Operations with a matrix of times applied. Matrices and operations on them. Matrix multiplication operation
Matrix rozmіrnostі is called a rectilinear table, which is folded greenery, roztashovanih in m rows that n stovptsi.
Matrix elements (first index i− row number, another index j− number of column) can be numbers, functions or even. The matrices signify the great letters of the Latin alphabet.
The matrix is called square, even though in it the number of rows is equal to the number of columns ( m = n). Which one has a number n is called the order of the matrix, and the matrix itself is called the matrix n th order.
Elements with the same indexes 
appease main diagonal square matrix, and the elements (to calculate the sum of indices, equal n+1)
− side by side diagonal.
solitary matrix It is called a square matrix, all elements of the head diagonal are equal to 1, and other elements are equal to 0. It is denoted by the letter E.
Nulyova matrix− the whole matrix, all elements are equal to 0. A zero matrix can be of any size.
Up to number linear operations on matrices be seen:
1) adding matrices;
2) multiplying matrices by a number.
The operation of adding matrices is reserved only for matrices of the same size.
Sumy two matrices BUTі IN called a matrix W, all the elements that are equal to the sums of the corresponding elements of the matrix BUTі IN:
.
Dobootcom Matrix BUT per number k called a matrix IN, all elements that are equal to similar elements of the given matrix BUT, multiply by the number k:
Operation multiple matrices be introduced for matrices that please the mind: the number of columns of the first matrix is more than the number of rows of the other.
Dobootcom Matrix BUT spaciousness on the matrix IN dimensionality is called a matrix W expansion, element i th row j th stovptsya koї dorіvnyuє sumі tvorіv elementіv i th row of the matrix BUT on visible elements j th column of the matrix IN:
Tvіr matrices (on the basis of creating real numbers) do not follow the order of the shifting law, that is. at the top of the hill BUT IN IN BUT.
1.2. Visionaries. The power of the appointees
Understanding the visionary introduced only for square matrices.
The number of the matrix of the 2nd order is called the number, as it is calculated according to the coming rule
.
3rd order matrix 
the number is called, as it is calculated according to the following rule:
First from the additions with the sign "+" є tvir elements, spread on the head diagonal of the matrix (). There are two more elements, ruffled at the tops of the tricots with a warp parallel to the head diagonal (i). With the “-” sign, add additional elements of the side diagonal () and elements that make tricots with bases parallel to this diagonal (i).
Tse calculation of the 3rd order is called the rule of tricks (or the rule of Sarrus).
The power of the appointees let's look at the butt of the vyznachniki in the 3rd order.
1. When replacing all the rows of the signpost on the column with the same numbers, like the rows, the signpost changes its meaning, tobto. rows and stovptsі vyznachnik rivnopravnі
.
2. When rearranging two rows (stovptsiv), the signer changes his sign.
3. If all the elements of the deyago row (stovptsya) are zero, then the signpost is equal to 0.
4. The overhead multiplier of all the elements of the row (stovptsya) can be blamed for the sign of the vyznachnik.
5. Vyznachnik, scho to avenge two identical rows (stowptsya), 0.
6. Vyznachnik, scho to avenge two proportional rows (stovptsya), leading to zero.
7. If the leather element of the same column (row) of the vyznachnik becomes the sum of two dodankіv, then the vyznachnik is more expensive than the sum of two vyznachniki, in one of them at the same stovpci (row) stand the first dodanki, and in the other - the other. Other elements in both are significant however. So,
.
8. The clerk does not change, just to the elements of a new row (rows), add the necessary elements of another row (rows), multiplied by the same number.
The coming power of the vyznachnik is connected with the concepts of the minor and the addition of algebra.
Minor the element of the arbiter is called the arbitrator, taking away from the given vykreslyuvannyam of that row and standing, on the retina of which element of rotting.
For example, the minor element of the signifier called the vyznazhnik.
Algebraic additions the sign element is called yoga minor, multiplications by, de i− row number, j− the number of the column, on the line of which there is an element. Algebra addendum is signified. For an element with a value of the 3rd order, algebraic addition
9. The signator of a richer sum of creative elements of any order (stovptsya) on the basis of their additions to algebra.
For example, the forerunner can be laid out behind the elements of the first row
,
otherwise
The authorities of the vyznachniks are zastosovuyutsya їh billing.
1st year, higher mathematics, vivechaemo matrices and the main ones above them. Here we systematize the main operations, which can be carried out with matrices. Why start learning about matrices? Zvichayno, from the simplest - the purpose, the main ones to understand and the simplest operations. Singing, the matrices will understand us, who will give them at least a little bit of an hour!
Designation of the matrix
matrix- This is a rectangular table of elements. Well, as simple as mine - a table of numbers.
Sound matrices are denoted by great Latin letters. For example, the matrix A , matrix B and so far. Matrices can be of different sizes: rectangular, square, also matrix-rows and matrix-stovpts, as they are called vectors. The size of the matrix is determined by the number of rows and columns. For example, let's write a rectilinear matrix expanded m on the n , de m - number of rows, and n - Kіlkіst stovptsіv.
Elementi, for yaks i=j (a11, a22, .. ) make up the main diagonal of the matrix, and are called diagonal.
What can be done with matrices? Store / withdraw, multiply by a number, multiply among yourself, transpose. Now about all the main operations on matrices in order.
Operations of folding and visualization of matrices
Let's go ahead, what can be folded more than a matrix of the same size. As a result, we will see a matrix of the same size. Fold (or see) matrices simply - it’s enough to put together their essential elements . Let's give an example. It is possible to fold two matrixes A and size two by two.
Vіdnіmannya vykonuєtsya for analogієyu, seldom with an opposite sign.
On a be-yak number, you can multiply a be-yak matrix. Shchab tse, you need to multiply by the number of skins її element. For example, we multiply the matrix A from the first butt by the number 5:
Matrix multiplication operation
Multiply among yourself not all matrices. For example, we have two matrices - A and B. Їx can be multiplied one by one only in that case, as the number of columns in matrix A is equal to the number of rows in matrix B. When this the skin element of the matrix, which should be in the i-th row and j-th column, will be more efficient sum of the creations of the corresponding elements in the i-th row of the first multiplier and the j-th column of the other. To understand the algorithm, let's write down how to multiply two square matrices:
І butt of real numbers. Let's multiply the matrices:
Matrix transposition operation
Matrix transposition - the whole operation, if the double rows and columns are replaced by months. For example, we transpose the matrix A from the first butt:
Significant matrix
Vyznachnik, about the determinant - one of the main ones to understand linear algebra. If people foresaw the lineage, and after them it happened to be vigilant and the benefactor. At the pіdbag, razbiratis z usіm tsim lie down for you, so the rest of the rivok!
Vyznachnik is a numerical characteristic of a square matrix, as it is necessary for the completion of rich tasks.
In order to fix the sign of the simplest square matrix, it is necessary to calculate the difference in the creations of the elements of the head and side diagonals.
The signifier of the matrix of the first order, so that it is composed of one element, more than the next element.
What about a three-by-three matrix? It’s already folded here, but you can turn around.
For such as Matrixi Value of Vysnivni Vyshіv's Creativnik Elentivniy He Hearts Dіagonalі і, івів ELEMENTІV, Scho lying on tricakers from the face of Parallelno Dіgonalі, Vіd Yakoi Dіdnimalі і і и доденко і и и и і і і і доднок сетники зарильная опильной офильной инагалі.
Luckily, it is practically rare to count the names of the matrices of the great roses.
Here we looked at the main operations with matrices. Obviously, in real life, you can once in a while and not strike a stress on the matrix system of equals, otherwise, on the contrary, you will get stuck with significantly folded vipadkas, if you happen to effectively knock your head. For such vipadkiv and іsnuє professional student service. Turn around for help, take back that report decision, enjoy the success of the teacher in the free hour.
Matrices. See the matrix. Operations on matrices and the yoga of power.
Significant matrix of the n-th order. N, Z, Q, R, C,
A matrix of the order m * n is called a rectilinear table of s numbers, which can be replaced by an m-row and n - stoptsiv.
Rivnist matrices:
Two matrices are called equal, so the number of rows and columns of one of them is equal to the number of rows and columns of the other and the other. el-ti tsikh matrices equal.
Note: El-ty, yakі may have the same indexes, є vіdpovіdnimi.
See matrix:
Square matrix: the matrix is called square, because the number of rows is equal to the number of columns.
Rectangular: the matrix is called rectangular, because the number of rows is not equal to the number of columns.
Row matrix: a matrix of order 1*n (m=1) can look like a11, a12, a13 and is called a row matrix.
Matrix stovpets:………….
Diagonal: the diagonal of a square matrix, which goes from the upper left corner to the lower right corner, which is formed by elements a11, a22 ... - is called the head diagonal. (definition: a square matrix with all the elements that add up to zero, the cream is quiet, which is spread out on the head diagonal, is called a diagonal matrix.
Single: the diagonal matrix is called single, because all the elements are placed on the head diagonal and add 1.
Upper tricut: A=||aij|| is called the upper tricot matrix, so aij=0. Think i>j.
Lower tricut: aij=0. i Zero: ce matrix El-ty as good 0. Operations on matrices. 1. Transposition. 2. Multiplication of a matrix by a number. 3. Folding matrices. 4. Multiple Matrices. The main sv-va podії over matrices. 1.A+B=B+A (commutativity) 2.A+(B+C)=(A+B)+C (associativity) 3.a(A+B)=aA+aB (distributivity) 4.(a+b)A=aA+bA (distributive) 5.(ab)A=a(bA)=b(aA) (asoots.) 6.AB≠BA (weekday room) 7.A(BC)=(AB)C (assoc.) Virobiv matrices are victorious. 8.A(B+C)=AB+AC (distributive) (B+C)A=BA+CA (distributive) 9.a(AB)=(aA)B=(aB)A The signifier of the square matrix is the signification of that yoga of power. Razkladannya vyznachnik behind rows and columns. Ways to calculate the nominees. If a matrix has order m>1, then the signifier of this matrix is a number. Algebraic additions Aij el-ta aij matrix A is called minor Mij, multiplications by the number THEOREM 1: Significant matrix A is a good sum of creations of all elements of a sufficient row (stovptsya) with their algebraic additions. The main characteristics of the appointees. 1. The matrix designator does not change at the hour of transposition. 2. When rearranging two rows (stovptsiv), the signifier changes the sign, but the absolute value of the yogo does not change. 3. Significant matrix that can have two identical rows (stowpts) equal to 0. 4. When multiplying a row (stovptsya) of a matrix by a number її, the signifier is multiplied by the whole number. 5. If one of the rows (stowpts) of the matrix is added to 0, then the index of the row of the matrix is equal to 0. 6. Even though all the elements of the i-th row (stowptsya) of the matrix are presented in the view of the sum of two additional matrices, then the same sign can be filed in the view of the sum of the sum of two matrices. 7. The appointee does not change, so to the elements of one column (row) add a double element of the other column (row) in front of a plurality. for the same number. 8. The sum of the additional elements of any column (row) of the signifier on the second algebraic addition of the elements of the next column (row) is equal to 0. https://pandia.ru/text/78/365/images/image004_81.gif" width="46" height="27"> Methods for calculating the principal: 1. For the purpose or by Theorem 1. 2. Brought to a tricot look. Significance of that power of the turning matrix. Calculation of the turnover matrix. Matrix alignment. Designation: A square matrix of order n is called a pivot to a matrix And of the same order i is assigned In order for the matrix A to be based on the reverse matrix, it is necessary and sufficient that the origin of the matrix A is 0. The dominance of the pivotal matrix: 1. Unity: for a given matrix A її is wrapped - unity. 2. matrix designator 3. The operation of taking the transposition and taking the matrix of the rotation. Matrix alignment: Let A and B be two square matrices of the same order. https://pandia.ru/text/78/365/images/image008_56.gif" width="163" height="11 src="> Understanding the linearity and independence of the matrix columns. The dominance of the linear fallacy and the linear independence of the system of partners. Stovptsі A1, A2 ... An are called linearly fallow, as it is not a trivial linear combination, which is closer to the 0th column. The columns A1, A2 ... An are called linearly independent, since they are not a trivial linear combination, which is equal to the 0th column. A linear combination is called trivial, because all the coefficients С(l) equal 0 and are not trivial in a different way. https://pandia.ru/text/78/365/images/image010_52.gif" width="88" height="24"> 2. in order for the columns to be linearly fallow, it is necessary and sufficient, so that they must be a linear combination of other columns. Bring 1 of the columns with a linear combination of other columns. https://pandia.ru/text/78/365/images/image016_38.gif "linearly fallow, then all the stations are linearly fallow. 4. Just as the system of pillars is linearly independent, whether the subsystem is so itself linearly independent. (Everything that is said about the stovptsiv is also true for the rows). Minori matrices. Basic minor. Matrix rank. The method is framed by the minors in the calculation of the rank of the matrix. The minor of the order up to matrix A is the signifier of the element of some sorting on the retina up to rows and up to rows of matrix A. Like all minors to the k-th order of the matrix A = 0, be it a minor to the order of up to + 1 to the same order as 0. Basic minor. The rank of matrix A is the order of the base minor. The method of framing minors: - We select a non-zero element of the matrix A (If there is no such element, then the rank A = 0) It is framed by the minor of the front 1st order by the minor of the 2nd order. (If this minor is not equal to 0, then the rank is >=2) If the rank of the first minor is 0, then the vibrations of the 1st order minor are framed by other minors of the 2nd order. (If all minors of the 2nd order = 0, then the rank of the matrix = 1). Matrix rank. Methods for determining the rank of a matrix. The rank of matrix A is the order of the 1st base minor. Calculation methods: 1) Method of oblyamіvnykh minorіv: - Choose a non-zero element of the matrix A (as there is no such element, then the rank = 0) - Framing the minor of the forward 1st order by the minor of the 2nd order. >r+1 Mr+1=0. 2) Bringing the matrix to a stepwise look: the whole method of foundations on elementary transformations. With elementary transformations, the rank of the matrix changes. The following transformations are called elementary transformations: Permutation of two rows (stovptsiv). The multiplication of all elements of the deyago stovptsya (rows) number is not =0. Supplement to all elements of the next row (row) of the elements of the next row (row), forward multiplied by the same number. The theorem about the basic minor. That sufficient intelligence is necessary for the equalness of the zero of the signifier. The base minor of the matrix A is the minor of the greatest pre-th order of the dominant view 0. Basis minor theorem: Basic rows (stovpts) are linearly independent. Whether a row (stovpchik) of matrix A is a linear combination of basic rows (stovptsiv). Rows and columns on the retina of which stand the basic minor are called basically basic rows and columns. a11 a12… a1r a1j a21 a22….a2r a2j a31 a32….a3r a3j ar1 ar2 ….arr arj ak1 ak2…..akr akj Necessary and sufficient mind to be equal to zero of the signifier: Sob vyznachnik of the n-th order = 0, necessary and sufficient, so that rows (stovptsі) were linearly fallow. Systems of linear lines, their classification and form of the record. Cramer's rule. Let's take a look at the system of 3-linear lines from the trio of nevidomimi: https://pandia.ru/text/78/365/images/image020_29.gif" alt="(!LANG:(!LANG:l14image048" width="64" height="38 id=">!}!} called the arbiter of the system. We add three more leaders in the coming rank: we replace the successor D in sequence 1, 2 and 3 of the pillars of the free members https://pandia.ru/text/78/365/images/image022_23.gif" alt="(!LANG:(!LANG:l14image052" width="93" height="22 id=">!}!} Proof. Later, let's take a look at the system of 3 equals from a trio of nevіdomimi. We multiply the 1st alignment of the system by the addition of the algebra A11 of the element a11, the 2nd alignment by A21 and the 3rd by A31: https://pandia.ru/text/78/365/images/image024_24.gif" alt="(!LANG:(!LANG:l14image056" width="247" height="31 id=">!}!} Let's look at the skin of the bow and the right part of the same line. According to the theorem about the arrangement of the arbitrator for the elements of the 1st column https://pandia.ru/text/78/365/images/image026_23.gif" alt="(!LANG:(!LANG:l14image060" width="324" height="42 id=">!}!} Similarly, it can be shown that i . Nareshti don't care to remember that Otzhe, otrimuemo jealousy:. Father, . Similarly, the equivalence and zvіdki і follow the assertion of the theorem. Systems of linear lines. Umov's summation of linear rivnyan. The Kronecker-Capelli theorem. The solution of the system of algebraic equalizations is called such a plurality of n numbers C1,C2,C3……Cn, as when substantiating y, the system is found on the space x1,x2,x3…..xn The system of linear alignments of algebra is called a joint system, as if it could not have one solution. A split system is called singing, because there is only one solution, and it is invisible, because there is an impersonal solution. Wash the summation of systems of linear algebraic lines. a11 a12 ……a1n x1 b1 a21 a22 ……a2n x2 b2 ……………….. .. = .. am1 am2…..amn xn bn THEOREM: In order for the system of m linear alignments with n to be invariably coherent, it is necessary and sufficient, so that the rank of the extended matrix is increased to the rank of matrix A. Note: This theorem gives more than a criterion for the basis of a solution, but does not indicate the method of seeking a solution. 10 meals. Systems of linear lines. The method of the basic minor is a wild way of examining all solutions of linear alignment systems. A=a21 a22…..a2n Basic minor method: Let the system be spilna that RgA=RgA'=r. Give the basic minor of the inscriptions at the upper left corner of the matrix A. https://pandia.ru/text/78/365/images/image035_20.gif" width="22" height="23 src=">…...gif" width="23" height="23 src= ">......gif" width="22" height="23 src=">......gif" width="46" height="23 src=">-…..-a d2 b2-a(2r+1)x(r+1)-..-a(2n)x(n) … = ………….. Dr br-a(rr+1)x(r+1)-..-a(rn)x(n) https://pandia.ru/text/78/365/images/image050_12.gif" width="33" height="22 src="> If the rank of the main matrix and the analyzed one is r=n, then in this case dj=bj і the system has only one solution. Uniform systems of linear lines. The system of linear equalities of algebra is called homogeneous, because all its free terms are equal to zero. AX=0 – homogeneous system. AX \u003d B is a heterogeneous system. Homogeneous systems for each bedroom. X1 = x2 = .. = xn = 0 Theorem 1. Homogeneous systems may have heterogeneous solutions, if the rank of the matrix of the system is less than the number of non-homogeneous ones. Theorem 2. Homogeneous system of n-linear equalities with n-incomplete maє zero solutions, if the sign of matrix A is equal to zero. (detA=0) The power of rozvyazkіv odnorodnyh systems. Whether it be a linear combination of a solution of a homogeneous system and solutions of a system. α1C1 +α2C2; α1 and α2 are decimal numbers. A(α1C1 + α2C2) = A(α1C1) + A(α2C2) = α1(AC1) + α2(AC2) = 0, i.e. k. (A C1) = 0; (AC2) = 0 There is no place for power for a heterogeneous system. Fundamental solution system. Theorem 3. Since the rank of the matrix system is equal to n-independent dorivnyu r, this system can have n-r linearly independent solutions. Let the basic minor at the upper left corner. Yakscho r< n, то неизвестные х r+1;хr+2;..хn называются свободными переменными, а систему уравнений АХ=В запишем, как Аr Хr =Вr C1 = (C11 C21 .. Cr1 , 1.0..0) C2 = (C21 C22 .. C2r,0, 1..0)<= Линейно-независимы. …………………….. Cn-r = (Cn-r1 Cn-r2.. Cn-rr ,0, 0..1) A system of n-r linearly independent solutions of a homogeneous system of linear equalities with n-independent ranks r is called a fundamental system of solutions. Theorem 4. Whether a solution to a system of linear alignments is a linear combination of a solution to a fundamental system. С = α1C1 + α2C2 + .. + αn-r Cn-r Yakscho r 12 meals. Zagalne rozvyazannya heterogeneous system. Sleep (zag. non-uniform.) \u003d Coo + Mid (private) AX = B (heterogeneous system); AX = 0 (ASoo) + ASch \u003d ASch \u003d B, since K. (ASoo) \u003d 0 Sleep = α1C1 + α2C2 +.. + αn-r Cn-r + Sch Gaus method. The price of the method of the last vines of the unknown (changing) - in that, with the help of the elementary transformations, the equal system is brought to the equal system of the stepwise look, for which, starting from the rest of the changes, to find the solution of the change. Let a ≠ 0 (if it is not so, then by rearranging the equals one reaches it). 1) including changing x1 from the other, third ... n-th rank, multiplying the first rank by the second number and adding the results to the 2nd, 3rd ... n-th rank, then take: We take the system equally strong. 2) turn off change x2 3) turn off x3 change, etc. Continuing the process of the subsequent switching off of the replacements x4; x5 ... xr-1 is taken for (r-1) crop. The number of zero remaining n-r in the equals means what the left part of it looks like: 0x1 +0x2+..+0xn If one wants one of the numbers vr+1, vr+2… not equal to zero, then the equality is super-efficient and the system (1) is not coherent. In such a rank, for any kind of coherent system, vr+1...vm is equal to zero. Remaining n-r equals in the system (1; r-1) є with the sameness and їх can not be taken to respect. There are two possibilities: a) the number of equals of the system (1; r-1) is equal to the number of unknowns, so r = n (the system looks tricky in this case). b) r The transition from the system (1) to the equal system (1; r-1) is called the direct move to the Gauss method. About the knowledge of the variable from the system (1; r-1) - a turning point to the Gauss method. The transformation of Gaus is carried out manually, using their equals, and with an expanded matrix of their coefficients. 13 meals. Similar matrices. Let's look at only square matrices of order n/ Matrix A is called a similar matrix (A~B), since there is such a non-singular matrix S that A=S-1BS. Power of such matrices. 1) Matrix A is similar to itself. (A~A) Like S=E, also EAE=E-1AE=A 2) If A ~ B, then B ~ A Yakscho A = S-1BS => SAS-1 = (SS-1) B (SS-1) = B 3) If A~B and one hour B~C, then A~C Given that A=S1-1BS1 and B=S2-1CS2 => A= (S1-1 S2-1) C(S2 S1) = (S2 S1)-1C(S2 S1) = S3-1CS3, de S3 = S2S1 4) Designators of similar matrices are equal. It is given that A ~ B, it is required to bring that detA=detB. A=S-1 BS, detA=det(S-1 BS)= detS-1* detB* detS = 1/detS *detB*detS (soon) = detB. 5) The ranks of similar matrices are changed. Vlasnі vektori i vlasnі values of matrices. The number λ is called the given value of the matrix A, because it is a non-zero vector X (matrix row) such that AX = X, the vector X is called the given vector of the matrix A, and the combination of all the given values is called the spectrum of the matrix A. The power of powerful vectors. 1) When multiplying the power vector by the number, we take the power vector from the same power values. AX = X; Х≠0 α X => A (α X) \u003d α (AX) \u003d α (λ X) \u003d \u003d λ (α X) 2) Wet vectors with pairwise-different wet values are linearly independent λ1, λ2,.. λk. Let the system be composed of one vector, let's make it inductive: C1 X1 + C2 X2 + .. + Cn Xn = 0 (1) - multiply by A. C1 AX1 + C2 AX2 + .. + Cn AXn = 0 С1 λ1 Х1 +С2 λ2 Х2 +.. +Сn λn Хn = 0 Multiply by λn+1 and see C1 X1 + C2 X2 + .. + Cn Xn + Cn +1 Xn +1 = 0 С1 λ1 Х1 +С2 λ2 Х2 + .. +Сn λn Хn+ Сn+1 λn+1 Хn+1 = 0 C1 (λ1 –λn+1)X1 + C2 (λ2 –λn+1)X2 +.. + Cn (λn –λn+1)Xn + Cn+1 (λn+1 –λn+1)Xn+1 = 0 C1 (λ1 –λn+1)X1 + C2 (λ2 –λn+1)X2 +.. + Cn (λn –λn+1)Xn = 0 Required schob С1 = С2 = ... = Сn = 0 Cn+1 Xn+1 λn+1 =0 Characteristically equal. A-λE is called the characteristic matrix for matrix A. In order for a non-zero vector X to be an arbitrary vector of the matrix A, which must match the arbitrary value of λ, it is necessary to make a difference between a homogeneous system of linear-algebraic equations (A - λE)X = 0 A non-trivial solution to the system can be, if det (A - XE) = 0 - it is characteristically equal. Firmness! Characteristic equalities of similar matrices are changed. det(S-1AS - λЕ) = det(S-1AS - λ S-1ЕS) = det(S-1 (A - λЕ)S) = det S-1 det(A - λЕ) detS= det(A - λЕ) Characteristic rich member. det(A – λЕ) - function of parameter λ det(A – λЕ) = (-1)n Xn +(-1)n-1(a11+a22+..+ann)λn-1+..+detA This polynomial is called the characteristic polynomial of the matrix A. Last: 1) As matrices A ~ B, then the sum of their diagonal elements is increased. a11+a22+..+ann = в11+в22+..+вnn 2) There are a lot of powerful values of similar matrices. Even though the characteristic equalization of the matrices is behaving, the stench is neobov'yazkovo similar. For matrix A For matrix B https://pandia.ru/text/78/365/images/image062_10.gif" width="92" height="38"> Det(Ag-λE) = (λ11 – λ)(λ22 – λ)…(λnn – λ)= 0 In order for the matrix A to be diagonalized to the order of n, it is necessary that the linearly independent wave vectors of the matrix A were used. Last. Although all the values of the matrix A are different, it is diagonalized. Algorithm for the knowledge of the power vectors and the power values. 1) folded characteristically equal 2) we know the roots of rivnyan 3) we put together a system of equalization for the designation of a wet vector. λi (A-λi E)X = 0 4) we know the fundamental solution system x1,x2..xn-r, de r - rank of the characteristic matrix. r = Rg(A - λi E) 5) the power vector, the power values λi are recorded in the view: X \u003d C1 X1 + C2 X2 + .. + Cn-r Xn-r, de C12 + C22 + ... C2n ≠ 0 6) check whether the matrix can be reduced to a diagonal look. 7) we know Ag Ag=S-1AS S= 15 meals. The basis of a straight line, a square, a space. https://pandia.ru/text/78/365/images/image065_9.gif" height="11">│, ││). 4.Orth vector. The orth of this vector is called a vector, which directs however with this vector and may have a module, which is the most common unit. Rivnі vectori mayut rіvnі orti. 5. Cut between two vectors. The smaller part of the area is surrounded by two interchanges, which go from one point and straight lines, however, with given vectors. Vector storage. Multiplying a vector by a number. 1) Adding two vectors https://pandia.ru/text/78/365/images/image065_9.gif" height="11">+ │≤│ │+│ │ 2) Reproduction of a vector by a scalar. The new vector, which can be called the sub-vector of that scalar, is: a) = increase the modulus of the multiplied vector by the absolute value of the scalar. b) directly concurrently with a multiplied vector, as if the scalar is positive, i as the opposite, as if the scalar is negative. λ a(vector)=>│ λ │= │ λ │=│ λ ││ │ Dominance of linear operations on vectors 1. Law of communicativeness. 2. The law of associativity. 3. Adding zero. a(vector)+ō= a(vector) 4. Addendum with protilogny. 5. (αβ) = α(β) = β(α) 6; 7. Law of distributivity. Viraz vector via yogo module i ort. The maximum number of linearly independent vectors is called a basis. The basis on the straight line is a vector. The basis on the plane is two non-calendar vectors. The basis of space is a system of three non-coplanar vectors. The coefficient of the distribution of a vector for a certain basis is called the components or the coordinates of the vector in the given basis. https://pandia.ru/text/78/365/images/image075_10.gif" height="11 src=">.gif" height="11 src="> vikonaty to add that multiplication by a scalar, then in the result be there any number of such diy otrimaemo: λ1 https://pandia.ru/text/78/365/images/image079_10.gif" height="11 src=">+...gif" height="11 src=">.gif" height="11 src="> are called linear-deposit, as it is a non-trivial linear combination, even?. λ1 https://pandia.ru/text/78/365/images/image079_10.gif" height="11 src=">+...gif" height="11 src=">.gif" height="11 src="> are called line-independent, as there is no non-trivial line combination. Dominance of linear fallow and independent vectors: 1) the system of vectors to replace the zero vector is linearly fallow. λ1 https://pandia.ru/text/78/365/images/image079_10.gif" height="11 src=">+...gif" height="11 src=">.gif" height="11 src="> will be linearly fallow, it is necessary that the vector be a linear combination of other vectors. 3) as part of the vector in the system a1(vector), a2(vector) ... ak(vector) is linear-deposit, then all vectors are linear-deposit. 4) as well as all vectors https://pandia.ru/text/78/365/images/image076_9.gif" https://pandia.ru/text/78/365/images/image082_10.gif" height="11 src=">.gif" height="11 src=">) Linear operations in coordinates. https://pandia.ru/text/78/365/images/image069_9.gif" height="12 src=">.gif" height="11 src=">.gif" height="11 src="> .gif" height="11 src=">.gif" width="65" height="13 src="> Power of the scalar creation: 1. Commutativity 3. (a;b)=0, even and only once, if the vectors are orthogonal, or if they are from vectors, they are more or less 0. 4. Distributivity (αa+βb;c)=α(a;c)+β(b;c) 5. Viraz the scalar creation a and b through їх coordinates https://pandia.ru/text/78/365/images/image093_8.gif" width="40" height="11 src="> https://pandia.ru/text/78/365/images/image095_8.gif" width="254" height="13 src="> When vykonannі wash (), h, l = 1,2,3 https://pandia.ru/text/78/365/images/image098_7.gif" width="176" height="21 src="> https://pandia.ru/text/78/365/images/image065_9.gif" height="11"> and the third vector is called which is pleased with the coming equals: 3. - rights The power of vector creativity: 4. Vector line of coordinate orts Orthonormal basis. https://pandia.ru/text/78/365/images/image109_7.gif" width="41" height="11 src="> https://pandia.ru/text/78/365/images/image111_8.gif" width="41" height="11 src="> Often 3 symbols are used to determine the orthonormal basis https://pandia.ru/text/78/365/images/image063_10.gif" width="77" height="11 src="> https://pandia.ru/text/78/365/images/image114_5.gif" width="549" height="32 src="> Yakscho is an orthonormal basis, then https://pandia.ru/text/78/365/images/image117_5.gif" width="116" height="15">- alignment of the straight line parallel to the OX axis 2) - alignment of the straight line parallel to the OS axis 2. Replace 2 straight lines. Theorem 1 A) Todi is necessary that enough mind if the stench is tinted at a glance: B) That is necessary and sufficient of the mind of that which is directly parallel to the mind: B) To the same necessary and sufficient mind of the one who is directly angry in one mind: 3. Move from the point to the straight line. Theorem. Move from a point to a straight line using a Cartesian coordinate system: https://pandia.ru/text/78/365/images/image127_7.gif" width="34" height="11 src="> 4. Cut between two straight lines. Umov perpendicularity. Let 2 direct assignments to a Cartesian coordinate system with large levels. https://pandia.ru/text/78/365/images/image133_4.gif" width="103" height="11 src="> Yakscho, then straight lines are perpendicular. 24 meals. The area near the space. Umov's vector and plane complonarity. Vіdstan vіd point to the plane. Umov parallelism and perpendicularity of two planes. 1. Umov's complonarity of a vector and a plane. https://pandia.ru/text/78/365/images/image138_6.gif" width="40" height="11 src="> https://pandia.ru/text/78/365/images/image140.jpg" alt="(!LANG:(!LANG:Nameless4.jpg" width="111" height="39">!}!} https://pandia.ru/text/78/365/images/image142_6.gif" width="86" height="11 src="> https://pandia.ru/text/78/365/images/image144_6.gif" width="148" height="11 src="> https://pandia.ru/text/78/365/images/image145.jpg" alt="(!LANG:(!LANG:Nameless5.jpg" width="88" height="57">!} !} https://pandia.ru/text/78/365/images/image147_6.gif" width="31" height="11 src="> https://pandia.ru/text/78/365/images/image148_4.gif" width="328" height="24 src="> 3. Cut between two flats. Umov perpendicularity. https://pandia.ru/text/78/365/images/image150_6.gif" width="132" height="11 src="> Yakshcho, then the planes are perpendicular. 25 meals. A straight line at the open space. Differently see the alignment of straight lines in the open space. https://pandia.ru/text/78/365/images/image156_6.gif" width="111" height="19"> 2. Vector of direct alignment in space. https://pandia.ru/text/78/365/images/image138_6.gif" width="40" height="11 src="> https://pandia.ru/text/78/365/images/image162_5.gif" width="44" height="29 src="> 4. Canonical straight line. https://pandia.ru/text/78/365/images/image164_4.gif" width="34" height="18 src="> https://pandia.ru/text/78/365/images/image166_0.jpg" alt="(!LANG:(!LANG:Without'яний3.jpg" width="56" height="51">!}!} Lecture 1. “Matrices and main functions over them. Visionaries
Appointment.
Matrix rosemary m
n, de m- Number of rows, n- The number of columns, called a table of numbers, spread out in the first order. Qi numbers are called matrix elements. The area of the skin element is unambiguously identified by the number of the row and the column, which can be found on the front of the veins. Matrix elements are assigneda ij, de i- row number, and j- Station number. A = Basic subdivisions over matrices.
The matrix can be folded as in one row, and in one column. Remember, the matrix can be folded from one element. Appointment.
If the number of columns of the matrix is equal to the number of rows (m=n), then the matrix is called square.
Appointment.
Matrix mind: called single matrix.
Appointment.
Yakscho a
mn
=
a
nm
, then the matrix is called symmetrical.
butt. Appointment.
Square matrix mind Storage and visual matrices are built up to the next operations on their elements. The supreme authority of these operations is those who stink reserved only for matrices of the same size. In this order, it is possible to designate the operation of folding that visual matrix: Appointment.
bag (retail) matrix є matrix, the elements of which are the sum (retail) of the elements of the output matrices. cij = aij
b ij Z \u003d A + B \u003d B + A. Operation plural (podіlu) the matrix, whether it be expanded by a certain number, is reduced to the multiple (divided) of the skin element of the matrix by the whole number. (A + B) \u003d A B A ( ) \u003d A A butt. Given matrix A = 2A = Matrix multiplication operation.
Appointment:
Tvorom A matrix is called a matrix, the elements of which can be calculated using the following formulas: A
B =
C;
From the induced designation, it can be seen that the operation of multiplying matrices is assigned only to matrices, the number of columns in the first place with some more expensive number of rows in another.
The power of the operation of multiplying matrices.
1) Multiple Matricesnot commutative
, then. AB VA navіt yakscho it is appointed to create insults. However, even though for some matrices of the relation AB = BA it is victorious, then such matrices are calledpermutable.
The most characteristic butt can be
a matrix, like a permutable one, be a different matrix of the same rozmіru. Only a few square matrices of the same order can be permutable. Obviously, for whatever matrices such power is conferred: A
O =
O;
O
A =
O,
de O - zero matrix. 2) Matrix multiplication operation associative, tobto. just as it is assigned to create AB and (AB) C, then it is assigned to BC and A (BC), and equality is to be won: (AB)C=A(BC). 3) Matrix multiplication operation distributive a hundred years before dodavannya, tobto. if there is a sense of using A (B + Z) and (A + B) Z, then it is obvious: (A + B) C = AC + PS. 4) If the dobutok AB is assigned, then for whatever number
correct spelling: (AB) = (
A)
B =
A(
B).
5) If the supplementary AB is assigned, then the supplementary B T A T is assigned and equality is awarded: (AB) T = B T A T, de index T denotes transposed matrix. 6) It is also respected that for any square matrices det(AB) = detA detB. What is det will be reviewed below. Appointment
.
Matrix B is called transposed matrix A, and transition from A to B transposition For example, the elements of the skin row of matrix A are written in the same order in the columns of matrix B. A = In other words, b ji = a ij. As a consequence of the forward power (5) it can be written that: (ABC ) T = C T B T A T , for the mind, scho is assigned to dobutok matrices ABC. butt.
Given matrix A = A T =
C =
butt. Find additional matrix A = і B = AB = VA = butt. Find doboot matrix A = AB = Visionaries(Determinants). Appointment.
Vyznachnik square matrix A = det A = M 1 to– the determinant of the matrix, otrimana z vihіdnymi vykrelyvannyam first row and k – st stovptsa. Follow the respect that the vyznachniki may only have square matrices, that is. matrices, in which the number of rows is equal to the number of columns. F det A = Vlasne kazhuchi, vyznachnik can be counted in order of cleanliness of the matrix, that is. the correct formula is: detA= Obviously, different matrices can be mothers of the same ones. The signifier of the single matrix is more expensive 1. For the assigned matrix A, the number M1k is called additional minor matrix element a 1 k. In this way, it is possible to create visnovok, that the skin element of the matrix can have its own additional minor. Dodatkovі minors are found only in square matrices. Appointment.
Dodatkovy minor an additional element of the square matrix a ij is more important than the matrix's designator, taken away from the output of the i-th row and the j-th column. Power1.
The important authority of the magistrates is the same spіvvіdnoshennia: det A = det A T; power
2.
det(A
B) = det A
det B. Power 3.
det (AB) =
detA
detB Power 4.
If you remember the square matrix with two rows (or stovptsya), then the sign of the matrix will change the sign without changing the absolute value. Power 5.
When multiplying the column (or row) of the matrix by the number її, the sign is multiplied by the whole number. Power 6.
As a matrix A, the rows are pure linearly deposited, її vyznachnik is closer to zero. Appointment:
The columns (rows) of the matrix are called linear fallow, which is a true linear combination, equal to zero, which can be non-trivial (not equal to zero) solution. Power 7.
Like a matrix to avenge a zero row, or a zero row, the primordial value is closer to zero. (It is obvious that it is possible for you to enter the vyznachnik yourself behind the zero row of the apostle.) Power 8.
The designator of the matrix does not change, just to the elements of one of the th row (stowptsya) to add (to see) the elements of the next row (stow), multiplied by a number that does not add up to zero. Power 9.
As for the elements, whether there is a row, or the matrix is correct sp_v_dnoshennia:d
=
d
1
d
2
,
e
=
e
1
e
2
,
f
= det(AB).
1st method: det A = 4 - 6 = -2; det B = 15 - 2 = 13; det(AB) = det A det B = -26. 2nd way: AB =
– 152 = -26.
Appointment. The matrix is called impersonal numbers, like a rectangular table, which is made up of rows and columns Briefly, the matrix is defined as follows: deelementi of the given matrix, i is the number of the row, j is the number of the column. Like in the matrix, the number of rows is equal to the number of columns ( m
=
n), then the matrix is called square
n th order, but otherwise - rectilinear.
Yakscho m=
1 ta
n >
1, then we take a one-row matrix what is it called row vector
, well m>1 ta n=1, then we take a one-column matrix what is it called column-vector
. A square matrix, which has all the elements, crim elements in the head diagonal, equals zero, is called diagonal.
Diagonal matrix, which elements of the head diagonal have equal ones, is called alone,
be appointed E.
The matrix, otrimana with a given replacement її row with the same number, is called transposed
to qiєї. Appointed. Two matrices and equals, which are equal among themselves, elements that stand on the same places, that is, at all i
і j(when the number of rows (stowpts) of the matrix Aі B may be the same). 1°. Sumy two matrices A=(a ij) that B=(b ij) with the same quantity m
rowkіv ta n the matrix is called C=(c ij), the elements that signify jealousy The sum of the matrices is C=A+B. butt. twenty . Dobootcom Matrix A=(a ij) per number λ
the matrix is called, in which skin element is more expensive to obtain a similar element of the matrix A per number λ
: λA=λ
(a ij)=(λa ij),
(i=1,2…,m; j=1,2…,n). butt. thirty . Dobootcom Matrix A=(a ij), what can m rowkіv ta k stoptsіv, on the matrix B=(b ij), what can k
rowkіv ta n stoptsіv, the matrix is called C=(c ij), what can m rowkіv ta n stovptsіv, yakої element c ij the sum of creative elements i th row of the matrix A
і j th column of the matrix B, then At what number of columns of the matrix A can match the number of rows in the matrix B. Otherwise tvir is not assigned. TV matrix is indicated A*B=C. butt. For dobootka matrices do not win equalness between matrices A*
B
і B*
A, In vipadu one of them may not be assigned. Reproduction of a square matrix, no matter what order on a different single matrix, does not change the matrix. butt. Let it be similar to the rule of multiplication of matrices we put the stars Let me give you a square matrix of the third order: Appointment.
The signifier of the third order, which matches the matrix (1), is the number that is denoted by the symbol that signifies jealousy To remember, how to create at the right part of equality (2) are taken with a "+" sign, and like a "-" sign. butt. Let's formulate the main powers of the magistrates in the third order, wanting to stench the stench of the magistrates, no matter what order. 1. The rozmіr of the vyznachnik will not be changed, so that rows and stovptsі commemorate the missions, tobto. 2. Permutation of two columns or two rows of the signifier adds the multiplier of yogo to -1. 3. If the leader can have two identical rows, or two identical rows, then the score is equal to zero. 4. Reproduction of all elements of one column or one row of the signifier on a be-yak number λ
equal to multiplying the signifier by the whole number λ
. 5. If all the elements of a certain row or a certain row of the signifier are equal to zero, then the sign itself is equal to zero. 6. If the elements of two columns, or two rows of the signpost are proportional, then the signpost is equal to zero. 7. Like a skin element n-th column ( n-th row) of the vyznachnik is the sum of two dodankіv, then the vyznachnik may have ideas when looking at the sum of two vyznachniks, of which one n-th column ( n-th row) to avenge the first zgadanih dodankiv, and the last - others; Elements that stand in other places, in the presence of three saints in one themselves. For example, 80. If you add to the elements of the next column (row) of the signifier the additional elements of the next column (row), multiplied by any kind of wild multiplier, then the value of the signifier will not change. For example, Minor The next element of the arbiter is called the arbiter, which is taken from the given arbiter for the row of that column, on the retina of such arrangements this element. For example, the minor element but 1 vyznachnik Δ
є vyznachnik 2nd order Algebraic additions to the deyago element of the signifier are called the minor of the element, multiplications by (-1) p, de R- the sum of the numbers of the row is the same, on the peretina of some sorting of the whole element. Yakshcho, for example, element but 2 to be on the peretina of the 1st column and the 2nd row, then for the new R\u003d 1 + 2 \u003d 3 and algebraic additions є 90. The signator of the richest sum of creative elements, such as the construction of the rows on their algebraic additions. one hundred . The sum of the creative elements of either the same row or the same row of the signifier on the algebraic addition of the key elements of the other row or the other row is equal to zero. Vinicate power, what is possible for a square matrix BUT choose a matrix for a day, such that multiplying a matrix by it BUT as a result, take a single matrix E, such a matrix is called reversible to matrix BUT. Appointment.
The matrix is called the gated square matrix A, so. Appointment.
A square matrix is called non-virgin, because it is a sign of zero. Otherwise, a square matrix is called a virogen. Whether a non-virogene matrix can be reversed. Elementary transformations of matricesє: rearrangement of two parallel rows of the matrix; multiplying all matrix elements by a number, not including zero; adding to all elements in the matrix row of the same elements of the parallel row, multiplied by the same number. matrix IN, taken from the matrix BUT for the help of elementary transformations, called equivalent
matrix. For a non-virgin square matrix third order reversal matrix BUT-1 can be calculated using this formula here Δ is the matrix BUT,A ij
- algebraic additions to elements a ij
matrices BUT. The row element of the matrix is called extreme
, yakscho vіn vіdmіnny vіd zero, and all elements of the row, which are left-handed vіd ny, are equal to zero. The matrix is called step-frequent
as the extreme element of the skin row is located to the right of the extreme element of the front row. For example: Chi is not a step; - steps.
= E
,
- symmetric matrix
called diagonal matrix.
; B= 
, know 2A+B.
, 2A + B = 
.
.
A E = E A = A
A(B+C) = AB+AC
; B = A T = 
;
, V = , Z = 
i number \u003d 2. Know AT + C.
;
A T B =
=
=
;
; A T B + C = +
=
.
.
= 
.
= 2
1 + 4
4 + 1
3 = 2 + 16 + 3 = 21.
, V = 
= 
= 
.
a number is called, which can be calculated for the elements of the matrix for the formula:
, de (1)
formula (1) allows you to calculate the matrix index by the first row, the formula for calculating the index by the first row is also valid:
(2)
, i = 1,2, ..., n. (3)
,
det (AB) = 7
18 - 8
19 = 126 –
,The leaders of that yoga of power.
