Determining whether a matrix is non-heterogeneous is a focus in advanced algebra teaching. Generally speaking, there are five common methods for determining non-heterogeneous matrices, namely:
(i) Calculation of determinant (refer to Chapter 1 of the Advanced White Paper);
(ii) Coarse factor method (refer to Section 2.2.3 of the Advanced White Paper);
(iii) Application of the theory of solving linear equations (refer to Part 2 of Section 3.2.6 of the Advanced White Paper);
(iv) Application of relatively prime polynomials (refer to Section 5.2.9 of the Advanced White Paper);
(v) Calculation of eigenvalues (refer to Section 6.2.1 of the Advanced White Paper).
The following is the seventh major question of the Grade 19 Advanced Algebra I midterm exam at the School of Mathematics, Fudan University. It focuses on the application of the theory of solving linear equations in proving the non-heterogeneity of matrices, that is, a square matrix is a non-heterogeneous matrix if and only if the corresponding system of homogeneous linear equations has zero solution.
The seventh major questionLet be a real antisymmetric matrix of order, be a real antisymmetric matrix of order, and be a real matrix of order. Prove: and are both nonheterogeneous matrices.
First, we quote a common conclusion, which gives a characterization of antisymmetric arrays and is one of the most essential properties of antisymmetric arrays.
Lemma (Example 2.3 of the Advanced White Paper)Assume that is a square matrix of order on the number field, then is an antisymmetric matrix if and only if for any , .
Proof of the seventh questionTaking any solution of the linear equation system , we have . Multiplying both sides of the equation on the left at the same time, we can get by the lemma
So , that is, there is , so it can be known from the theory of solving linear equations that is a non-heterogeneous matrix. Regarding the non-heterogeneity of the matrix, we give three different proof methods.
Proof method oneBased on the particularity of the matrix , we use the reduction formula in reverse to construct a matrix of order . It can be seen from the reduction formula
Therefore, to prove that is not different, we only need to prove that is not different. Taking any solution of , we have
The first equation above is left multiplied by , the second equation is left multiplied by , and then the two equations are subtracted. Using the lemma and noting that , we finally get , so , , and thus it can be seen from the theory of solving the linear equation system that is a non-heterogeneous matrix.
Proof method twoBy left-multiplying the second block row of the block matrix in proof method 1 (the second type of block elementary transformation), the block matrix can be obtained
Note that is a real antisymmetric matrix, so from the first conclusion, it can be seen that is a non-heterogeneous matrix, and thus is also a non-heterogeneous matrix.
Proof method threeDirectly prove the non-heterogeneity of the matrix . Take any solution of the linear equation system , that is, we have
By left-multiplying both sides of the equation at the same time, we can get by the lemma
Order , then
Substitute equation (2) into equation (1) to get
So , , and thus it can be known from the theory of solving linear equations that is a non-heterogeneous matrix.
The conclusion of the seventh question has the following three extensions.
Promotion 1Let be a positive definite (semi-definite) real symmetric matrix of order, and be a real antisymmetric matrix of order, then
proveThe case of a positive definite matrix is Example 9.100 of the Advanced White Paper. It is noted that a diagonal matrix whose main diagonal elements are all greater than zero is a positive definite real symmetric matrix, so Example 3.78 of the Advanced White Paper is a special case of Example 9.100. They both give the generalization of the first conclusion of the seventh topic. The case of a positive semidefinite array can be obtained from the case of a positive definite array through the perturbation method.
definitionLet be a real square matrix of order, if for any , , it is called subpositive definite matrix; if for any , , it is called subpositive semipositive matrix. In the same way, the definitions of subnegative definite matrix and subseminegative definite matrix can be given, that is, is a subnegative definite matrix (subseminegative definite matrix) if and only if is a subpositive definite matrix (subsemipositive definite matrix). It is easy to prove: is a subpositive definite matrix (subpositive semidefinite array) if and only if is a positive semidefinite array (positive semidefinite array); if is a subpositive definite array, then is also a subpositive definite array. For specific proof details and other related conclusions, please refer to Answer Question 14 on page 411 of the Advanced White Paper.
propositionAssume that is a real square matrix of order, then is a subpositive definite matrix (subpositive semidefinite matrix) if and only if , where is a positive definite (semidefinite) real symmetric matrix, and is a real antisymmetric matrix. In particular, if is a subpositive definite matrix (subpositive semidefinite matrix), then ().
proveThe sufficiency can be obtained by directly verifying the definition of subpositive definite matrix (subpositive semidefinite matrix) by using the lemma. The necessity can be obtained by
And the equivalent definition of subpositive definite matrix (subpositive semidefinite matrix) is obtained. The second conclusion is obtained by generalizing 1.
Promotion 2Suppose is a subpositive definite matrix of order, is a subpositive definite matrix of order, is a real matrix of order, then is also a subpositive definite matrix of order. In particular, .
proveSince is a subpositive definite array, we know that is also a subpositive definite array, so is a subpositive definite array. And is a subpositive definite array, so from the definition, we know that is also a subpositive definite array. The second conclusion is obtained from the proposition.
Returning to the seventh question, it can be seen from the proposition that and are both subpositive definite arrays, and from generalization 2 we can see that is also a subpositive definite array, so
This gives a generalization of the second conclusion of the seventh major question. Finally, we give a generalization of the fourth question of the 2017 Ninth National College Student Mathematics Contest preliminaries.
Promotion 3Let be a real square matrix of order, satisfying . Proof: If is a subpositive definite matrix or a subnegative definite matrix, then is a non-heterogeneous matrix.
proveJust prove that is a subpositive definite matrix. Taking any solution of the linear equations , we have
It can be seen from the proposition that , so are all non-heterogeneous arrays. Assume , substitute into (3) and simplify it, we can get
By left-multiplying both sides of equation (4) at the same time, we can get
From the subpositive definiteness of , we can get and , so from the solution theory of linear equations, it can be seen that is a non-heterogeneous matrix.
References
[1] 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 复旦大学数学学院19级高等代数I期中考试第七大题的三种证法及其推广 (published 2020-11-03). WeChat original ↗