Mathematics II PEAK 2018 – 2019 (2)

• Mathematics, College of Arts and Sciences, The University of Tokyo
• Exercise sesseion with Minitest and Tutorial by assistant professors and TAs for Mathematics I, II PEAK:

  Time:	5th period of Thursdays during the semester
  Place:	KIBER 109  (Room 109 of 駒場国際教育研究棟)

• 2019/07/28   On marking the final exam:
  • Problem 1 (1) By row addition, one finds that the determinant is 120.
  • Problem 1 (2) The roots are z = ± (1 + √3) i) .
  • Problem 2 (1) The resulting vectors are
    (1,-1, 0)^T, (1/2, 1/2,-1)^T, (2/3,-2/3, 2/3)^T.
  • Problem 2 (2) The matrix is found to be

    1/3	-2/3	-2/3
    -2/3	1/3	-2/3
    -2/3	-2/3	1/3

  • Problem 3 (1) The characteristic polynomial is found to be t³-12t+16=(t-2)²(t+4). Hence the signature is given by (n₊ ,n_- ,n₀)=(2,1,0).
  • Problem 3 (2) By solving (A-2E)x=0 and (A+4E)x=0, pick up bases of Ker (A-2E) and Ker (A+4E). Then applying Gram-Schmidt process to the former, and normalizing the vectors so obtained, one finds A=Q^TDQ, where D is the diagonal matrix with diagonal entries 2,2,-4 and Q is the orthogonal matrix given, for intance, by
    

    1/√2	-1/√3	1/√6
    0	1/√3	2/√6
    1/√2	1/√3	-1/√6

    The columns of Q should be taken to be orthonormal.
  • Problem 4 (1) The minimal polynomial is (t-1)².
  • Problem 4 (2) It follows from (1) that 1 is the only eigenvalue. By solving the equation (A-E)x=0, one finds that the vectors (1,1,1,1)^T, (1,0,0,1)^T, (0,1,0,1)^T, for instance, form a basis of the eigenspace.
  • Problem 4 (3) It follows from (2) that there are 3 Jordan blocks. Since the size of A is 4×4, exactly one of the blocks is of size 2 by (1). Therefore, the Jordan canonical form is given by J₂(1) ⊕ J₁(1) ⊕ J₁(1). Namely

    1	1	0	0
    0	1	0	0
    0	0	1	0
    0	0	0	1

  • Problem 4 (4) For instance, the transition matrix P is given by

    1	1	1	0
    1	0	0	1
    1	0	0	0
    1	0	1	1

    in accordance with the order of the Jordan blocks chosen in (3) as above.
• 2019/07/03   Correction.
  • In obtaining the number of Jordan blocks of size =d, subtract the number of > d from the number of ≥ d.

The schedule may change.

 Academic Calender 2019

1. 2019/04/10 (Wed)  Determinant of square matrices
2. 2019/04/17 (Wed)  Expansion of determinants
3. 2019/04/24 (Wed)  Applications of determinants
4. 2019/04/27 (Sat)   Orthonormal bases
5. 2019/05/08 (Wed)  Principal axes of quadratic forms
6. 2019/05/15 (Wed)  Diagonalization of square matrices
7. 2019/05/22 (Wed)  Complex numbers
8. 2019/06/05 (Wed)  (Trial Exam and Addendum)
9. 2019/06/12 (Wed)  Generalized eigenspaces
10. 2019/06/19 (Wed)  Jordan canonical forms
11. 2019/06/26 (Wed)  Exponential of square matrices
12. 2019/07/03 (Wed)  (Trial Exam and Addendum)
13. 2019/07/10 (Wed)  Diagonalization of Hermitian matrices
14. 2019/07/24 (Wed)  Exam for S semester. 
    
15. 2019/10/**  Make-up Exam for S semester. 
    

The following topics will be covered by the course.

1. Determinants and regularity of square matrices
2. Laplace expansion of determinants
3. Signature of quadratic forms
4. Complex numbers and fundamental theorem of algebra
5. Characteristic and minimal polynomials of square matrices
6. Eigenspaces and generalized eigenspaces
7. Linear differential equations with constant coefficients
8. Jordan canonical forms
9. Exponentials of square matrices
10. Diagonalization of Hermitian forms