The following are four major questions from the 20th-level Advanced Algebra I midterm exam of the School of Mathematics, Fudan University. Their solution ideas for Advanced Algebra I are as follows: The first question of the fourth major question uses matrix multiplication to directly calculate, the second question uses the solution theory of linear equations to discuss; the fifth major question uses the reduced order formula of determinants to calculate; the sixth major question uses Sylvester's inequality of matrix rank to prove; the seventh major question uses matrix operations to prove; Among them, the Advanced Algebra I solutions to the fourth, fifth, and seventh questions all have a shortcoming, which is that "the amount of calculation is too large and the solution is not concise enough." Below we will give the Advanced Algebra II solutions to these questions. Since this is a "dimensionality reduction blow", the calculation process is greatly simplified. In addition, we also provide the generalizations of the sixth and seventh questions respectively.
4. The linear space composed of all square matrices of order on the counting field is . For , consider .
(1) If , , find a set of basis of .
(2) If , try to determine all possible values of .
White Paper Example 7.22Let be a square matrix of order on the number field, representing the linear space composed of all polynomials of . Prove: The necessary and sufficient condition for the minimal polynomial of is equal to the characteristic polynomial is . If the necessary and sufficient conditions are established, then a set of bases of is .
Answer to the fourth question(1) Since is a friend matrix of the polynomial , the minimal polynomial of is equal to the characteristic polynomial of . From Example 7.22 in the white paper, we can see that a set of bases of is .
(2) Let the minimal polynomial of the square matrix of order be , and the characteristic polynomial be . If , then we can know from the Cayley-Hamilton theorem, and then we can know from Example 7.22 of the white paper, . If , then we can set , so , at this time, and thus .
5. Assume that the compound square matrix of order is irreversible. Prove that: there are at most two complex numbers , making it irreversible.
White Paper Example 6.13Assume that the overall eigenvalues of the order matrix are , and verify: the overall eigenvalues of are , , , .
Proof of the fifth questionAssume that the total eigenvalues of are , from which we can see that there is at least one . Let's assume , then the total eigenvalues of are and , so there are at most two mutually different eigenvalues, namely and , and the conclusion is proved.
6. Assume that is a square matrix of order, prove:
Promotion 1Assume that is a complex square matrix of order, let the eigenvalue of is and the number of Jordan blocks with order greater than or equal to be , define in the same way. Proof: Either is true, or is true.
Proof of promotion 1Since , we can see from the proof process of Example 7.33 of the white paper that the eigenvalue of is equal to the number of Jordan blocks whose order is greater than or equal to , so
The same principle can be proved, so the conclusion is proved.
NoteIn the teaching paper [2], we proved that: the Jordan blocks of non-zero eigenvalues of and are exactly the same; the order difference of the largest Jordan blocks of the eigenvalues of and is less than or equal to . The above generalization proves: if , then the difference in the number of Jordan blocks of the eigenvalues of and belongs to . The above information can help us determine the relationship between the Jordan standard forms of and .
Proof of the sixth questionSince the number of Jordan blocks of order is less than or equal to , the conclusion can be obtained by generalizing 1.
7. Let be a real square matrix of order, where is an upper triangular matrix whose main diagonal elements are all greater than zero, and satisfy . Prove: (1) must be a symmetric matrix;
(2) is a diagonal matrix if and only if ;
(3) .
White Paper Example 6.65Suppose , , have no common eigenvalues in the complex field. Prove: For any , there is a unique solution to the matrix equation in .
White Paper Example 9.101Assume that is a real matrix of order, known to be positive definite, verify: , the necessary and sufficient condition for the equality sign to be true is that is a symmetric matrix.
Proof of the seventh major question (1) Consider the matrix equation
Obviously, the eigenvalues of are all positive real numbers, and the eigenvalues of are all negative real numbers, so they have no common eigenvalues. From Example 6.65 in the white paper, we can see that there is a unique solution to the matrix equation . By transposing both sides of the equation at the same time, we can get , that is, is also the solution of the matrix equation. From the uniqueness, we can get that , that is, is a symmetric matrix.
(2) If is a diagonal matrix (also a symmetric matrix), then is also the solution of the matrix equation, and can be obtained from the uniqueness, so that . On the contrary, if , then
From Example 2.7 in the white paper, we can get that , that is, is both an upper triangular matrix and a symmetric matrix, and thus a diagonal matrix.
(3) Note that is a positive definite matrix, so it can be obtained from Example 9.101 of the white paper, and thus we can get .
Promotion 2Let be a real square matrix of order, in which the eigenvalues of are all positive real numbers, and satisfy . Prove:
(1) It must be a symmetric matrix;
(2) is a symmetric array if and only if , and also if and only if ;
(3) , the equality sign holds if and only if .
Proof of promotion 2(3) The necessary and sufficient conditions for the establishment of the medium sign require the use of Example 9.101 in the white paper, and the rest can be obtained by the same proof as in the seventh question.
References
[1] Advanced Algebra White Paper: Edited by Yao Musheng and Xie Qihong, Learning Method Guide: Advanced Algebra (Third Edition), Fudan University Press, 2015.
[2] Xie Qihong, Talking about the teaching series of advanced algebra from AB and BA, Advanced Mathematics Research, 2015, 18(6), 29–32.
Machine-translated from the Chinese original 复旦大学数学学院20级高等代数I期中考试大题的高等代数II解法及其推广 (published 2021-10-15). WeChat original ↗