Determinant theory is an important theory in advanced algebra courses. Although its status is far less than that of matrices, determinants are widely used, such as: solving linear equations (Cramer's rule), determining the non-isotropy of matrices, calculating matrix rank (discriminant method of subformula of rank, theorem 3.6.2 of advanced textbooks), calculation of matrix characteristic polynomials, simplification of quadratic forms and determination of positive definiteness of real symmetric matrices (theorem of advanced textbooks) 8.4.3), etc. In addition, determinant theory can also be applied to the variable substitution of heavy integrals, the solution of differential equations and the study of celestial mechanics, etc.
This article aims to demonstrate the relevant application of determinant theory in the simplification of quadratic forms through the discussion of three exercises. First, we give the following lemma, which is a key step in determining the standard form of quadratic forms by determinant calculation.
Lemma (Example 8.11 of the Advanced White Paper)Assume that is a symmetric matrix of order on the number field, and is the accompanying quadratic form. Assume that the first sequential main subformula of is non-zero, then after non-heterogeneous transformation, it can be transformed into the following standard form:
proveMake an induction on the order. When , the conclusion obviously holds. Assume that the conclusion holds for . Let
Since , we can perform the following symmetric block elementary transformation (contract transformation) on :
Since the third type of block elementary transformation does not change the value of the determinant, so
There is
According to the inductive hypothesis, there are non-heterogeneous matrices such that
Exercise 1 (14th question per week for Grade 19 Advanced Level II)Find the standard form of the following real quadratic forms:
solutionSuppose a square matrix of order , where , is a real symmetric matrix accompanied by a real quadratic form. The answer to question 13 in the first chapter of the Advanced White Paper is to find the determinant . Let us quote the evaluation process in the white paper (the evaluation method is not unique): multiply the row of by by and add to the row of () in turn, you can get a lower triangular matrix whose main diagonal elements are respectively . Since the third kind of elementary transformation does not change the value of the determinant, .
Solution 1 (Symmetric elementary transformation)Note that through several elementary row transformations of the third kind it can be transformed into a lower triangular matrix and the symmetry of , so it is not difficult to verify that through several symmetric elementary column transformations of the third kind it can be transformed into a diagonal matrix . Specifically, there are the following symmetric elementary transformations:
Therefore, the standard form of is .
Solution 2 (determinant calculation)From the previous calculation of , we can know the sequential principal formula of , and then from the lemma, we can get the standard form of .
Exercise 2 (14th question per week for Grade 17 Advanced Level II)Find the standard form of the following real quadratic forms:
solutionSuppose a square matrix of order , where , is a real symmetric matrix accompanied by a real quadratic form. The answer to question 14 in the first chapter of the Advanced White Paper is to find the determinant . Let us quote the evaluation process in the white paper (the evaluation method is not unique): multiply the row of by by and add to the row of () in turn, and then add the column to the front column in turn, you can get an upper triangular matrix, Its main diagonal elements are respectively . Since the third kind of elementary transformation does not change the value of the determinant, so .
Solution 1 (Symmetric elementary transformation)Sequentially multiply the -th row by and add to the -th row (), symmetrically multiply the -th column by and add to the -th column (), to obtain a symmetrical claw matrix; then multiply the -th row by and add to the -th row (), symmetrically multiply the -th column by and add to the -th column (); finally, we get the diagonal matrix . Therefore, the standard form of is .
Note that the sequential main formula of , , so the lemma cannot be directly used to obtain the standard form. Three methods are given below to overcome this difficulty.
Solution 2 (determinant calculation + properties of determinant)Consider , that is, add the -th row of to the - -th row, and then add the -th column to the -th column, to obtain a symmetric array . Obviously, it is in agreement with , so only the standard form of is required. From the properties of the determinant, we can see that the sequential principal subformula of , , so according to the lemma, the standard form of can be obtained as .
Solution 3 (Determinant calculation + variant of lemma)In the induction process of the lemma proof, the starting point is set to , so the quadratic form can be simplified to:
Let , , then the quadratic form can finally be simplified to .
Solution 4 (determinant calculation + eigenvalue + perturbation method)It can be known from the orthogonal similarity standard form theory that there is an orthogonal matrix such that
where is the overall eigenvalues of . For any positive number ,
Note that the main subform of the sequence is . Since
Therefore , and , so it can be obtained that , has the same sign as () and has the same sign as (). Note that , , so it can be seen from the lemma that it agrees with , so there are positive eigenvalues and negative eigenvalues. Let's assume , , so we can know from the hypothesis that , , so it also agrees with , so the canonical standard form is .
Exercise 3 (Answer to Question 7 of Chapter 8 of the Advanced White Paper)Find the standard form of the following real quadratic forms:
The following four solutions are given to solve the real symmetric matrix associated with the quadratic form.
Solution 1 (Symmetric elementary transformation)We use induction to determine the contract standard form of , where . Determine , obviously . Assume that the upper left corner of the matrix has been transformed into a diagonal matrix , where , we determine . Multiply the row by and add to the row (eliminate the -th element), then multiply the column by and add to the column (eliminate the -th element), and finally we can get
It can be obtained by . The recursion relation (*) can be obtained after sorting
It follows that . Therefore, the standard form of is .
Solution 2 (determinant calculation)According to Example 1.23 of the Advanced White Paper, we can see that the sequential principal form of is , and then from the lemma, we can get the standard form of .
Solution 3 (combination method) Use the combination method to turn into
Therefore, is at least semi-positive definite form. Furthermore, if the above formula is equal to zero, then obviously there is , so it is positive definite form, and its canonical standard form is .
Solution 4 (Eigenvalue)From Example 6.40 of the Advanced White Paper, we can see that all eigenvalues of are . Note that all eigenvalues are , so is a positive definite matrix, and thus is a positive definite form, and its canonical standard form is .
References
[1] Advanced algebra textbooks: edited by Yao Musheng, Wu Quanshui, and Xie Qihong, Advanced Algebra (Third Edition), Fudan University Press, 2014.
[2] Advanced Algebra White Paper: Edited by Yao Musheng and Xie Qihong, Learning Method Guide: Advanced Algebra (Third Edition), Fudan University Press, 2015.
Machine-translated from the Chinese original 从三道习题看行列式计算在二次型化简中的应用 (published 2020-10-12). WeChat original ↗