Entrance Examination for Master’s Program
Graduate School of Mathematics
Nagoya University
2021 Admission
Part 2 of 2February 7, 2021, 10:00 〜12:00

Note:
1. Please do not turn pages until told to do so.
2. The problem sheet consists of the cover page and 3 single-sided pages. Afterthe exam has begun, please ﬁrst conﬁrm that the number of pages and theirprinting and order are correct. Please report any problem immediately.
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3. There are a total of 3 problems labeled 1  , 2  , and 3  respectively.
Please answer all 3 problems.
4. The answering sheet consists of 3 single-sided pages. Please conﬁrm thenumber of pages, and please do not remove the staple.
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5. Pleasewritethe answers to problems 1  , 2  ，and 3  on pages 1  ,2  , and 3  of the answering sheet, respectively.
6. Please write name and application number in the space provided
on each of the 3 pages in the answering sheet．
7. The back side of the 3 pages in the answering sheet may also be used. Ifused, please check the box at the lower right-hand corner on the front side.
8. If the answering sheet staple is torn, or if additional paper is needed forcalculations, please notify the exam proctor.
9. After the exam has ended, please hand in the 3 page answering sheet. The
problem sheet and any additional sheets used for calculations may be taken
home.

Notation:
The symbols Z, Q, R, and C denote the sets of integers, rational numbers, real
numbers, and complex numbers, respectively.

(Master’s Program Entrance Exam for 2021; Part 2)

11 Consider an n × n real matrix A such that A = A, where A denotes the transpose of2tt
x1y1

A. For any elements x =  ... , y =  ...  of Cn , let ⟨x, y⟩ denote the standardxnHermitian inner productn∑yn

xj yj . Answer the following questions.

j=1(1) Let v be an eigenvector of A corresponding to an eigenvalue α. Show thatα2 = α by considering ⟨v, Av⟩.
(2) Show that eigenvectors of A corresponding to distinct eigenvalues are orthogonal.
Below we consider n = 3 and assume that A satisﬁes the following conditions:“ A2 = tA holds, A is invertible and is not the identity matrix. ”(3) Show that at least one of the eigenvalues of A is real, and calculate all theeigenvalues of A. Also show that A is diagonalizable by a unitary matrix.

(4) Give an example of a matrix A satisfying the conditions.

(February 7, 2021)(over)

(Master’s Program Entrance Exam for 2021; Part 2)2

2 Deﬁne I = [0, 1]. A function φ deﬁned on I is said to be of bounded variation on I if
there exists M > 0 such thatn−1∑|φ(tj+1 ) − φ(tj )| ≤ M

j=0holds for any partition
0 = t0 < t1 < · · · < tn = 1of I. Answer the following questions.
(1) Consider a function f that is diﬀerentiable on an open interval including I andsuch that its derivative f ′ is bounded on I. Show that f is of bounded variation
on I.
(2) Determine whether the following functions are of bounded variation on I or not.
Explain the reasons.
1 2(x ̸= 0)x sin(i) g(x) =x 0(x = 0)

(February 7, 2021)

(ii) h(x) =x cos 01(x ̸= 0)
x(x = 0)(over)

(Master’s Program Entrance Exam for 2021; Part 2)3



3Answer the following questions.
(1) Consider a real-valued function f : U → R over an open set U (̸= ∅) of Rnthat is of class C 1 and such that its partial derivatives∂f ∂f∂f,, ...,
are∂x1 ∂x2∂xn

identically zero on U . Show that f is a constant function if U satisﬁes thefollowing condition:“ any two points of U can be connected by a curve of class C 1 inside U . ”(2) A real-valued function f : X → R over a topological space X (̸= ∅) is a locallyconstant function if for any p ∈ X there exists an open set V containing p suchthat f is a constant function on V . Show that if X is connected, then a locallyconstant function on X is a constant function on X.

(February 7, 2021)(end)