up all the results. So in the example shown, an element from the third row and the second column is obtained as the sum of the element-wise products of the third row A and second column B

Fig.7 Element of the product of matrices

The product of matrices depends on the order, i.e. AB ≠ B.A., at least for dimensional reasons. They say that it is non-commutative. However, the product of matrices is associative. This means that ABC = (AB)C = A(B.C.). In addition, it is also distributive, i.e. A(B+C) = AB+A.C.. It's obvious that A.O. = O.

1.4. Square matrices

If the number of matrix columns is equal to the number of its rows ( I = J=N), then such a matrix is ​​called square. In this section we will consider only such matrices. Among these matrices, matrices with special properties can be distinguished.

Single matrix (denoted I, and sometimes E) is a matrix in which all elements are equal to zero, with the exception of diagonal ones, which are equal to 1, i.e.

Obviously A.I. = I.A. = A.

The matrix is ​​called diagonal, if all its elements except diagonal ones ( a ii) are equal to zero. For example

Rice. 8 Diagonal matrix

Matrix A called the top triangular, if all its elements lying below the diagonal are equal to zero, i.e. a ij= 0, at i>j. For example

Rice. 9 Upper triangular matrix

The lower triangular matrix is ​​defined similarly.

Matrix A called symmetrical, If A t = A. In other words a ij = a ji. For example

Rice. 10 Symmetric matrix

Matrix A called orthogonal, If

A t A = A.A. t = I.

The matrix is ​​called normal If

1.5. Trace and determinant

Next square matrix A(denoted by Tr( A) or Sp( A)) is the sum of its diagonal elements,

For example,

Rice. 11 Matrix trace

It's obvious that

Sp(α A) = α Sp( A) And

Sp( A+B) = Sp( A)+ Sp( B).

It can be shown that

Sp( A) = Sp( A t), Sp( I) = N,

and also that

Sp( AB) = Sp( B.A.).

Another important characteristic of a square matrix is ​​its determinant(denoted det( A)). Determining the determinant in the general case is quite difficult, so we will start with the simplest option - the matrix A dimension (2×2). Then

For a (3×3) matrix the determinant will be equal to

In the case of the matrix ( N× N) the determinant is calculated as the sum 1·2·3· ... · N= N! terms, each of which is equal

Indexes k 1 , k 2 ,..., k N are defined as all possible ordered permutations r numbers in the set (1, 2, ..., N). Calculating the determinant of a matrix is ​​a complex procedure, which in practice is carried out using special programs. For example,

Rice. 12 Matrix determinant

Let us note only the obvious properties:

det( I) = 1, det( A) = det( A t),

det( AB) = det( A)det( B).

1.6. Vectors

If the matrix consists of only one column ( J= 1), then such an object is called vector. More precisely, a column vector. For example

One can also consider matrices consisting of one row, for example

This object is also a vector, but row vector. When analyzing data, it is important to understand which vectors we are dealing with - columns or rows. So the spectrum taken for one sample can be considered as a row vector. Then the set of spectral intensities at a certain wavelength for all samples should be treated as a column vector.

The dimension of a vector is the number of its elements.

It is clear that any column vector can be turned into a row vector by transposition, i.e.

In cases where the shape of the vector is not specifically specified, but is simply said to be a vector, then they mean a column vector. We will also adhere to this rule. A vector is denoted by a lowercase, upright, bold letter. A zero vector is a vector all of whose elements are zero. It is designated 0 .

1.7. The simplest operations with vectors

Vectors can be added and multiplied by numbers in the same way as matrices. For example,

Rice. 13 Operations with vectors

Two vectors x And y are called colinear, if there is a number α such that

1.8. Products of vectors

Two vectors of the same dimension N can be multiplied. Let there be two vectors x = (x 1 , x 2 ,...,x N)t and y = (y 1 , y 2 ,...,y N) t . Guided by the row-by-column multiplication rule, we can compose two products from them: x t y And xy t. First work

called scalar or internal. Its result is a number. It is also denoted by ( x,y)= x t y. For example,

Rice. 14 Inner (scalar) product

Second piece

called external. Its result is a matrix of dimension ( N× N). For example,

Rice. 15 External work

Vectors whose scalar product is zero are called orthogonal.

1.9. Vector norm

The scalar product of a vector with itself is called a scalar square. This value

defines a square length vector x. To indicate length (also called the norm vector) the notation is used

For example,

Rice. 16 Vector norm

Unit length vector (|| x|| = 1) is called normalized. Nonzero vector ( x ≠ 0 ) can be normalized by dividing it by length, i.e. x = ||x|| (x/||x||) = ||x|| e. Here e = x/||x|| - normalized vector.

Vectors are called orthonormal if they are all normalized and pairwise orthogonal.

1.10. Angle between vectors

The scalar product determines and cornerφ between two vectors x And y

If the vectors are orthogonal, then cosφ = 0 and φ = π/2, and if they are colinear, then cosφ = 1 and φ = 0.

1.11. Vector representation of a matrix

Each matrix A size I× J can be represented as a set of vectors

Here every vector a j is j the th column, and the row vector b i is i th row of the matrix A

1.12. Linearly dependent vectors

Vectors of the same dimension ( N) can be added and multiplied by a number, just like matrices. The result will be a vector of the same dimension. Let there be several vectors of the same dimension x 1 , x 2 ,...,x K and the same number of numbers α α 1 , α 2 ,...,α K. Vector

y= α 1 x 1 + α 2 x 2 +...+ α K x K

called linear combination vectors x k .

If there are such non-zero numbers α k ≠ 0, k = 1,..., K, What y = 0 , then such a set of vectors x k called linearly dependent. Otherwise, the vectors are called linearly independent. For example, vectors x 1 = (2, 2)t and x 2 = (−1, −1) t are linearly dependent, because x 1 +2x 2 = 0

1.13. Matrix rank

Consider a set of K vectors x 1 , x 2 ,...,x K dimensions N. The rank of this system of vectors is the maximum number of linearly independent vectors. For example in the set

there are only two linearly independent vectors, for example x 1 and x 2, so its rank is 2.

Obviously, if there are more vectors in a set than their dimension ( K>N), then they are necessarily linearly dependent.

Matrix rank(denoted by rank( A)) is the rank of the system of vectors of which it consists. Although any matrix can be represented in two ways (column or row vectors), this does not affect the rank value, because

1.14. Inverse matrix

Square matrix A is called non-degenerate if it has a unique reverse matrix A-1, determined by the conditions

A.A. −1 = A −1 A = I.

The inverse matrix does not exist for all matrices. A necessary and sufficient condition for non-degeneracy is

det( A) ≠ 0 or rank( A) = N.

Matrix inversion is a complex procedure for which there are special programs. For example,

Rice. 17 Matrix inversion

Let us present the formulas for the simplest case - a 2×2 matrix

If matrices A And B are non-degenerate, then

(AB) −1 = B −1 A −1 .

1.15. Pseudoinverse matrix

If matrix A is singular and the inverse matrix does not exist, then in some cases you can use pseudoinverse matrix, which is defined as such a matrix A+ that

A.A. + A = A.

The pseudoinverse matrix is ​​not the only one and its form depends on the method of construction. For example, for a rectangular matrix you can use the Moore-Penrose method.

If the number of columns is less than the number of rows, then

A + =(A t A) −1 A t

For example,

Rice. 17a Pseudo-inversion of a matrix

If the number of columns is greater than the number of rows, then

A + =A t( A.A. t) −1

1.16. Multiplying a vector by a matrix

Vector x can be multiplied by a matrix A suitable size. In this case, the column vector is multiplied on the right Ax, and the vector row is on the left x t A. If the vector dimension J, and the matrix dimension I× J then the result will be a vector of dimension I. For example,

Rice. 18 Multiplying a vector by a matrix

If matrix A- square ( I× I), then the vector y = Ax has the same dimension as x. It's obvious that

A(α 1 x 1 + α 2 x 2) = α 1 Ax 1 + α 2 Ax 2 .

Therefore, matrices can be considered as linear transformations of vectors. In particular Ix = x, Ox = 0 .

2. Additional information

2.1. Systems of linear equations

Let A- matrix size I× J, A b- dimension vector J. Consider the equation

Ax = b

relative to the vector x, dimensions I. Essentially, it is a system of I linear equations with J unknown x 1 ,...,x J. A solution exists if and only if

rank( A) = rank( B) = R,

Where B is an extended matrix of dimensions I×( J+1), consisting of a matrix A, supplemented by a column b, B = (A b). Otherwise, the equations are inconsistent.

If R = I = J, then the solution is unique

x = A −1 b.

If R < I, then there are many different solutions that can be expressed through a linear combination J−R vectors. System of homogeneous equations Ax = 0 with square matrix A (N× N) has a nontrivial solution ( x ≠ 0 ) if and only if det( A) = 0. If R= rank( A)<N, then there are N−R linearly independent solutions.

2.2. Bilinear and quadratic forms

If A is a square matrix, and x And y- vector of the corresponding dimension, then the scalar product of the form x t Ay called bilinear form defined by matrix A. At x = y expression x t Ax called quadratic form.

2.3. Positive definite matrices

Square matrix A called positive definite, if for any nonzero vector x ≠ 0 ,

x t Ax > 0.

Similarly defined negative (x t Ax < 0), non-negative (x t Ax≥ 0) and negative (x t Ax≤ 0) certain matrices.

2.4. Cholesky decomposition

If the symmetric matrix A is positive definite, then there is a unique triangular matrix U with positive elements, for which

A = U t U.

For example,

Rice. 19 Cholesky decomposition

2.5. Polar decomposition

Let A is a non-singular square matrix of dimension N× N. Then there is a unique polar performance

A = S.R.

Where S is a non-negative symmetric matrix, and R is an orthogonal matrix. Matrices S And R can be defined explicitly:

S 2 = A.A. t or S = (A.A. t) ½ and R = S −1 A = (A.A. t) −½ A.

For example,

Rice. 20 Polar decomposition

If matrix A is degenerate, then the decomposition is not unique - namely: S still alone, but R maybe a lot. Polar decomposition represents the matrix A as a combination of compression/extension S and turn R.

2.6. Eigenvectors and eigenvalues

Let A is a square matrix. Vector v called eigenvector matrices A, If

Av = λ v,

where the number λ is called eigenvalue matrices A. Thus, the transformation that the matrix performs A above the vector v, comes down to simple stretching or compression with a coefficient λ. The eigenvector is determined up to multiplication by a constant α ≠ 0, i.e. If v is an eigenvector, then α v- also an eigenvector.

2.7. Eigenvalues

At the matrix A, dimension ( N× N) cannot be more than N eigenvalues. They satisfy characteristic equation

det( A − λ I) = 0,

which is an algebraic equation N-th order. In particular, for a 2×2 matrix the characteristic equation has the form

For example,

Rice. 21 Eigenvalues

Set of eigenvalues ​​λ 1 ,..., λ N matrices A called spectrum A.

The spectrum has various properties. In particular

det( A) = λ 1 ×...×λ N,Sp( A) = λ 1 +...+λ N.

The eigenvalues ​​of an arbitrary matrix can be complex numbers, but if the matrix is ​​symmetric ( A t = A), then its eigenvalues ​​are real.

2.8. Eigenvectors

At the matrix A, dimension ( N× N) cannot be more than N eigenvectors, each of which corresponds to its own eigenvalue. To determine the eigenvector v n need to solve a system of homogeneous equations

(A − λ n I)v n = 0 .

It has a non-trivial solution, since det( A −λ n I) = 0.

For example,

Rice. 22 Eigenvectors

The eigenvectors of a symmetric matrix are orthogonal.

Eigenvalues ​​(numbers) and eigenvectors.
Examples of solutions

Be yourself

From both equations it follows that .

Let's put it then: .

As a result: – second eigenvector.

Let's repeat important points solutions:

– the resulting system certainly has a general solution (the equations are linearly dependent);

– we select the “y” in such a way that it is integer and the first “x” coordinate is integer, positive and as small as possible.

– we check that the particular solution satisfies each equation of the system.

Answer .

There were quite enough intermediate “checkpoints”, so checking equality is, in principle, unnecessary.

In various sources of information, the coordinates of eigenvectors are often written not in columns, but in rows, for example: (and, to be honest, I myself am used to writing them down in lines). This option is acceptable, but in light of the topic linear transformations technically more convenient to use column vectors.

Perhaps the solution seemed very long to you, but this is only because I commented on the first example in great detail.

Example 2

Matrices

Let's train on our own! An approximate example of a final task at the end of the lesson.

Sometimes you need to do additional task, namely:

write the canonical matrix decomposition

What is it?

If the eigenvectors of the matrix form basis, then it can be represented as:

Where is a matrix composed of coordinates of eigenvectors, – diagonal matrix with corresponding eigenvalues.

This matrix decomposition is called canonical or diagonal.

Let's look at the matrix of the first example. Its eigenvectors linearly independent(non-collinear) and form a basis. Let's create a matrix of their coordinates:

On main diagonal matrices in the appropriate order the eigenvalues ​​are located, and the remaining elements are equal to zero:
– I once again emphasize the importance of order: “two” corresponds to the 1st vector and is therefore located in the 1st column, “three” – to the 2nd vector.

Using the usual algorithm for finding inverse matrix or Gauss-Jordan method we find . No, that's not a typo! - before you is a rare event, like a solar eclipse, when the reverse coincided with the original matrix.

It remains to write down the canonical decomposition of the matrix:

The system can be solved using elementary transformations, and in the following examples we will resort to this method. But here the “school” method works much faster. From the 3rd equation we express: – substitute into the second equation:

Since the first coordinate is zero, we obtain a system, from each equation of which it follows that .

And again pay attention to the mandatory presence of a linear relationship. If only a trivial solution is obtained , then either the eigenvalue was found incorrectly, or the system was compiled/solved with an error.

Compact coordinates gives the value

Eigenvector:

And once again, we check that the solution found satisfies every equation of the system. In subsequent paragraphs and in subsequent tasks, I recommend taking this wish as a mandatory rule.

2) For the eigenvalue, using the same principle, we obtain the following system:

From the 2nd equation of the system we express: – substitute into the third equation:

Since the “zeta” coordinate is equal to zero, we obtain a system from each equation of which a linear dependence follows.

Let

Checking that the solution satisfies every equation of the system.

Thus, the eigenvector is: .

3) And finally, the system corresponds to the eigenvalue:

The second equation looks the simplest, so let’s express it and substitute it into the 1st and 3rd equations:

Everything is fine - a linear relationship has emerged, which we substitute into the expression:

As a result, “x” and “y” were expressed through “z”: . In practice, it is not necessary to achieve precisely such relationships; in some cases it is more convenient to express both through or and through . Or even “train” - for example, “X” through “I”, and “I” through “Z”

Let's put it then:

We check that the solution found satisfies each equation of the system and writes the third eigenvector

Answer: eigenvectors:

Geometrically, these vectors define three different spatial directions ("back and forth"), according to which linear transformation transforms non-zero vectors (eigenvectors) into collinear vectors.

If the condition required finding the canonical decomposition, then this is possible here, because different eigenvalues ​​correspond to different linearly independent eigenvectors. Making a matrix from their coordinates, diagonal matrix from relevant eigenvalues ​​and find inverse matrix .

If, by condition, you need to write linear transformation matrix in the basis of eigenvectors, then we give the answer in the form . There is a difference, and the difference is significant! Because this matrix is ​​the “de” matrix.

Problem with more simple calculations For independent decision:

Example 5

Find eigenvectors of a linear transformation given by a matrix

When finding your own numbers, try not to go all the way to a 3rd degree polynomial. In addition, your system solutions may differ from my solutions - there is no certainty here; and the vectors you find may differ from the sample vectors up to the proportionality of their respective coordinates. For example, and. It is more aesthetically pleasing to present the answer in the form, but it’s okay if you stop at the second option. However, there are reasonable limits to everything; the version no longer looks very good.

An approximate final sample of the assignment at the end of the lesson.

How to solve the problem in the case of multiple eigenvalues?

The general algorithm remains the same, but it has its own characteristics, and it is advisable to keep some parts of the solution in a more strict academic style:

Example 6

Find eigenvalues ​​and eigenvectors

Solution

Of course, let’s capitalize the fabulous first column:

And, after factoring the quadratic trinomial:

As a result, eigenvalues ​​are obtained, two of which are multiples.

Let's find the eigenvectors:

1) Let’s deal with a lone soldier according to a “simplified” scheme:

From the last two equations, the equality is clearly visible, which, obviously, should be substituted into the 1st equation of the system:

You won't find a better combination:
Eigenvector:

2-3) Now we remove a couple of sentries. In this case it may turn out either two or one eigenvector. Regardless of the multiplicity of the roots, we substitute the value into the determinant which brings us the next homogeneous system of linear equations:

Eigenvectors are exactly vectors
fundamental system of solutions

Actually, throughout the entire lesson we did nothing but find the vectors of the fundamental system. It’s just that for the time being this term was not particularly required. By the way, those clever students who missed the topic in camouflage suits homogeneous equations, will be forced to smoke it now.

The only action was to remove the extra lines. The result is a one-by-three matrix with a formal “step” in the middle.
– basic variable, – free variables. There are two free variables, therefore, there are also two vectors of the fundamental system.

Let's express the basic variable in terms of free variables: . The zero factor in front of the “X” allows it to take on absolutely any values ​​(which is clearly visible from the system of equations).

In the context of this problem, it is more convenient to write the general solution not in a row, but in a column:

The pair corresponds to an eigenvector:
The pair corresponds to an eigenvector:

Note : sophisticated readers can select these vectors orally - simply by analyzing the system , but some knowledge is needed here: there are three variables, system matrix rank- one, which means fundamental decision system consists of 3 – 1 = 2 vectors. However, the found vectors are clearly visible even without this knowledge, purely on an intuitive level. In this case, the third vector will be written even more “beautifully”: . However, I warn you that in another example, a simple selection may not be possible, which is why the clause is intended for experienced people. In addition, why not take, say, as the third vector? After all, its coordinates also satisfy each equation of the system, and the vectors linearly independent. This option, in principle, is suitable, but “crooked”, since the “other” vector is a linear combination of vectors of the fundamental system.

Answer: eigenvalues: , eigenvectors:

A similar example for an independent solution:

Example 7

Find eigenvalues ​​and eigenvectors

An approximate sample of the final design at the end of the lesson.

It should be noted that in both the 6th and 7th examples a triple of linearly independent eigenvectors is obtained, and therefore the original matrix is ​​representable in the canonical decomposition. But such raspberries do not happen in all cases:

Example 8

Solution: Let’s create and solve the characteristic equation:

Let's expand the determinant in the first column:

We carry out further simplifications according to the considered method, avoiding the third-degree polynomial:

– eigenvalues.

Let's find the eigenvectors:

1) There are no difficulties with the root:

Don’t be surprised, in addition to the kit, there are also variables in use - there is no difference here.

From the 3rd equation we express it and substitute it into the 1st and 2nd equations:

From both equations it follows:

Let then:

2-3) For multiple values ​​we get the system .

Let's write down the matrix of the system and, using elementary transformations, bring it to a stepwise form:

www.site allows you to find . The site performs the calculation. In a few seconds the server will give the correct solution. The characteristic equation for the matrix will be an algebraic expression found using the rule for calculating the determinant matrices matrices, while along the main diagonal there will be differences in the values ​​of the diagonal elements and the variable. When calculating characteristic equation for the matrix online, each element matrices will be multiplied with corresponding other elements matrices. Find in mode online only possible for square matrices. Finding operation characteristic equation for the matrix online reduces to calculating the algebraic sum of the product of elements matrices as a result of finding the determinant matrices, only for the purpose of determining characteristic equation for the matrix online. This operation occupies a special place in the theory matrices, allows you to find eigenvalues ​​and vectors using roots. The task of finding characteristic equation for the matrix online consists of multiplying elements matrices followed by summing these products according to a certain rule. www.site finds characteristic equation for the matrix given dimension in mode online. Calculation characteristic equation for the matrix online given its dimension, this is finding a polynomial with numerical or symbolic coefficients, found according to the rule for calculating the determinant matrices- as the sum of the products of the corresponding elements matrices, only for the purpose of determining characteristic equation for the matrix online. Finding a polynomial with respect to a variable for a quadratic matrices, as a definition characteristic equation for the matrix, common in theory matrices. The meaning of the roots of a polynomial characteristic equation for the matrix online used to determine eigenvectors and eigenvalues ​​for matrices. Moreover, if the determinant matrices will be equal to zero, then characteristic equation of the matrix will still exist, unlike the reverse matrices. In order to calculate characteristic equation for the matrix or find for several at once matrices characteristic equations, you need to spend a lot of time and effort, while our server will find in a matter of seconds characteristic equation for matrix online. In this case, the answer to finding characteristic equation for the matrix online will be correct and with sufficient accuracy, even if the numbers when finding characteristic equation for the matrix online will be irrational. On the website www.site character entries are allowed in elements matrices, that is characteristic equation for matrix online can be represented in general symbolic form when calculating characteristic equation of the matrix online. It is useful to check the answer obtained when solving the problem of finding characteristic equation for the matrix online using the site www.site. When performing the operation of calculating a polynomial - characteristic equation of the matrix, you need to be careful and extremely focused when solving this problem. In turn, our site will help you check your decision on the topic characteristic equation of a matrix online. If you do not have time for long checks of solved problems, then www.site will certainly be a convenient tool for checking when finding and calculating characteristic equation for the matrix online.

Instructions

The number k is called an eigenvalue (number) of the matrix A if there is a vector x such that Ax=kx. (1) In this case, the vector x is called the eigenvector of matrix A, corresponding to the number k. In the space R^n (see Fig. 1), matrix A has the form as in the figure.

It is necessary to set the task of finding the vectors of matrix A. Let the eigenvector x be given by coordinates. In matrix form, it will be written as a column matrix, which for convenience should be represented as a transposed row. X=(x1,x2,…,xn)^T. Based on (1), Ax-khx=0 or Ax-kEx=0, where E is the identity matrix (ones on the main diagonal, all other elements are zeros). Then (A-kE)x=0. (2)

Expression (2) of linear homogeneous algebraic equations has a non-zero solution (eigenvector). Therefore, the main determinant of system (2) is equal to zero, that is, |A-kE|=0. (3) The last equality of the eigenvalue k is called the characteristic equation of matrix A and in expanded form has the form (see Fig. 2).

Substituting the root k of the characteristic equation into system (2), a homogeneous system of linear equations with a singular matrix (its determinant is zero). Each non-zero solution of this system is an eigenvector of the matrix A corresponding to a given eigenvalue k (that is, the root of the characteristic equation).

Example. Find the eigenvalues ​​and vectors of matrix A (see Figure 3). Solution. The characteristic equation is presented in Fig. 3. Expand the determinant and find the eigenvalues ​​of the matrix, which are the given equation (3-k)(-1-k)-5=0, (k-3)(k+1)-5=0, k^2-2k -8=0. Its roots are k1=4, k2=-2

a) Eigenvectors corresponding to k1=4 are found by solving the system (A-4kE)х=0. In this case, only one of its equations is required, since the determinant of the system is obviously equal to zero. If we put x=(x1, x2)^T, then the first equation of the system is (1-4)x1+x2=0, -3x1+x2=0. If we assume that x1=1 (but not zero), then x2=3. Since a homogeneous system with a singular matrix has as many non-zero solutions as desired, the entire set of eigenvectors corresponding to the first eigenvalue x =C1(1, 3), C1=const.

b) Find the eigenvectors corresponding to k2=-2. When solving the system (A+2kE)x=0, its first equation is (3+2)x1+x2=0, 5x1+x2=0. If we put x1=1, then x2=-5. The corresponding eigenvectors x =C2(1, 3), C2=const. The total set of all eigenvectors of a given matrix: x = C1(1, 3)+ C2(1, 3).

Sources:

• Piskunov N.S. Differential and integral calculus. M., 1976, - 576 p.
• find eigenvalues ​​and matrix vectors

Matrices, which are a tabular form of recording data, are widely used when working with systems of linear equations. Moreover, the number of equations determines the number of rows of the matrix, and the number of variables determines the order of its columns. As a result, solving linear systems is reduced to operations on matrices, one of which is finding the eigenvalues ​​of the matrix. Their calculation is carried out using the characteristic equation. Eigenvalues ​​can be defined for a square matrix of order m.

Instructions

Write down a given square A. To find its eigenvalues, use the characteristic equation resulting from the condition for a nontrivial solution of a linear homogeneous system, represented in this case by a square matrix. As follows from Cramer, a solution exists only if its determinant is equal to zero. Thus, we can write the equation | A - λE | = 0, where A is the given value, λ is the required numbers, E is the identity matrix in which all elements on the main diagonal are equal to one, and the rest are equal to zero.

Multiply the desired variable λ by the identity matrix E of the same dimension as the given original A. The result of the operation will be a matrix where the values ​​of λ are located along the main diagonal, the remaining elements remain equal to zero.