Hermite quadratic form H quadratic form
As we mentioned in the article (1) of the Hermite quadratic form, most of the related discussions of Hermite quadratic form in matrix theory can be directly used for reference in linear algebra.
Therefore, to discuss the H quadratic form, it will also be transformed into H-array related issues.
1. Relevant conclusions and definitions
Establish a basic impression of the H quadratic form and the relationship with the H array.
1. Correspondence between H quadratic form and H matrix: H matrix uniquely corresponds to a quadratic form
If A and B are both H matrix , and for any X∈C n , X H AX = X H BX, then A = B
2. What is the relationship between the H matrix of the two quadratic forms of the invertible transformation?
Let f(X) = X H AX, g(Y) = Y H BY, C is an invertible matrix, if X = CY, f(X) = g(Y), then B = C H AC
[Proof]
Substitute the relationship of X = CY into the quadratic form corresponding to f(X), and according to the equation relationship of f(X) = g(Y), the corresponding conclusion can be derived.
3. Conjugate Contractual Relationship
The proposed conjugate contract relationship can also be analogous to the "contract" relationship in linear algebra-if there is an invertible matrix C such that B = C T AC, then matrix A and matrix B are in contract.
So it can be proved that the conjugate contractual relationship also satisfies reflexivity, symmetry, and transitivity .
- Reflexivity: I H AI = A , I is the unit matrix, which must be reversible
- Symmetry: If there is an invertible matrix C that satisfies C H AC = B, then C -H BC -1 = A
- Transitivity: there are invertible matrices M, N, M H AM = B, N H BN = C , then C = N H BN = N H M H AMN = (MN) H A(MN)
ps The product of the invertible matrix is still Invertible matrix
2. Hermite quadratic standard form
As mentioned earlier when discussing related conclusions and theorems, it is possible to perform reversible linear transformations on quadratic forms.
So we began to think about whether we can find a suitable linear transformation so that the coefficients of the cross term after the quadratic transformation are all reduced to 0.
1. Definition of standard form
(1) The standard form of Hermite quadratic form
Corresponding quadratic form cross-term coefficients are all zero and only contain square terms. Such quadratic form is called standard form.
ps For any quadratic form, the coefficients of its standard form should be real numbers.
The teacher did not demonstrate why. My understanding is that the standard form of each quadratic form is "unique", and the eigenvalues of each Hermite quadratic form are real numbers , from an arbitrary quadratic form to a standard form. The change of is reversible , so it will not change its characteristic space .
(2) The standard form of Hermite matrix
[Strengthen] Here again, the correspondence between the quadratic form and the matrix is emphasized again
. The invertible linear transformation of the Hermite quadratic form ↔ the conjugate contract transformation of the H matrix corresponding to the Hermite quadratic form
If there is an invertible matrix C for an H matrix A, such that C H AC = Λ (conjugate transformation of matrix A), where Λ is a diagonal matrix , and the elements on the diagonal are d in the figure above i (i = 1,2,...n).
Then we call this diagonal matrix Λ the canonical matrix of A.
2. Existence of Hermite canonical form
(1) Problem description:
Form ①: For any given Hermite quadratic form, is there a reversible linear transformation that turns it into the corresponding standard form?
Form ②: For any given H matrix A, is there an invertible matrix C, so that the conjugate contract transformation of A can get a diagonal matrix?
(2) Problem solving:
The basic idea can also be used in linear algebra to prove the existence of the quadratic form of real matrices and related methods and theorems .
①Use the matching method to transform the quadratic form
②Make the corresponding elementary transformation on the H matrix
③Apply the theorem
- <In linear algebra> For any real symmetric matrix A, there must be an orthogonal matrix C such that C T AC = Λ
- <In matrix theory> For any H matrix A, there must be a unitary matrix C such that C H AC = Λ
ps The second theorem can refer to the "H matrix " in " [Matrix Theory] Hermite Quadratic Form (1)" Proof of Theorem 2 and Theorem 3 in "The Nature of "
3. Theorem of inertia
We have discussed the existence of canonical forms before, and now we are going to discuss the uniqueness of canonical forms.
(1) Generally speaking, a quadratic canonical form is not unique
For example, after the following figure, we performed in block 1 of reversible conversion f (x) has been the standard quadratic form of the y, the coefficients D I .
But in box 2, we can use z to substitute y as a whole to obtain the standard form of z, and the coefficients are 4d i .
Judging from the numerical value of the standard form, the results are different, that is to say, different standard forms can be obtained.
(2) Looking for invariants
- Positive inertia index: the number of coefficients greater than 0 in the various coefficients of the standard form
- Negative inertia index: the number of coefficients of the standard form that are less than 0
- Rank: the number of items that are not 0 in the coefficients of the standard form
The above theorem tells us that if we require that the standard form of a quadratic form be transformed into a diagonal element that can only take 0 or 1 or -1, then the standard form given by any quadratic form must be unique.
4. Canonical Form
(1) Definition
For the matrix corresponding to the standard form, the conjugate contract can be further changed so that the elements on the diagonal of the matrix are only 0 or 1 or -1.
Such a matrix is called the canonical form of the original matrix A.
Among them, in the normal matrix obtained, the number of +1 (that is, p) is the positive inertia index, and q is the negative inertia index.
(2) Theorem
Use the normal form to prove the necessary and sufficient conditions of the conjugate contract of two matrices.
The necessity is obvious, so I won't repeat it.
Now briefly discuss sufficiency.
Because A and B have the same positive and negative inertia indices, assuming that the positive and negative inertia indices of A and B are p and q respectively, then:
A will be the
same as the normal matrix conjugate contract B drawn in the figure above Conjugate with the canonical matrix in the figure above.
Because of the symmetry and transitivity of the conjugate contract relationship, it can be seen that the matrices A and B are also the relationship of the conjugate contract.
Suppose the normal form matrix of the above figure is C, then it can be seen that A is conjugated to C and B is conjugated to C.
According to symmetry , then C conjugate contracts with B
according to transitivity , then A conjugate contracts with C conjugate contracts with B
Note: Because positive inertia index + negative inertia index = rank, these three know the two to be one , so the "positive inertia index" and "negative inertia index" in the theorem can be replaced with any two of these three conditions. The theorem still holds.
[Example]
First divide n+1 categories according to the rank, and then solve the possible value of the positive inertia index according to the possibility of each rank, and add all of them to the total number of possible situations.
3. Positive definiteness of Hermite quadratic form
The discussion of the positive definiteness of the H quadratic form is basically the same as the discussion of the positive definiteness of the real quadratic form, and the theorem will not be proved here.
1. Definition
2. Judgment method of positive definiteness
(1) Judgment of the positive definiteness of the diagonal matrix
The above figure briefly explains the necessity of the theorem:
Knowing that the matrix D is positive definite, to prove that each element on the diagonal is positive, you only need to take each basis vector as X 0 and participate in X 0 H DX The operation of 0 results in the element value on each diagonal.
Judgment method 1 is described in the above figure using the H matrix, and can also be described in the language of the H quadratic form:
A quadratic form is positive definite ↔ the coefficient before any square term is positive .
(2) The conjugate contract relationship does not affect the positive definiteness of the matrix
This theorem does not give a proof, but it can be understood with the help of the theorem "Two mutually conjugated matrices have the same positive inertia index and negative inertia index".
Since both A and B have the same positive (negative) inertia index, if A is positive definite, it means that the positive inertia index of A is the order of A, and the positive inertia index of B is also the order of B. B is positive definite.
The judgment method can also be described in quadratic language:
If there is a quadratic form f(X) = X H AX, f(Y) = Y H BY, if there is a reversible linear transformation X = CY, so that f(X) is always equal to g(Y), then f(X ) Is positive definite ↔ g(Y) is positive definite .
(3) The positive definiteness of any given matrix can be judged with the help of a diagonal matrix
Judging method three is essentially a combination of one and two.
3. Theorem
(1) Theorem description
(2) Theorem proof and understanding
① ↔ ②
Because A is an H matrix, there is a unitary matrix U such that U H AU = Λ, where the elements on each diagonal of the diagonal matrix of Λ are the eigenvalues of A.
And because A is positive definite, and the previous sentence also reflects that A and Λ are conjugate contracts, according to the judgment condition 3, it can be known that the elements on each diagonal of Λ should also be greater than zero, that is, the eigenvalues of A are all Is greater than 0.
① ↔ ③
Because A is positive definite, for n-th order square matrix A, its positive inertia index is n. A normal form with a positive inertia index of n should have elements on the main diagonal with all 1, which is an identity matrix I.
So the canonical form of A is the identity matrix I, and a matrix and its canonical form must be conjugate contract. (definition)
①↔④ → ③↔④
To prove the equivalence of ① and ④, only need to prove the equivalence of ③ and ④.
Through ③, the conjugate contract between A and I is known, so by definition, there is an invertible matrix P, P H IP = A, that is, A = P H IP = P H P to
prove that the
above reasoning process is equivalent. It means that it can be pushed from ① to ④, or from ④ to ①.
It can also be proved from the perspective of definition that ① can be derived from ④:
From the above figure, according to the definition of positive definiteness, for any non-zero vector Z 0 , Z 0 H AZ 0 > 0 should be satisfied , and A = P H P Substitution, the final result is the standard inner product of the PZ 0 vector. To prove that the inner product is positive, you only need to prove that PZ 0 is not a zero vector.
Because P is an invertible matrix and Z 0 is also a non-zero vector, the result of PZ 0 must be a non-zero vector in the column space of the P matrix.
Or it can be deduced by contradiction, adding PZ 0 = θ, then there is P -1 PZ 0 = P -1 θ = θ, which is Z0 = θ, a contradiction arises.
(3) Explanation
These equivalence theorems are also often used to determine whether a certain matrix is a positive definite matrix. The most commonly used ones are Article ② (eigenvalue) and Article ④ (A = P H P).
[Example]
When it is required to solve the positive definiteness of a certain matrix, we must first prove whether the matrix is an H matrix. We only discuss the so-called positive definiteness within the scope of the H matrix .
When k is a real number, the above proof process is completely valid.
Because A = I-kα H α is a matrix polynomial about the matrix α H α, if you want to find the eigenvalues of the A matrix, you must first find the eigenvalues of the matrix α H α.
According to an example we have done before, the matrix α H α has an n-1 multiplicity of 0 eigenvalues and another eigenvalue of <α,α> = 1 (because ||α|| = 1).
In "[Matrix Theory] Similar Standard Type of Matrix (1)" , there is a detailed process for solving the sample problems in part 2..3.(4).
Then according to the characteristics of the matrix polynomial, it can be concluded that the eigenvalues of matrix A should be 1-k and 1 (n-1 times). To make A positive definite, all eigenvalues are positive, so k< 1 and k∈R .
[Example]
[1]: Seeing that A is an H matrix, it will conditionally reflect and use the theorem-A can be transformed into a diagonal matrix through unitary transformation. Because it is positive definite, all diagonal elements are positive
[2]: Construction A diagonal matrix Λ 1 , now think of a way to write A in the form of a certain matrix square.
[3]: Let S = UΛ 1 U H , now only need to prove that the S matrix is positive definite.
To demonstrate a positive definite matrix -
①S H matrix is, by definition there are S H = S
2s characteristic values are all positive numbers, because UΛ = S . 1 the U- H , S, and so Lambda . 1 conjugated contract, and Lambda . 1 of The diagonal elements are all positive numbers.
The certificate is complete.
[Example]
Because A is positive definite, A must be an H array.
Because A is an H matrix, A can be transformed into a diagonal matrix by unitary transformation, and because it is proved that the main diagonal elements on the diagonal matrix are all positive numbers.
U is a unitary matrix, and A is also a unitary matrix, (the product of the unitary matrix is still a unitary matrix) so the diagonal matrix Λ is also a unitary matrix.
According to the definition, we know that Λ H Λ = I, so the elements on the diagonal are all 1. That is, Λ = I.
Finally, because U H AU = I → A = UIU H = UU H = I, the
certificate is completed.
Through the above three sample questions, the reader should have established a conditioned reflex .
As long as you see that a matrix is an H matrix, you will think that you can use a unitary matrix to turn it into a diagonal matrix. If the matrix is still positive definite, then the elements on the main diagonal of the diagonal matrix should all be positive.
The other derivations and proofs can be calculated on this diagonal matrix, which is more convenient for calculation.