Linear Algebra (6)-Quadratic Form

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Preface

I have always been puzzled about the relationship between quadratic form and linear algebra, leading to a series of knowledge points that I often forget because I don't understand. Here we sort out the quadratic form, hoping to deepen the impression of the quadratic form.

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What is the quadratic form?

The purpose of quadratic form is to use matrix to study quadratic equation.
So it has little to do with the essence of linear algebra, and it can be regarded as an application of linear algebra.

Quadratic : n variables is a homogeneous quadratic polynomial called for the n variables quadratic form .

So we naturally think that since the use of matrices to study quadratic equations, the quadratic form must be represented by a matrix.

Matrix representation:

Note: because $x_1{x}_2=x_2{x}_1$, So the quadratic corresponding matrix must be a symmetric matrix, and the quadratic corresponding matrix is ​​uniquely determined only when the corresponding matrix is ​​symmetric. And our chapter mainly studies real matrices, so quadratic forms are real symmetric matrices at this stage .

This is connected with the previous section, the real symmetric matrix must be similar to diagonalization, and there is $Q^{-1}AQ=Q^{T}AQ=\Lambda$ . This is again related to the contract matrix we are going to talk about (the real symmetric matrix A must be both similar and contract to the diagonal matrix).

Quadratic representation

It is to classify homogeneous quadratic polynomials. Quadratic form $f(x_1,x_2,x_3,x_4,...,x_n)$ Can be divided into square terms ($x_i^2$) And mixed terms ( $x_i{x}_j$)。

• The quadratic form with only square terms and no mixed terms is called standard form .
• Only the square term does not have a mixed term, and the coefficients of the square term are only $1,0,-\; When\; it\; is1$ , it is called thecanonical form.

Theorem 1 : Any quadratic form $f$ , there must be anorthogonal transformation$x=Qy$ , where Q is an orthogonal matrix, making the quadratic form into a standard form.
Because the quadratic form is a real symmetric matrix, and there is$Q^{-1}AQ=Q^{T}AQ=\Lambda$ . There is only oneΛ Λ in themiddle$\Lambda$ is the standard type, and the coefficients of the standard type are the eigenvalues ​​of the matrix.

Because the quadratic form is a real symmetric matrix, and there is $Q^{-1}AQ=Q^{T}AQ=\Lambda$ . There is only oneΛ Λ in themiddle$\Lambda$ is the standard type, and the coefficients of the standard type are the eigenvalues ​​of the matrix.

Theorem 2 : Any quadratic form $f$ ,there must be areversible linear transformationx = C yx = Cythrough thematching method$x=Cy$ , where Q is an invertible matrix, making the quadratic form into a standard form.

There are two methods to transform quadratic form into standard form, orthogonal transformation and matching method . Through orthogonal transformation, the coefficients of the standard type are just the eigenvalues ​​of the matrix, and through the matching method, the coefficients of the matrix are not necessarily eigenvalues. That is , the canonical form of the quadratic form is not unique, and the canonical form of the quadratic form is unique.

To be proficient in these two transformation methods, this is a classic calculation problem in the quadratic form.

Contract matrix and contract quadratic form

Definition: Let A and B be two n-order matrices, if there is an invertible matrix C, make $C^{T}AC=B$ , it is said that A contract is in B.

Necessary and sufficient conditions: matrix A and B contract if and only if the corresponding quadratic forms have the same positive and negative inertia exponents (inertia theorem) and $r(A)=r(B)$。

The following is an excerpt from here on the analysis of matrix equal rank, equivalence, similarity, and contract.

1. The du difference between matrix equivalence, similarity and contract:

1. Equivalence, similarity and contract are all equivalent relationships.

2. The matrix is ​​similar or the contract must be equivalent, and vice versa is not necessarily true.

3. Matrix equivalence, only needs to satisfy that the two matrices can be obtained through a series of invertible transformations, that is, the multiplication of several invertible matrices.

4. If the matrices are similar, there is an invertible matrix P such that $AP=PB$。

5. Matrix contract, there is an invertible matrix P such that $P^{T}AP=B$。

6. When the above matrix P is an orthogonal matrix, that is, $P^T=P^{-1}$ , then there is A and B that satisfy both similarity and contractual relationship.

2. Matrix equivalence, similarity, and relationship between contracts:

1. Matrix equal rank is a necessary condition for similarity, contract, and equivalence, and similarity, contract, and equivalence are sufficient conditions for equal rank.

2. Matrix equivalence is a necessary condition for similarity and contract, and similarity and contract are sufficient conditions for equivalence.

3. The matrices are similar, there is no necessary and sufficient relationship between the contracts, there are similar but not contracted matrices, and there are also contracts but not similar matrices.

4. In summary: similar => equivalent, contract => equivalent, and equivalent => equal rank.

Three, say one more sentence:

Matrix equivalent :

1. In terms of the same type of matrix.

2. Generally related to elementary transformation.

3. Rank is the invariant of matrix equivalence, and the essence of the equivalence of two homogeneous matrices is rank equality.

The matrix is ​​similar:

1. For the phalanx.

2. Equality of rank is a necessary condition.

3. The essence is that the two have the same constant factor (superclass).

Matrix contract:

1. For the square matrix, it is generally a symmetric matrix.

2. Equality of rank is a necessary condition.

3. The essence is that the ranks are equal and the positive inertia indices are equal, that is, the standard type is the same.

Through the above comparison, it can be seen that the equivalence relationship is the weakest among the three relations. Contract and similarity are equivalent relations of special blocking. If the two matrices are similar or contract, the two matrices must be equivalent, and vice versa. Similarity and contract cannot be deduced from each other, but if two real symmetric matrices are similar, they must be contractual.

Positive definite quadratic form, positive definite matrix

The necessary and sufficient conditions for the positive definite judgment matrix A:

1. A's positive inertia index $p=r=n$。
2. The contract between matrix A and matrix E, that is, $A=D^{T}ED=D^{T}D$。
3. All eigenvalues ​​of A are greater than 0.
4. All sequence principals of A are greater than zero.

The necessary conditions for the positive definite judgment matrix A:

1. The main diagonal element of A>0
2. $|A|>0$

Quadratic question type

• Judge whether the two matrices are contract matrices. The matrix looks at the positive and negative inertia indices, and the quadratic form looks at the standard positive and negative inertia indices.
• Judge whether the square and quadratic form is a positive definite matrix, that is, judge that each sub-form cannot be zero at the same time, that is, the matrix formed has only zero solutions.
• To determine whether an abstract matrix is ​​a positive definite matrix, it needs to meet the conditions of invertibility, symmetry and all eigenvalues ​​greater than 0, or use the definition.