The quadratic type is standard type 20, and the quadratic type is standard type

x2 is the square term and x1x3 is the mixed term. In order to eliminate the mixed term x1x3, you can make x1 = y1 + y3 and x3 = y1-y3 to formulate the square term with the square difference. Then let x2=y2 to get the standard shape.

In this linear substitution, the coefficient determinant of y is not zero, so the linear substitution is non-degenerate.

1. The matching method is used in the case of square terms.

F(x1,x2,x3)=x1 2-2x2 2-2x3 2-4x1x2+12x2x3 as the standard solution: f=x1 2-2x2 2-2x3 2-4x1x2+12x2x3 --Concentrate the items with x1 in the first square term, and then return more and make up for less = (x1-2x2) 2 -6x2 2-2x3 2+12x2x3 --Then the same process of the term with x2 = (x1-2x2) 2 -6(x2-x3) 2+4x3 2 2, the case without the square term, such as f(x1,x2,x3) = x1x2+x2x3 makes x1=y1+y2, x2=y1-y2 substitute for the square term, continue to deal with the first case 3, eigenvalue.

Method: Write a quadratic matrix, find the eigenvalues of the matrix, and find the corresponding eigenvectors.

The matrix is semi-definite.

Sum definite determination: a matrix of real symmetry.

A positive definite A contract in the unit matrix.

The eigenvalues of a are all greater than 0 x'The positive inertia exponent of ax = n a and the order principals and sub-formulas are all greater than 0 Real symmetric matrix a semi-positive definite a contracts with the block matrix (er, o; o, o) and ra are all eigenvalues greater than or equal to 0, and at least one eigenvalue is equal to 0 x'The positive inertia index of ax p < n

For example, there is an ellipsoid in space, because the coordinate origin and coordinate direction are not on the main axis, the ellipsoid expression equation is very complex, and the quadratic standard type is to move the reference coordinates to the main axis of the ellipsoid, while keeping the shape of the original ellipsoid unchanged.

Generally, the x-axis and y-axis are perpendicular, that is, orthogonal, and the relationship between the vector base obtained by the orthogonal transformation is the relationship between the vector base obtained by the orthogonal transformation, if the x-axis and the y-axis are non-90 degree angles, this is similar to the relationship between the vector base obtained by the contract or similar transformation, the image will be distorted, and when the vector base obtained by the contract and the similar transformation are the same, it will be an orthogonal relationship, that is, an orthogonal transformation.

Contract, similarity, and orthogonal transformations are all coordinate transformations in which the coordinates move to the center of the ellipsoid, so that its expression has only a square term, that is, a diagonal eigenvalue.

contract, similar transformation, the ellipsoid may become flat or may become a sphere, because the vector datum length may change;

In orthogonal transformation, the shape and size of the ellipsoid are kept constant, and the characteristic of orthogonal transformation is to keep the length of the vector constant.

Contract p t and similar p (-1) transformations can only be diagonalized, i.e., moved to the main axis, and the expression has only a square term, but it does not guarantee that the ellipsoid shape does not change.

There are countless contract transformations, the coordinates are converted to the main axis, and there are also countless similar transformations, the coordinates are converted to the main axis, only when the contract transformation and the similar transformation are equal, it is the orthogonal transformation p, and the quadratic standard type is realized.

I don't think you're at the right level to understand the geometry of doing this.

With an orthogonal transformation, the orthogonal transformation keeps the length (norm) of the vector constant, and also keeps the angle between the two vectors constant, a bit like a rigid body. This is essentially doing one more rotation and quadratic onto the spindle. There is a theorem (Schur's theorem) that is also relevant to this problem.

This is complicated because quadratic forms are very important.

A little superficial insight from the individual.

The orthogonal transformation method is to convert the quadratic type into the standard type, and the techniques are orthogonal transformation and matching method: orthogonal transformation. Zhenghong stove.

Orthogonal Transformation Steps:

1. Expressing the quadratic form as a matrix f=x tax, and finding the matrix a.

2. Find all the eigenvalues of a 1, 2,..n。

3. Find the eigenvectors a1, a2 and ,.. corresponding to the eigenvaluesan。

4. Orthogonalize and unitize the eigenvectors to obtain b1, b2 ,..bn, denote c = (b1,b2,..bn)。

5. For the orthogonal transformation x=cy, the standard type of f f=k1y1+k2y2+.knyn。

The quadratic form refers to the quadratic polynomial of n variables called quadratic, that is, in a polynomial, the number of unknowns is arbitrary, but the number of collisions of each term is 2.

In mathematics, an algebraic formula consisting of the addition of several monomials is called a polynomial (if there is subtraction: subtracting a number is equal to adding its opposite). Each monomial in a polynomial is called a term of a polynomial.

Standard type of quadratic type: matching method; Contract Conversion Law; Eigenvalue method.

The difference between the standard type and the normative type of the quadratic type is that the coefficients are different, the transformation is not the same, and all items are different.

First, the coefficients are different.

1. Standard type: The coefficient of the standard type can be any constant.

2. Normative type: The coefficient of normative type can only be -1,0,1.

Second, the transformation is different.

1. Standard type: There can be more than one standard type of the same real symmetric matrix a.

2. Normative type: The normative type of the same real symmetric matrix a is unique.

3. All items are different.

1. Standard type: All terms of the standard type are square terms, and the coefficient of all square terms is 1.

2. Normative type: All items of normative type are square terms.

The standard type of the quadratic type is not unique.

The standard type of a quadratic type is not unique, but the normative type is unique. The square or Zheng state method of finding the standard type is to follow the steps of diagonalization of the real symmetry matrix, take the quadratic matrix as the real symmetry matrix, find the q, and then do the orthogonal transformation x=qy (xy is the column vector), and replace each xi in the vector group with yi according to q, and the standard type can be obtained.

If the quadratic form has only a square term, the quadratic type is said to be the standard type.

If the coefficients are only 1, -1 and 0 in the standard type, then it is called the normative type of the quadratic type, because in the standard type, the number of 1, -1, 0 is determined by the positive and negative inertia inertia indices, and the matrix of the contract has the same positive and negative inertia refers to the same number of clusters, so the matrix of the mutual contract multiplied by the same vector group must have the same normative type of the quadratic type.

In addition, to find the positive and negative inertia inertia indices of a quadratic type, the number of eigenvalues that are positive is the positive inertia index, that is, the number of 1 in the normative type.

The orthogonal transformation method is as follows:

1. The quadratic old grip bright type is expressed as the matrix form f=x tax, and the matrix a is obtained.

2. Find the eigenvalue of the width of a, n.

3. Find the eigenvectors a, a, andan。

4. Orthogonalize and unitize the eigenvectors to obtain b, b, .,bn, note c = (b, b, skin and. bn)。

5. For the orthogonal transformation x=cy, then the standard type of f f=k y +k y +knyn。

The essence and significance of quadratic standardization:

1.Essence: The essence of quadratic standardization is the diagonalization of contracts, not the diagonalization of similarities.

The reason why orthogonal matrices can be similarly diagonalized: first, because the transposition of orthogonal matrices is equal to the inverse, and similarity is the same thing as contract. The second reason is that the eigenvectors of the symmetric matrix are orthogonal under the standard orthogonal basis vector, and there is no loss.

Note that the orthogonality mentioned here is orthogonal in the standard orthogonal basis, i.e., orthogonal normalized coordinate system, and not orthogonal in the geometric space corresponding to the above quadratic type.

It must be clear and unambiguous, not confused.

2.Significance: Normalization can clearly see the symmetry axis of the quadratic function, and whether it has an intersection with the x-axis, and it is easier to know that x is better to find y.

The difference between the standard type and the normative type of the quadratic type is that the coefficients are different, the transformation is different, and all items are different.

First, the coefficients are different.

1. Standard type: The coefficient of the standard type can be any constant.

2. Normative type: The coefficient of normative type can only be -1,0,1.

Second, the transformation is different.

1. Standard type: There can be more than one standard type of the same real symmetric matrix a.

2. Normative type: The normative type of the same real symmetric matrix a is the only one.

3. All items are different.

1. Standard type: All terms of the standard type are square terms, and the coefficient of all square terms is 1.

2. Normative type: All items of normative type are square terms.

Linear algebra is a branch of mathematics, and its research objects are vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations of finite dimension.

Vector space is an important topic in modern mathematics, therefore, linear algebra is widely used in abstract algebra and functional analysis, and linear algebra can be concretely represented through analytic geometry.

The theory of linear algebra has been generalized to operator theory. Since nonlinear models in scientific research can often be approximated as linear models, linear algebra is widely used in the natural and social sciences.

f x +2x +5x +2x x +2x x +8x [x +2x (x +x )]2x +5x +8x x (x +x +x ) x +x ) 2x +5x +8x x (x +x +x ) x +6x x +4x (x +x +x ) x +3x ) 5x ,make y x +x +x ,y x +3x , y x , then x y -y +2y , x y -3y , x y , so under this linear transformation.

f＝y₁²+y₂²-5y₃².