In my defense statement, I introduced my thesis from four aspects:?

1, the concepts of quadratic form and positive definite quadratic form needed in this paper; ?

2. The properties and judging method of positive definite quadratic form:?

3. The properties and judging method of semi-positive definite quadratic form:?

Second, the defense analysis:

The first part mainly introduces the concepts of quadratic form and positive definite quadratic form used in this paper.

The second part introduces the four-way judgment method of positive definite quadratic form.

The third part is the key part of the article. By consulting the data, I compared with the judgment method of positive definite quadratic form, and summarized the main judgment methods in 4.

In the last part, nine applications of positive definite quadratic form are summarized according to the property judgment method of positive definite quadratic form.

Three. Questions raised in the defense and key points to be answered:

What are the characteristics of matrix determinant values of 1 and positive definite quadratic forms? ?

A: The matrix of positive definite quadratic form is a positive definite matrix, and its determinant value is greater than zero. ?

Fourth, the judgment method:

This paper mainly introduces four judgment methods, namely:?

The necessary and sufficient condition of 1 and quadratic semi-positive definite is that all the coefficients of its standard form are non-negative; ?

2. The necessary and sufficient condition of quadratic semi-positive definite is that its positive inertia index is equal to rank;

3. The necessary and sufficient condition of quadratic semi-positive definite is that the eigenvalues of its matrix are all non-negative; ?

4. The necessary and sufficient condition of quadratic semi-positive definite is that all principal components of its matrix are nonnegative. Secondly, it can also be judged by the definition of semi-positive definite quadratic form.

5. Although not mentioned in the document, the more closely related issue is:

1. This paper mainly introduces the properties and judgment methods of positive definite and semi-positive definite quadratic forms. But in practical application, the related concepts of positive definite matrix are often used.

2, such as (the application of positive definite quadratic form in solving linear least squares problem), this part of knowledge is not discussed in this paper. Therefore, it is necessary to further summarize the properties of positive definite matrix and combine it with the content of this paper to make this part systematic.