Characteristically equal. Ways of folding the characteristic alignment. Wonder what is "Characteristic equal" in other dictionaries

Appointment. The characteristic alignment of the linear operator f is called the alignment of the mind, de - be it deisne number, A is a matrix of a linear operator, E is a single matrix of the same order.

Rich Member called characteristic rich term matrix A (linear operator f). The matrix view has the following characteristic look:

or

.

Also, by equating the characteristic rich term to zero, we take the equal step n, De yak unknowingly speaks λ, ​​otrimuemo the value of yogo roots - the characteristic numbers of this matrix. Characteristic roots play a great role in rich branches of mathematics. Let’s take a look at one of the characteristics of the characteristic root - an even more important tool when following linear spaces, and navіt pіd h vyshіshennya rich applied problems of linear algebra.

The collection of the necessary roots of the characteristic alignment is called the spectrum of the operator f(skin roots are examined from the same multiplicity, which can be characteristic equal).

butt. Know the characteristic roots of the matrix.

Adding a matrix

Bringing the characteristic rich term to zero, it is possible square alignment

Todi root equal to dorivnyu .

Appointment. Let f linear operator in space ta - any non-zero vector, for which equality is valid

de - deisne number. Then the vector is called the own vector of the operator and the matrix is ​​assigned - to the own values, or to the own number of transformations. It seems to him that the power vector should lie to its own value.

Good vectors play a great role both in mathematics itself and in its additions. For example, resonance, at the same time, the frequencies of the system's vibrations, zbіgayutsya with frequent vibrations of external forces. Mathematics's power vectors and corrections are perfect for systems of differential equations.

Theorem. Thus, the linear operator f at the basis (first basis) has the matrix A and at the basis (another basis) - the matrix Y, maє space equal: .

Also, when passing to a new basis, the characteristic rich term of the linear operator does not change.

◌ If T is the matrix of transition from the first basis to another, then . Let's remake the right part of equivalence ●

Theorem. In order for the number 0 from the field P to be the given values ​​of the vector space L n over P, it is necessary and sufficient for the number 0 to be the characteristic root of the operator f.

Doc. I. Necessity. Come on λ 0 operator value f, then in L nІsnuє vlasny vector , such, scho .

Come on – yoga coordinate row at the current basis, then

From the other side, because , de is the matrix of the linear operator in the given basis, then

Equating the right parts (1) and (2) we take:

(3)

Equality (3) means that the number vector with coordinates є rozv'yazkom offensive system rivnyan (4).

(4)

The vector of vіdminniy vіd zero (because vіn vlasny), so the system (4) can be non-zero solution, also її vyznachnik vіvnyuє 0.

(5)

and it means i vyznachnik, which transposes, returns 0.

(6)

in such a manner, λ 0 - The root of the characteristic jealousy.

ІІ. Availability. Come on λ 0 - The characteristic root of the operator in the effective basis . Let us know that λ 0 є available values ​​of the operator A.

Right, that's right λ 0 is a characteristic root, then equality (6) is victorious, also equality (5), but it is still significant that the system (4) can be non-zero solutions.

We choose a non-zero solution of the system (4): a numerical vector . Todі vykonuyutsya jealousy (3).

Let's look at the vector , and for the new one there is equality (2) i, due to the formula At. Zvіdsi viplivaє іvnіst , which means that the vector є in the operator's own vector , which vypovіdat vosne value λ 0 . It was necessary to finish it. The theorem has been completed.

Respect. In order to know the power of the operator, it is necessary to add and split equal (5). In order to know the vectors of the operator, it is necessary to add the system equal (4) and to know the fundamental set of solutions for the system.

To control the correctness of the calculation of their values ​​(stinks can be avoided, complex), two facts are fixed:

1) , where the rest of the sum of the following matrix is ​​the sum of the diagonal elements.

2) .

butt. Know the power values ​​and power vectors .

Equating to zero is acceptable. .

3) . , .

Come on - free change, then Otrimuєmo vector .

Right. Please recheck for the vector.

.

The correct mode of the circuit to lie in the energy reservoir, depends only on the structure of the circuit and the parameters of the її elements. Why is it obvious that the root of the characteristic equalization p1, p2, ... pn will be the same for all change functions(strumіv i naprug).

The characteristic equalization can be put together by different methods. The first method is the classical one, if the characteristic equalization develops strictly similarly to the differential one for the classical scheme. During the development of transitional processes in a folding circuit, a system of “m” differential equalities is formed according to Kirchhoff’s laws for the lancet circuit after switching. Shards of the root of the characteristic equivalence are equal to all the changes, then the solution of the system of differential equivalences is victorious as if it were a change (to choose). As a result, the decisions are not uniformly equal to one change. Fold characteristically equal in a similar way to an omitted differential and signify the same root.

butt. Fold characteristically equal and signify the first root for the changes in the scheme of fig. 59.1. Parameters of the elements in the task of the global looker.

The system of differential equations according to Kirchhoff's laws:

We decompose the system of equalities according to change i3, as a result, we take non-homogeneous differential equalities:

Another way of folding the characteristic equalization field is to add zero to the head variable of the Kirchhoff equalization system for the larger warehouse variables.

Let's go to the warehouse for a full strumu may look iksv = Akept, then:

The system of equalizations for large warehouses should come out of the system of differential equalizations of Kirchhoff by replacing the similar ones by the multiplier p, and the integral - by 1/p. For example, what is being looked at, the equalization system for free warehouse maє looks like:

Characteristically equal that yoga root:

The third way of folding the characteristic alignment (engineering) is based on adding zero to the input operator support of the scheme as it were.

The operator support of the element comes from the th complex support by simply replacing the multiplier jω by p, also

For example, what can be seen:

The third way is the most simple and economical, because more often than not, it gets stuck during the construction of transitional processes in electric lances.

The root of the characteristic equalization characterizes the transitional process in a circuit without energy sources. Such a process proceeds from the expenditure of energy and thus fades away in an hour. Why is it obvious that the root of the characteristic equal can be negative, or the mother of a negative speech part.

In the wild type, the order of differential equalization, which describes the transitional process in the scheme, and, therefore, the steps of the characteristic equalization, that number of roots equals the number of independent cob minds, as well as the number of independent accumulating energy (coil L and capacitors C). If in the circuit of the lancet there are parallel capacitors C1, C2, ... or in series the coils L1, L2, ..., then when the transitional processes are opened, they are to be replaced by one equivalent element CE = C1 + C2 + ... or LE = L1 + L2 + ...

In this way, the savage view of the solution for whether it is a change during the development of a transitional process can be more folded into an analysis of the lancet scheme, without folding that rozvyazannya system of differential equalizations.

For example, what can be seen more.

The characteristic equalization is formed for the lancer after commutation. It can be taken off in the following ways:

• directly from the improvement of the differential equality of the mind (2) (div. lecture No. 24), tobto. by way of exclusion from the system of equalities, which describe the electromagnetic camp of the Lanziug on the basis of the first and the other laws of Kirchhoff, all unknown values, crim one, which should be recorded equal to (2);
• with a path of victorious viraz for the input support of the lancet on the sinusoidal stream;
• s urakhuvannyam vyslovlyuvannya head vyznachnik.

According to the first method of the previous lecture, the differential equalization of the voltage on the capacitors for the subsequent R-L-C-lance was taken away, on the basis of which the characteristic equalization is recorded.

Slid zaznachit, scho, oskіlki linіyny lancets ohopleniya single transitional process, the roots of the characteristic equalization є spіlnymi for all vіlny warehouse voltages and strumіv gіlok circuit, the parameters of which are included to the characteristic equalization. To that, after the first way of folding the characteristic equivalence, I’ll change it, if it’s written down, maybe it’s ok.

The development of the other and the third way of folding the characteristic equalization can be seen from the butt of the lancet fig. one.

The folding of the characteristic alignment for the method of the input support of the field in the step:

the entrance opera of the lansyug is recorded on the snake stream;

jw is replaced by the p operator;

otrimaniy viraz dorivnyuє zero.

Rivnyannia

zbіgaєtsya s characteristic.

The next thing to say is that the input opera can be written down at any time, I will explore whether it’s a big scheme. With this, the active bipolar is replaced by a passive one by analogy with the method of an equivalent generator. The Danish way of folding the characteristic equalization of transferring power to the scheme of magnetic connections; for the obviousness of such, it is necessary to set them up ahead of time.

For the lanceug in Fig. 1 shodo

.

Replacing jw with p and equating viraz depletion to zero, we write

When folding the characteristic equalization on the basis of the head designator, the number of equalizations of algebra, on the basis of such wines, is recorded, adding to the number of unknown warehouse streams. Algebraicization of the external system of integro-differential equalities, folding, for example, on the basis of Kirchhoff's laws, or by the method of contour streams, is carried out by replacing the symbols of differentiation and integrating into the multiplier and into the operator nar. Characteristically equalizes by way of bringing the recorded signifier to zero. Oskіlki viraz head vyznachnik deposited in the right parts of the system of heterogeneous rivnyan, yogo folding can be carried out the basis of the system rіvnyan, records for new streams.

For the lanceug in Fig. 1 algebraic alignment system based on the method of contour strumming can be seen

Zvіdsi viraz for the head designer of the qієї system

Setting D to zero, we take away the result similar to (1).

The global method of rozrahunka of transitional processes by the classical method

In the past, the method of rozrahunka of transitional processes by the classical method includes the following steps:

Apply the rosette of transitional processes using the classic method

1. Transitional processes in R-L lansyug when connected to the power supply

Such processes can be carried out, for example, when connected to the power supply of electric magnets, transformers, electric motors, etc.

Let's look at two points:

Zgіdno with a reviewed technique for the struma in the lance in fig. 2 can be written

Characteristically equal

the stars are fast .

in such a manner,

Substituting (4) and (5) in relation to (3), we write

.

Vіdpovidno to the first law of commutation. Todi

,

In this rank, the strum of the lancer in the transitional process is described to equals

,

and the voltage on the coils of inductance - by viraz

.

The same type of curves and other similar solutions, representations in fig. 3.

With another type of dzherel vimushena, the warehouse is repaid for an additional symbolic method:

,

Viraz vіlnoї skladovoї to deposit in the type of dzherel naprugi. Otzhe,

.

Oscilki, then

In this rank, the rest is taken

Analysis of taken virase (6) shows:

If it is significant for the magnitude, then for the first period the storage warehouse does not change. At this peak, the maximum value of the strum of the transitional process can completely change the amplitude of the strum to the regime that has risen. As can be seen from Fig. 4, de

maximum struma can be approximately in . At the boundary at .

In this way, for a linear lancet, the maximum value of the struma of the transitional mode cannot exceed the sub-amplitude of the driven struma: .

It is similar for a linear lance with a capacitor: if at the moment of switching the voltage is greater than its amplitude value and the constant hour of the lance is high, then after about half the period the voltage on the capacitor reaches its maximum value, as it is not possible to overvoltage: .

2. Switching processes when the inductance coil is connected to the living room

When the key of the lancet is unlocked in fig. 5 vimushena warehouse struma through the inductance coil.

Characteristically equal

,

stars і .

Vіdpovidno to the first law of commutation

.

In this order, viraz for struma in transitional mode

that voltage on the inductance coil

.

(7)

Analysis (7) shows that when the lances are broken, which can be replaced by inductive elements, they can cause large overvoltages, which, without special inputs, can lead the device out of tune. Right, for module of voltage on the inductance coil at the remaining moment of switching in many times overestimated the voltage of the dzherel: . For the voltage of the resistor R, which is to be extinguished, a voltage is applied to the contacts of the key, which are opened, as a result of which an arc is formed between them.

3. Charge and discharge of a capacitor

When the key is turned to position 1 (Fig. 6), the process of charging the capacitor starts:

.

Vimushen storage voltage on the capacitor.

3 characteristic alignment

root . Zvіdsi good time.

) BUT = ||aik||n 1 for the values ​​of the diagonal elements. Tsey vyznachnik є rich term clearly X - characteristic rich term. At the open look of the X. century. write like this:

de S1 = a 11 + a 22 +... ann- T.sv. matrix trace, S2- the sum of all head minors of the 2nd order, then minors in the form i k) і etc., and S n- Significant matrix BUT. Korinnya H. v. λ 1 , λ 2 ,..., λ n are called the free values ​​of the matrix BUT. For a real symmetric matrix, as well as for a Hermitian matrix, all k deisn, for deissic skew-symmetric matrix all λ k purely apparent numbers; in different orthogonal matrices, as well as unitary matrices all | k| = 1.

H. c. zustrіchayutsya at the most intriguing blinds of mathematics, mechanics, physics, technology. In astronomy, the storms of the planets often come before X. at.; zvіdsi and other names for the X. century. - age equal.

2) X. century linear differential alignment with constant coefficients

a 0λ y (n) + a 1 y (n-1) +... + a n-1 y" + a n y = 0

Algebraic equation that comes out of a given differential equation after changing the function at and її similar similar steps of the value of λ, so equal

a 0λ n + a 1λ n-1 + ... + a n-1 y" + a n y = 0.

To what eve to come when you see a private decision in mind at = se λ X for which differential alignment. For a system of linear differential lines

H. c. sign up for help

H. c. matrices A =

Great Radianska Encyclopedia. - M: Radianska Encyclopedia. 1969-1978 .

Marvel at the same "Characteristic equalness" in other dictionaries:

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characteristically equal- b?dingoji lygtis statusas T sritis automatika atitikmenys: angl. characteristic equation; performance equation vok. characterize Gleichung, f; Stammgleichung, f rus. characteristic alignment, n pranc. équation caractéristique, f … Automatikos terminų žodynas

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