Linear algebra: Fifth chapter similarity Matrix and two-order (2) two-type and its standard trimming method contract transformation Method positive definite two-time type

section Fourth two and its standard form A. Mathematical Concepts

1. Two-time type

A two-time homogeneous function called n variables

is a two-time type.

2. Matrix form of two-times type

3. Rank of two-time type

The rank =r of f (a).

4. Standard form of two-time type

A standard form (or French) that has a two-time type of two-order type, which is said to contain only squares. two. Principles, formulas and laws

1. Set the reversible linear transformation x=cy, the F into a standard shape, i.e.

In fact, the symmetry matrix A is a form of a diagonal array L.

2. either give the invertible matrix C, b=cTAC, if a is a symmetric matrix, b is also a symmetric matrix, and R (B) =r (A).

3. any two-type, there is always an orthogonal transformation x=Py, so that f into a standard shape, where the matrix of f is the eigenvalues. three. Analysis of key points and difficulties

The focus of this section is to use the orthogonal transformation of the two-form into a standard shape, because the orthogonal transformation is the transformation of the Pao, so the orthogonal transformation of two to form the standard shape is the most widely used in the future. The difficulty is how to use the orthogonal transformation of two times form into a standard shape, the steps are more difficult, but the regularity is very strong, if catch its law, it is easy to two times into a standard shape. Four. Typical examples

Example 1 for an orthogonal transformation x=Py, will be transformed into a standard shape, and write out its standard shape.

Solution : ① writes F as a matrix

② to find the eigenvalues of matrix A

Get all the eigenvalues of a

③ a characteristic vector of a

Was

The characteristic vector of the solution

Was

The characteristic vector of the solution

，

④ because it is orthogonal, the unit must be

，

⑤, make x=Py as orthogonal transform, f into standard form

General steps for two-time type into a standard type:

1. Write f in the form of a matrix.

2. All eigenvalues of a are obtained.

3. The eigenvector of a is obtained.

4. The characteristic vectors corresponding to the different eigenvalues are orthogonal and only need to be unit-structured;

The K-linearly independent feature vectors corresponding to K-valued eigenvalues are orthogonal to 22 orthogonal unit vectors by the Schmitt standard.

5. X=py the F into a standard form by x=py22 orthogonal unit vectors into orthogonal matrix Pand orthogonal transformation.

。

section Fifth-balancing method and contract transformation method I. Mathematical Concepts

If there is a reversible matrix P, then the Matrix a and matrix B are called contracts. Two. Principles, formulas and Rules

1. If contract A b, R (a) =r (b);

2. Any symmetric matrix can be contracted to a diagonal matrix;

3. If, and C reversible,A symmetry, then

Among them is the elementary phalanx.

Because

i.e. (1)

(2)

Combined (1) and (2), the

three. Analysis of key points and difficulties

The focus of this section is to use the trimming method and the contract transformation method to form two of the form into a standard form, the difficulty is the embodiment of the specific criteria of the application of the method. four. Typical examples

Use the Trim method to

into a standard shape, and writes out the reversible linear transformation used.

Solution :

Make

The

The reversible line-line transformation used is

Example 2. The Contract transformation method is used to

  

into a standard shape, and writes out the reversible linear transformation used.

solution : Matrix of two-second-type F

Because

So

，

Then X=cy the F into a standard shape.

Sixth section positive definite two times type A. Mathematical Concepts

1. Positive definite two times type

is provided with a real two-type, if any x≠0 have f (x) >0 (obviously f (0) =0), then the F is positive definite two times, and said Matrix a is positive definite, kee a> 0.

2. Negative-set two-time type

For the real two-type, if any x≠0 has f (x) <0, then said F is negative two times, and that the Matrix a is negative, remember that a<0. Two. Principles, formulas and Rules

Inertia

Has a real two-time type, its purpose is r, there are two real reversible transformations

X=cy, and X=pz

Make

And

The number of positive numbers is equal.

determination of positive definite two-time type

The sufficient and necessary condition for positive definite of real two-order is that the n coefficients of its standard shape are all positive.

determination of positive definite matrices

The sufficient and necessary condition for the symmetric matrix a to be positive definite is that the characteristic value ofa is all positive.

The sufficient and necessary condition for the symmetric matrix a to be positive definite is that the first-order principal-type ofa is positive. That

。

judgement of negative definite matrix

A symmetric matrix a is a sufficient and necessary condition for a negative-definite matrix: the odd-order principal is negative, and even-order master is positive. That

three. Analysis of key points and difficulties

The focus of this section is to determine the positive nature of the two-type, to understand the practical significance of positive definite two-type, the difficulty is the proof of positive qualitative judgement theorem of positive definite two and its two-type positive qualitative proof. four. Typical examples

When the value is, two- time

is positive definite two times type.

solution : The Matrix A of the two-time F is

Due to 3>0,

That

At that time, F was a positive definite two-time type.

Since the two-order F and the symmetric matrix are one by one correspondence, it is necessary to prove that the corresponding symmetric matrix positive definite is proved by the two-type positive definite, and the other is to prove that the symmetric matrix is a positive definite matrix, and only the two-order corresponding to it is positive definite two-time type.

Example 2 . Set a matrix, if R (a) =n, then aTa is a positive definite matrix.

because, therefore, aTa is a symmetric matrix, and because R (a) =n, then, for any n Willi Vector x≠0, then

Therefore, the positive definite two-order type, so aTa is a positive definite matrix.

Example 3 : A 3-order real symmetry matrix A satisfies, and R (A) = 2, (1) write the standard shape under the orthogonal transformation, (2) the positive qualitative determination; (3) Seeking in | | x| | = 1 o'clock the maximum and minimum value of F.

Solution : Set is the eigenvalues of a , andx is a characteristic vector corresponding to the eigenvalues of a , thus,

By known, and x≠0, so

Because R (A) = 2, it is the 3 eigenvalues of a .

and A+E is still a real symmetric matrix, and the eigenvalue is 3,3,1, thus the standard shape under the orthogonal transformation

(1)

(2) because the coefficient of the standard form F is all positive, so f is positive definite two time type.

(3) as

That is 1≦f≦3 | | x| | When =1, the maximum value of F is 3 and the minimum value is 1.

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