1. Determinant
1. Digital determinant
- Numeral Determinant Calculation
- Block calculations with zero subtypes
2. The nature of the determinant
|A|=|A^T|- Swap the row and column, the value of the determinant changes sign
- Propose or multiply with common factors
- Adding K times one row to another row leaves the value of the determinant unchanged.
- The determinant can be split according to a certain row or a certain column
3. Abstract determinant
-
n-order or higher-order determinant
Regular key determinants must be mastered
-
Contains specific numbers, possibly expanded or recursive
-
Generally, the ones with the same value are grouped aside and combined for calculation
4. Computational properties
|A*|=|A|n-1
|A**|=|A| (n-1) way
A matrix is an orthogonal matrix, and the value of the determinant is <0, then its eigenvalues must have -1.
2. Matrix
1. Basic operation of matrix
Classic example:
- A has rank 1
- (E+A) The expansion of the binomial theorem for n
- |A|n
- Binomial Theorem Expansion Coefficient Summation
For sample questions, please see the video of Century Higher Education.
2. Matrix power operation
Generally, the cumulative multiplication of P -1 BP is used
3. Elementary transformation of matrix
Matrix A undergoes a finite number of elementary changes to obtain B, then A and B are equivalent; the relations brought about by equivalence are only simultaneous reversibility, the same rank, and equal determinant values, excluding the corresponding relationship of eigenvalues. (So in the matrix, the place where the similarity is "bigger" than the equivalent is that the corresponding eigenvalues of the two similar matrices are also equal)
4. Adjoint matrix and invertible matrix
- Pay attention to the correspondence between the elements of the companion matrix and the original matrix
- Generally involves adding the identity matrix to participate in the simplification operation
- A few formulas to remember
Tips: Find the value of the determinant after some matrix operations, do not be affected by the determinant, first take the matrix operation in the "absolute value" to simplify, generally it is an abstract matrix, and the result will come out after the simplification is correct
Importance of rank (forgettable property):
- |r(A)-r(B)|<=r(A+/-B)<=r(A)+r(B)
5. Matrix Equation
Refer to the following part of the linear equation system
3. Vector
1. Vector operations
- Addition (subtraction as addition of negative numbers)
- Number multiplication (division as multiplication of fractions)
- inner product (unique operation on vectors)
- Vectors are orthogonal (inner product is 0)
2. Linear correlation problem
It can also be understood as a linearly independent problem
-
Definition: Discussion of zero solution and non-zero solution (less used, easy to understand)
-
rank:
Full rank === "linearly independent
Dissatisfied with the rank === "linear correlation
-
Determinant:
From rank it can be proposed that:
- |A|=0, linear correlation
- |A|!=0, linearly independent
Important conclusions:
-
n+1 n-dimensional vectors must be linearly related
-
Linear correlation is originally the concept of the relationship between two or more vectors, but if there is only one vector, the concept of linear independence must be mentioned, then there is a 0 vector linear correlation, and a non-zero vector linear correlation. (generally don't say that)
-
When a new vector is added to a group of vectors, the possibility of its correlation becomes larger; when a new element is added to a group of vectors, the possibility of its irrelevance becomes larger.
-
Equivalent sets of vectors have the same rank, but sets of vectors with the same rank are not necessarily equivalent
A relationship: Two vectors are orthogonal and must be linearly independent, but linearly independent may not be orthogonal.
3. Linear representation problem
It can be understood that the following linear vector group solution problems
- Unique representation: r(A)=r(A|b)=n
- Multiple representations: r(A)=r(A|b)<n
- Cannot represent: r(A)<r(A|b)
4. Maximum linearly independent groups
The vector where the rank pivot is located constitutes a maximal linearly independent group
4. Linear equations
1. Homogeneous linear equations
First, understand the concept:
General Form and Vector Form of Equations
The solution of the system of equations (what does it matter if there is a solution or not)
Basic solution system and general solution
The basic solution system is a representative, and the general solution contains all the basic solution systems
Number of basic solution vectors
The number of vectors of the basic solution system + r(A) = n (the number of unknown quantities)
Here n is the number of x in the general form, and generally the number of columns in the vector form
2. Inhomogeneous linear equations
Conditions for a solution:
- has a unique solution
- has infinite solutions
- No solution
Corresponding to the linear representation problem of the third part
Properties of the solution:
- Homogeneous solution + nonhomogeneous solution is still nonhomogeneous solution
- The non-homogeneous general solution structure is the corresponding homogeneous general solution + a special solution of the non-homogeneous equation
- When seeking a special solution, select all free variables to be 0, and a corresponding special solution can be obtained
The core issue here is these two corresponding knowledge points, one is the existence condition of the solution, and the other is to find the general solution.
Five. Eigenvalue, eigenvector, similarity matrix
1. Eigenvalues and eigenvectors
- Matrix of order n, that is, (square matrix) has eigenvalues
- Eigenvectors are not 0
step:
-
1. Find the eigenvalues
The fast method is a combination of rows and columns. If you only change rows, you will feel that the calculation is very complicated.
-
2. Find the corresponding eigenvector according to the eigenvalue
I always forget that it solves the eigenvectors according to the homogeneous equations corresponding to the eigenvalues
2. Similarity Matrix
Properties of similarity matrices:
- 1. Personality: A~A
- 2. Symmetry:
A~B ====> B~A- 3. Transitivity
3. Orthogonal matrix
-
Definition: A*A T =A T *A=E
-
A is an orthogonal matrix whose determinant has a value of 1 or -1.
-
A is an orthogonal matrix, and its inverse matrix and accompanying matrix are also orthogonal matrices
-
If A and B are both orthogonal matrices, then AB and BA are also orthogonal matrices
4. Real symmetric matrix
- are real numbers
- Symmetric matrix
Compared with ordinary square matrices, the eigenvalues of ordinary equations may be complex numbers, while the eigenvalues of real symmetric matrices must be real numbers.
| comparison point | Ordinary square | real symmetric matrix |
|---|---|---|
| Eigenvalues | possible plural | certain real number |
| Eigenvectors corresponding to different eigenvalues | linear independent | Linearly independent + mutually orthogonal |
| similar diagonalization | uncertain | must |
| Orthogonal similar diagonalization | cannot | able |
Orthogonal similar diagonalization: Similar diagonalization matrices are orthogonal matrices
Six. Secondary type
1. Standardization of quadratic form (combination method)
- 1. Let x1=y1+y2, x2=y1-y2, x3=y3, simplify (skip if it contains square term)
- 2. With x1
- 3. With x2
- 4. With x3
- 5. Describe the reversible linear transformation and the final quadratic canonical form
2. Standardization of quadratic form (orthogonal transformation method)
- 1. Write the matrix form of the quadratic form
- 2. Find the eigenvalues of the matrix
- 3. Find the eigenvector corresponding to the eigenvalue
- 4. Orthogonalization and unitization
- 5. Write the final reversible linear change
3. Inertia theorem and matrix contract
Reversible linear transformation does not change the positive and negative inertia exponents of the quadratic form
A reversible linear transformation to get B, then AB contract
2021 real test notes
- Choice 3: Taylor expansion
- Choice 4: Limit expression of integral over 0 to 1
- Choice 7: Understanding of block matrix rank
- Fill in 3: the parity and symmetry of the integral
- Fill in 4: Euler's equation (unpopular)
- Fill in 5: Abstract determinant calculation problem