Advanced Algebra One, determinant 1.1 to find the third-order determinant: The calculation method: calculated by the figure, the solid line is a positive sign, the dotted lines for the minus 1.2 to find the nth-order determinant 1) Calculation of the permutation of the number of examples 1:32154 Reverse order Number:Answer: 0+1+2+0+1=4 Example 2: N (n-1) ... 21 Reverse Order Number: Answer: N (n-1)/22) Calculation formula: Reverse number is odd to take negative, for even take positive. Attention is
by RowSequential fetch: The properties of a1p1,a2p2,...,anpn1.3 determinant D 1, D=DT2, Interchange determinant 2 rows (columns), d change the sign. 3, D in the existence of 2 rows (columns) exactly the same, D is 04, d in a row (column) in which the element is all 0, then D is 05, D has two rows (columns) corresponding elements in proportion, then D is 06, the determinant of a row (column) of all elements of the same multiplied by K, equals K*d, that is, you can refer to the determinant outside 7, D to a row plus k* another line, then D invariant proof: 1.4 Elements AIJ algebraic cofactor AIJ1. FormulaAIj= ( -1)I+j*mIJ where mij is the cofactor of D, which is removed by the determinant D AIJ the n-1-order determinant of the remaining (n-1) elements that remain in the original position after the row J column is removed. 2, the determinant: the determinant of the expansion of the Law D=ai1ai1+ai2ai2+...+ainain (i=1,2,...,n) Example: two, Matrix 2.1 operation Law Addition: 1) Exchange Law: A+B=B+A2) Binding law: (a+b) +c=a+ (b + C) Number of multiplication: 1) Binding law: (λμ) a=λ (ΜA)2) Distribution Rate:λ (a+b) =λa+Λb, (λ+μ) a=λa+ΜA2.2 Matrix multiplication 2.2.1 formula
2.2.2 Operation Law 1) Binding law: (AB) C=a (BC) 2) Distribution law: A (B+c) =ab+ac, (b+c) A=BA+CA3)λ (AB) = (Λa) B=a (λb)4) AE=A5) A power operation:AKAL=aK+l, (AK)L=akl2.3 transpose matrix at Operating Law 1) (at) t=a2) (a+b) T=AT+BT3) (λa) t=Λat,λ is an array
4)(AB) T=btat extended to (A1A2 ... Al) T=altal-1t ... AlTOperation Law of 2.4.1 determinant Det (A) of 2.4 square matrices1) | at|=| a|
2)|Λa|=λn| A|3) | ab|=| ab|2.4.2 Inverse matrix 1) formulaset A is n-order matrix, if there is an n-order square matrix B soab=ba=eIt is said that a is reversible and B is an inverse matrix of a. 2) a adjoint matrix A
* Where the element aij in a * is | An algebraic cofactor of AIJ in a|. 3) Nature1) AA*=a*a=| a| E2) (A-1) -1=a3) (Λa) -1=1/Λa-1(λ! =0)4) | a-1|=| A|-15) (at) -1= (A-1) T6) A reversible, B reversible, AB also reversible, and (AB) -1=b-1a-1
- Singular matrices: A is a phalanx, | a|=0;
- non-singular matrices : A is a phalanx, | a|! =0;
If matrix A is reversible, then | a|! =0
2.5 The rank of the matrix and the elementary transform 2.5.1 rank matrix A's R-order subtype D is not equal to 0, and all r+1 of a is 0, then R is called the rank of matrix A and is recorded as R (a) =r. Where the rank of 0 matrices is 0. 2.5.2 Elementary Transformation 2.5.2.1 Definition 1) ri<-->rj2 to a couple of lines k! =0 times all the elements in a row kri3) multiplies the K of all elements of a row to the corresponding element on another line RI+KRJEquivalence matrix: Matrix A Elementary transformation gets the matrix B. Recorded as a~b2.5.2.2 property 1) A primary line transformation to a is equivalent to multiplying the left of am*n by the elementary matrix pm*m;an Elementary column transformation to a is equivalent to multiplying the right of a by the elementary matrix QN*n2A reversible matrix A can be identified as a product of a finite number of elementary matrices A=p1p2 ... Pl, which can beP-1 (a| E) = (e| A-1)
A-1 (a| B) = (e| A-1B)Example 1: Seeking A-1
Answer: Use the formulaP-1(a| E) = (e| A-1)Example 2: AskA-1B
Answer: Use the formulaA-1(a| B) = (e| A-1B)The eigenvalues and eigenvectors of a square matrix 3.1 definition 1) eigenvalues and eigenvectorsset A is an n-order phalanx, λ is a number, if the equation or
There is a non-0 solution vector x, which is called λ as acharacteristic value, the corresponding x is the corresponding value of the eigenvalue.feature Vectors。 2) characteristic equation This is the homogeneous linear equation of n unknown x,n equation, and it has the sufficient and necessary condition of the non-0 solution is the determinant of coefficientsthatThe above formula is based onλis the one-dimensional equation of unknown, called the characteristic equation of square A: == Because the solution of the characteristic equation is a characteristic value λ.The characteristic equation has the constant solution in the plural range, the number is the number of the equation (the root is calculated by the weight number), so the N-order matrix A has n eigenvalues. 3.2 Solving eigenvalues and eigenvectors Examples: 3.3 properties 1) The characteristic vectors belonging to different eigenvalues are linearly independent. 2)λ3)similarity: is the equivalent matrix before
- Set A, B are n-order matrices, if there is a full rank matrix p, so that
It is said that a is similar to B, which is recorded as A~b, and the full rank matrix P is called a similarity transformation matrix that changes A to B.
- Similar matrices have the same characteristic polynomial and thus have the same eigenvalues.
- If the order matrix and the diagonal matrix
Similar, thenthat is, the n eigenvalues of A. 4)Characteristic valuedoes not have any relation to the order of the Matrix,you can take any large value. It is only in principal component analysis and factor analysis that we emphasizeMaximum characteristic valueλ. Reference: 1,Gary Li, astrologer, Li Zhiming,"Advanced Algebra of Engineering", Science press2, http://course.tjau.edu.cn/xianxingdaishu/jiao/5.htm
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"Mathematical Statistics" advanced algebra
