# Mathematics II PEAK 2016 – 2017 (2)

## Atsushi Matsuo

Continued from Mathematics II PEAK 2016 – 2017 (1)
Previous Year Mathematics II PEAK 2015 – 2016 (2)   Mathematics II PEAK 2015 – 2016 (2)
NOTICES   CLASS SCHEDULE   COURSE INFORMATION   LECTURE PLAN   TUTORIAL   PAST NOTICES
• NOTICES

• 2017/10/19 On marking the Make-up exam.

Problem 1 (1): The determinant is 12. (2): For instance, the following matrix works:
 1/sqrt(3) 2/sqrt(6) 0 −1/sqrt(3) 1/sqrt(6) 1/sqrt(2) 1/sqrt(3) −1/sqrt(6) 1/sqrt(2)
where sqrt(x) denotes the square root of x. Many other matrices work fine. Note that the columns of the matrix have to be not only orthogonal but also orthonormal.

Problem 2 (1): The matrix is given by
 et e2t−et 0 0 e2t 0 0 e2t− et et
(2): It is because, by square completion, Q(x,y,z) = 2(x−y/2−z/2)2 + (3/2) (y−z)2 + 0z2, where the transformation (x,y,z) to (x−y/2−z/2,y−z,z) is invertible.

Problem 3 (1): t3−t2. (2): The Jordan canonical form is given by
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0
or  1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
or  0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1
Some students erroneously calculated as t3−t2=t (t−1) (t + 1) and found a wrong eigenvalue −1.

Problem 4 (1): Use the definition of unitary matrices. (2): No student received a positive point.

Problem 5: No student received a positive point.

• 2017/09/27 Make-up exam for S semester.
• Make-up exam will take place on October 6, 08:40 — 10:10.
• 2017/07/19
• On marking the exam. (revised on 07/22, 07/26, 08/18)

Your points may be reduced in case you have not explained your answer by sentences. It is also the case when you have used logical symbols in bad ways.

Problem 1 (1): The determinant is 1.
Although the determinant is easily calculated by elementary row or column operations, many students have strangely chosen the much harder way of calculating it by Laplace expansion.
Some students created and used a wrong formula for determinants of 4 by 4 matrices imitating the Sarrus rule for determinants of 3 by 3 matrices. Creating own wrong formulas is a typical symptom of weak students.

Problem 1 (2): The signature is (2,1) as readily seen by the square completion
Q(x, y, z) = (x + 2y + 2z) 2 - 4 (y + 5z/4) 2 + z2/4,
where the transformation of variables is invertible.
Some students erroneously calculated the signature by using a sum of more than 3 square terms ignoring the condition on the transformation of variables occuring in the square completion. Calculating without thinking of conditions is also a typical symptom of weak students.
The characteristic polynomial of the representation matrix is calculated as chA(t)=t3+t2-11t+1. This polynomial has 3 real roots but they do not have simple expressions by rational numbers or square roots of rational numbers. So a solution given in terms of roots with simple explicit expressions is incorrect. Note that one can find the signs of roots by considering the graph of the function y = x3+x2-11x+1. But no student seems to have tried this method.

Problem 2 (1): The characteristic polynomial is calculated as chA(t) = t3-9t = t (t-3) (t+3). Thus the eigenvalues are λ = 0, 3, -3. The matrix is diagonalized by a real orthogonal matrix as Q-1AQ = diag(0,3,-3) with Q given for instance by
 2/3 2/3 1/3 1/3 -2/3 2/3 -2/3 1/3 2/3
Here diag(0,3,-3) means the diagonal matrix with the diagonal entries 0, 3, -3. The order of the eigenvalues does not matter as far as the eigenvectors are arranged in Q according to the order of the eigenvalues in the diagonal matrix.
Since the question requires the diagonalization by a real orthogonal matrix, the columns of Q must be normalized so that their norms are 1. Some students seem to have overlooked this condition stated in the question.
Your points will be reduced if you have not explicitly written out the diagonalization Q-1AQ=D but have only answered the matrix Q, for one cannot read off from your answer whether the order of the columns of Q is in accordance with the order of the diagonal entries of the matrix D.

Problem 2 (2): The Jordan canonical form is given by J = J3(2) ⊕ J2(2) ⊕ J1(2).
You must explain the reason why the Jordan canonical form is determined as above from the conditions given in the question. It is possible that no point will be given if the explanation is incorrect or not accurate even though the Jordan canonical form is correctly given as above.
Note that we eventually obtain dim Ker (A-2E)2 = 5 and dim Ker (A-2E)3 = 6. Some students have erroneously evaluated these dimensions first and then tried to determine the Jordan canonical form.

Problem 3 (1): A basis is given, for example, by 1, et, e-t.
Your points may be reduced in case you have not explained your answer by sentences.

Problem 3 (2): The exponential is given by
 et cos t -et sin t et sin t et cos t
No point will be given if you tried to diagonalize the given matrix but did not get the correct eigenvalues 1+i, 1-i. No point will also be given if you only spell out a method but did not apply it by actual calculation to the specific situation of the question. .
One can verify one's answer by explicitly calculating the both sides of (d/dt) exp tA = A exp tA.

Problem 4 (1): The Jordan canonical form is given by J2(-1) ⊕ J2(-1). Indeed, by calculating A2, one easily finds A2=-2A-E, and this relation together with the fact that A is not a scalar multiple of E implies that the minimial polynomial of A is mA(t)=(t+1)2. Now by checking dim Ker (A+E), one obtains sufficient data to determine the Jordan canonical form.
Many students have gone into the extremely hard calculation of the characteristic polynomial by Laplace expansion. Sticking to a specific method is again a typical symptom of weak students.

Problem 4 (2): For instance, the following matrices work fine:
 1 1 1 0 1 0 0 1 1 0 1 0 1 0 0 0
 1 0 0 0 0 0 1 0 1 0 0 -1 0 -1 1 1
 1 0 0 1 0 1 1 0 1 0 0 0 0 0 1 1
There are infinitely many other matrices which satisfy the requirement of the question. One can verify one's answer by explicitly calculating the both sides of AP=PJ and by checking the linear independence of the columns of P. Being indifferent to veryfing correctness of one's result is yet another typical symptom of weak students.
No point will be given if you have failed to answer (1) correctly.

Problem 5: No student but one received a positive point.

• 2017/07/13
• Corrections.

Problem 1 (2) line 2: t-2 should be t-1.
Problem 2 (2) Notes ii) line 5 and 6: Ker A should be Ker (A-E).
Problem 2 (2) Notes ii) line 6: a complement should be the complement.

• CLASS SCHEDULE

The schedule may change.

1. 2017/04/05 (Wed)  Lecture 11. Determinants and regularity of square matrices.
2. 2017/04/19 (Wed)  Lecture 12. Laplace expansion.
3. 2017/04/26 (Wed)  Lecture 13. Signature of quadratic forms.
4. 2017/05/10 (Wed)  Lecture 14. Complex numbers and fundamental theorem of algebra
5. 2017/05/17 (Wed)  Lecture 15. Characteristic and minimal polynomials of square matrices.
6. 2017/05/24 (Wed)  Lecture 16. Eigenspaces and generalized eigenspaces.
7. 2017/05/31 (Wed)  Exercise Session 3.
8. 2017/06/07 (Wed)  Lecture 17. Jordan canonical forms.
9. 2017/06/14 (Wed)  Lecture 18. Linear differential equations with constant coefficients
10. 2017/06/21 (Wed)  Lecture 19. Exponentials of square matrices.
11. 2017/06/28 (Wed)  Lecture 20. Diagonalization of Hermitian matrices.
12. 2017/07/05 (Wed)  Exercise Session 4.
13. 2017/07/12 (Wed)  Lecture 21. Direct sum decomposition with respect to the minimal polynomial.
14. 2017/07/19 (Wed)  Exam for S semester.
Time and Place: 13:10—14:40  in Room 121
15. 2016/10/06 (Fri)  Make-up Exam for S semester.
Time and Place: 08:40—10:10  in Room ***

• LECTURE PLAN (tentative)

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

• TUTORIAL

Tutorial by assistant professors will take place as shown below.

Date: As marked by pink in the tables below.
Time: 5th period for Monday and Wednesday, and 11:15-13:00 for Friday.
Place: Room 109 of  ¶ð¾ì¹ñºÝ¶µ°é¸¦µæÅï

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