We fix an integer $d \in \mathbb{N}^*$, and we equip $\mathbb{R}^d$ with the usual inner product denoted $\langle \cdot, \cdot \rangle$ and the associated Euclidean norm $\|\cdot\|$. We denote $C := \{x \in \mathbb{R}^d \mid \|x\| \leq 1\}$ the closed unit ball of $\mathbb{R}^d$. Let $M$ be a nonzero real symmetric matrix of size $d \times d$ such that $\forall x \in \mathbb{R}^d,\, \langle x, Mx \rangle \geq 0$. We define $f(x) := -\frac{1}{2}\langle x, Mx \rangle$, so that $\nabla f(x) = -Mx$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $$x_{n+1} := P_C(x_n - \tau \nabla f(x_n)), \quad \text{with} \quad P_C(x) := \begin{cases} x & \text{if } \|x\| \leq 1, \\ x/\|x\| & \text{otherwise.} \end{cases}$$ How does the sequence behave when $\|x_0\| < 1$?

We fix an integer $d \in \mathbb{N}^*$, and we equip $\mathbb{R}^d$ with the usual inner product denoted $\langle \cdot, \cdot \rangle$ and the associated Euclidean norm $\|\cdot\|$. We denote $C := \{x \in \mathbb{R}^d \mid \|x\| \leq 1\}$ the closed unit ball of $\mathbb{R}^d$. Let $M$ be a nonzero real symmetric matrix of size $d \times d$ such that $\forall x \in \mathbb{R}^d,\, \langle x, Mx \rangle \geq 0$. We define $f(x) := -\frac{1}{2}\langle x, Mx \rangle$, so that $\nabla f(x) = -Mx$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $$x_{n+1} := P_C(x_n - \tau \nabla f(x_n)), \quad \text{with} \quad P_C(x) := \begin{cases} x & \text{if } \|x\| \leq 1, \\ x/\|x\| & \text{otherwise.} \end{cases}$$ Show that there exists a hyperplane $H \subset \mathbb{R}^d$ such that, for all $x_0 \in \mathbb{R}^d \setminus H$, we have $\lim_{n \rightarrow \infty} f(x_n) = \min\{f(x) \mid x \in C\}$.

The tribonacci numbers $\left\{ T _ { n } \right\}$ are defined for non-negative integers $n$ as follows.
$$\left\{ \begin{array} { l } T _ { 0 } = T _ { 1 } = 0 \\ T _ { 2 } = 1 \\ T _ { n + 3 } = T _ { n + 2 } + T _ { n + 1 } + T _ { n } \quad ( n \geq 0 ) \end{array} \right.$$
Answer the following questions.
(1) Find the matrix $A$ that satisfies Eq. (1.1) for all non-negative integers $n$.
$$\left( \begin{array} { l } T _ { n + 3 } \\ T _ { n + 2 } \\ T _ { n + 1 } \end{array} \right) = A \left( \begin{array} { l } T _ { n + 2 } \\ T _ { n + 1 } \\ T _ { n } \end{array} \right)$$
(2) Find the rank and the characteristic equation, i.e., the equation that eigenvalues satisfy, of the matrix $A$.
(3) Let $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 }$ denote the eigenvalues of the matrix $A$. Express an eigenvector corresponding to each of the eigenvalues using $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 }$.
(4) Prove that the matrix $A$ has only one real number eigenvalue. Letting $\lambda _ { 1 }$ correspond to this eigenvalue, prove that $1 < \lambda _ { 1 } < 2$.
(5) Prove that $T _ { n }$ can be expressed as $T _ { n } = c _ { 1 } \lambda _ { 1 } ^ { n } + c _ { 2 } \lambda _ { 2 } ^ { n } + c _ { 3 } \lambda _ { 3 } ^ { n }$ using constant complex numbers $c _ { 1 } , c _ { 2 } , c _ { 3 }$. You do not need to find values of $c _ { 1 } , c _ { 2 } , c _ { 3 }$ explicitly.
(6) Prove $\lim _ { n \rightarrow \infty } \frac { T _ { n + 1 } } { T _ { n } } = \lambda _ { 1 }$.

Suppose that three-dimensional vectors $\left( \begin{array} { c } x _ { n } \\ y _ { n } \\ z _ { n } \end{array} \right)$ satisfy the equation
$$\left( \begin{array} { l } x _ { n + 1 } \\ y _ { n + 1 } \\ z _ { n + 1 } \end{array} \right) = A \left( \begin{array} { l } x _ { n } \\ y _ { n } \\ z _ { n } \end{array} \right) \quad ( n = 0,1,2 , \ldots )$$
where $x _ { 0 } , y _ { 0 } , z _ { 0 }$ and $\alpha$ are real numbers, and
$$A = \left( \begin{array} { c c c } 1 - 2 \alpha & \alpha & \alpha \\ \alpha & 1 - \alpha & 0 \\ \alpha & 0 & 1 - \alpha \end{array} \right) , \quad 0 < \alpha < \frac { 1 } { 3 }$$
Answer the following questions.
(1) Express $x _ { n } + y _ { n } + z _ { n }$ using $x _ { 0 } , y _ { 0 }$ and $z _ { 0 }$.
(2) Obtain the eigenvalues $\lambda _ { 1 } , \lambda _ { 2 }$ and $\lambda _ { 3 }$, and their corresponding eigenvectors $\boldsymbol { v } _ { \mathbf { 1 } } , \boldsymbol { v } _ { \mathbf { 2 } }$ and $\boldsymbol { v } _ { \mathbf { 3 } }$ of the matrix $A$.
(3) Express the matrix $A$ using $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } , \boldsymbol { v } _ { 1 } , \boldsymbol { v } _ { 2 }$ and $\boldsymbol { v } _ { 3 }$.
(4) Express $\left( \begin{array} { l } x _ { n } \\ y _ { n } \\ z _ { n } \end{array} \right)$ using $x _ { 0 } , y _ { 0 } , z _ { 0 }$ and $\alpha$.
(5) Obtain $\lim _ { n \rightarrow \infty } \left( \begin{array} { l } x _ { n } \\ y _ { n } \\ z _ { n } \end{array} \right)$. (6) Regard
$$f \left( x _ { 0 } , y _ { 0 } , z _ { 0 } \right) = \frac { \left( x _ { n } , y _ { n } , z _ { n } \right) \left( \begin{array} { l } x _ { n + 1 } \\ y _ { n + 1 } \\ z _ { n + 1 } \end{array} \right) } { \left( x _ { n } , y _ { n } , z _ { n } \right) \left( \begin{array} { l } x _ { n } \\ y _ { n } \\ z _ { n } \end{array} \right) }$$
as a function of $x _ { 0 } , y _ { 0 }$ and $z _ { 0 }$. Obtain the maximum and the minimum values of $f \left( x _ { 0 } , y _ { 0 } , z _ { 0 } \right)$, where we assume that $x _ { 0 } ^ { 2 } + y _ { 0 } ^ { 2 } + z _ { 0 } ^ { 2 } \neq 0$.

Problem 2

Answer the following questions about the square matrix $A$ of order 3:
$$A = \left( \begin{array} { c c c } 3 & 0 & 1 \\ - 1 & 2 & - 1 \\ - 2 & - 2 & 1 \end{array} \right)$$
I. Find all eigenvalues of $A$. II. Find the matrix $A ^ { n }$, where $n$ is a natural number. III. The square matrix $\boldsymbol { B }$ of order 3 is diagonalizable and meets $\boldsymbol { A } \boldsymbol { B } = \boldsymbol { B } \boldsymbol { A }$. Prove that any eigenvector $\boldsymbol { p }$ of $\boldsymbol { A }$ is also an eigenvector of $\boldsymbol { B }$. IV. Find the square matrix $\boldsymbol { B }$ of order 3 that meets $\boldsymbol { B } ^ { 2 } = \boldsymbol { A }$, where $\boldsymbol { B }$ is diagonalizable and all eigenvalues of $\boldsymbol { B }$ are positive. V. The square matrix $\boldsymbol { X }$ of order 3 is diagonalizable and meets $\boldsymbol { A } \boldsymbol { X } = \boldsymbol { X } \boldsymbol { A }$. When $\operatorname { tr } ( \boldsymbol { A } \boldsymbol { X } ) = d$, find the maximum of $\operatorname { det } ( \boldsymbol { A } \boldsymbol { X } )$ as a function of $d$.
Here, $d$ is positive real and all eigenvalues of $X$ are positive. In addition, $\operatorname { tr } ( M )$ is the trace (the sum of the main diagonal elements) of the square matrix $\boldsymbol { M }$, and $\operatorname { det } ( \boldsymbol { M } )$ is the determinant of the matrix $\boldsymbol { M }$.

I. Suppose that $\lambda$ is an eigenvalue of a regular matrix $\boldsymbol { P }$, prove that:

1. $\lambda$ is not zero.
2. $\lambda ^ { - 1 }$ is an eigenvalue of $\boldsymbol { P } ^ { - 1 }$ and $\lambda ^ { n }$ is an eigenvalue of $\boldsymbol { P } ^ { n }$, where $n$ is a positive integer.

II. Suppose $\boldsymbol { P }$ is an orthogonal matrix. When the following symmetric matrix $\boldsymbol { A }$ can be diagonalized by $\boldsymbol { P }$, find the matrix $\boldsymbol { P }$ and obtain the diagonalized matrix.
$$A = \left( \begin{array} { c c c } 2 & - 1 & 1 \\ - 1 & 2 & - 1 \\ 1 & - 1 & 2 \end{array} \right)$$
III. When a matrix $\boldsymbol { P }$, and vectors $\boldsymbol { r }$ and $\boldsymbol { x }$ are given as
$$\boldsymbol { P } = \left( \begin{array} { c c c } 1 & 1 & 1 \\ p & p ^ { 2 } & p ^ { 3 } \\ q & q ^ { 2 } & q ^ { 3 } \end{array} \right) , \quad \boldsymbol { r } = \left( \begin{array} { c } r \\ r ^ { 2 } \\ r ^ { 3 } \end{array} \right) , \quad \boldsymbol { x } = \left( \begin{array} { c } x \\ y \\ z \end{array} \right) ,$$
where $p , q$, and $r$ are non-zero real numbers that differ from each other.

1. Find the condition that $p$ and $q$ must satisfy in order for $\boldsymbol { P }$ to be a regular matrix.
2. When $\boldsymbol { P } ^ { \mathrm { T } } \boldsymbol { x } = \boldsymbol { r }$ has a single solution, obtain $\boldsymbol { x }$. Here, $\boldsymbol { P } ^ { \mathrm { T } }$ is the transposed matrix of $\boldsymbol { P }$.

IV. The matrix $\boldsymbol { P } _ { n }$ is an $n$-th order square matrix ( $n \geq 2$ ), as shown below, where $p$ and $q$ are real numbers that differ from each other.
$$\boldsymbol { P } _ { n } = \left( \begin{array} { c c c c c c } p + q & q & 0 & \cdots & 0 & 0 \\ p & p + q & \ddots & \ddots & \vdots & \vdots \\ 0 & p & \ddots & \ddots & 0 & \vdots \\ \vdots & 0 & \ddots & \ddots & q & 0 \\ \vdots & \vdots & \ddots & \ddots & p + q & q \\ 0 & 0 & \cdots & 0 & p & p + q \end{array} \right)$$

1. Obtain the recurrence formula satisfied by the determinant of $\boldsymbol { P } _ { n }$, $\left| \boldsymbol { P } _ { n } \right|$.
2. Express the determinant $\left| \boldsymbol { P } _ { n } \right|$ in terms of $p , q$, and $n$, using the recurrence formula in Question IV.1.

Problem 2
I. Answer the following questions about the matrix $\boldsymbol { P }$: $$\boldsymbol { P } = \left( \begin{array} { c c c } 0 & 0 & \frac { 3 } { 2 } \\ 2 & 0 & 0 \\ 0 & \frac { 1 } { 3 } & 0 \end{array} \right)$$

1. Obtain all eigenvalues of the matrix $\boldsymbol { P }$ and the corresponding eigenvectors with unit norms.
2. Obtain $P ^ { 2 }$ and $P ^ { 3 }$.

II. Let $\boldsymbol { A }$ be the real matrix given by the block diagonal matrix: $$\boldsymbol { A } = \left( \begin{array} { c c c c c } 0 & 0 & c & 0 & 0 \\ a & 0 & 0 & 0 & 0 \\ 0 & b & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & e \\ 0 & 0 & 0 & d & 0 \end{array} \right)$$ Express succinctly the necessary and sufficient condition on $a , b , c , d$, and $e$, such that there exists a positive integer $m$ for which $\boldsymbol { A } ^ { m }$ is the identity matrix (proof is not required).
III. The matrix $M$ is a square matrix of order 12 with all elements taking either 0 or 1, such that each row and column has exactly one element being 1. Let $k _ { 0 }$ be the minimum value of the positive integer $k$ such that $M ^ { k }$ is the identity matrix. For all possible matrices $M$, give the maximum value of $k _ { 0 }$ (proof is not required).

Consider the following matrix $\boldsymbol { A }$ :
$$A = \left( \begin{array} { c c c } 1 & - 2 & - 1 \\ - 2 & 1 & 1 \\ - 1 & 1 & \alpha \end{array} \right)$$
where $\alpha$ is a real number. In the following, the transpose of a vector $\boldsymbol { v }$ is denoted by $\boldsymbol { v } ^ { \mathrm { T } }$.
I. Obtain $\alpha$ when the sum of the three eigenvalues of the matrix $A$ is 7.
II. Obtain $\alpha$ when the product of the three eigenvalues of the matrix $\boldsymbol { A }$ is $- 16$.
III. Let $\| \boldsymbol { A } \|$ be the maximum of $\boldsymbol { x } ^ { \mathrm { T } } \boldsymbol { A x }$ for the set of real vectors $\boldsymbol { x } = \left( \begin{array} { l } x _ { 1 } \\ x _ { 2 } \\ x _ { 3 } \end{array} \right)$ that satisfy $\boldsymbol { x } ^ { \mathrm { T } } \boldsymbol { x } = 1$. Obtain $\alpha$ when $\| \boldsymbol { A } \| = 4$.
IV. In the following questions, $\alpha = 4$.

1. Obtain all eigenvalues of the matrix $\boldsymbol { A }$ and their corresponding normalized eigenvectors.
2. Find the range of $\boldsymbol { y } ^ { \mathrm { T } } \boldsymbol { A } \boldsymbol { y }$ for the real vectors $\boldsymbol { y } = \left( \begin{array} { l } y _ { 1 } \\ y _ { 2 } \\ y _ { 3 } \end{array} \right)$ that satisfy $\boldsymbol { y } ^ { \mathrm { T } } \boldsymbol { y } = 1$ and $y _ { 1 } - y _ { 2 } - 2 y _ { 3 } = 0$.
3. Find the range of $\boldsymbol { z } ^ { \mathrm { T } } \boldsymbol { A } \boldsymbol { z }$ for the real vectors $\boldsymbol { z } = \left( \begin{array} { c } z _ { 1 } \\ z _ { 2 } \\ z _ { 3 } \end{array} \right)$ that satisfy $\boldsymbol { z } ^ { \mathrm { T } } \boldsymbol { z } = 1$ and $z _ { 1 } + z _ { 2 } + z _ { 3 } = 0$.

Consider expressing the following matrix $\boldsymbol { A }$ in a form of $\boldsymbol { A } = \boldsymbol { P } \boldsymbol { D } \boldsymbol { P } ^ { - 1 }$, using a diagonal matrix $\boldsymbol { D }$ and a regular matrix $\boldsymbol { P }$. Here, $a$ is a real number.
$$A = \left( \begin{array} { l l l } 2 & 1 & 0 \\ 1 & 3 & a \\ 0 & a & 2 \end{array} \right)$$
I. When $a = 1$, find a diagonal matrix $D$.
II. When $a = 1$, prove $\boldsymbol { x } ^ { \mathrm { T } } \boldsymbol { A } \boldsymbol { x } > 0$ for any three-dimensional non-zero real vector $\boldsymbol { x }$. $\boldsymbol { x } ^ { \mathrm { T } }$ represents the transpose of $\boldsymbol { x }$.
III. Find the condition of $a$ which satisfies $\boldsymbol { x } ^ { \mathrm { T } } \boldsymbol { A } \boldsymbol { x } > 0$ for any three-dimensional non-zero real vector $\boldsymbol { x }$.
IV. Assume that $a$ satisfies the condition obtained in Question III.
For a real vector $\boldsymbol { b } = \left( \begin{array} { c } a \\ 0 \\ - 1 \end{array} \right)$, express the minimum value of the function $f ( \boldsymbol { x } ) = \boldsymbol { x } ^ { \mathrm { T } } \boldsymbol { A } \boldsymbol { x } - \boldsymbol { b } ^ { \mathrm { T } } \boldsymbol { x }$ by using $a$.

Problem 2

Answer the following questions about a real symmetric matrix, $\boldsymbol { A }$ :
$$A = \left( \begin{array} { l l l } 0 & 1 & 2 \\ 1 & 0 & 2 \\ 2 & 2 & 3 \end{array} \right)$$
I. Find all the different eigenvalues of matrix $\boldsymbol { A } , \lambda _ { 1 } , \cdots , \lambda _ { r } \left( \lambda _ { 1 } < \cdots < \lambda _ { r } \right)$.
II. Find all the eigenspaces $W \left( \lambda _ { 1 } \right) , \cdots , W \left( \lambda _ { r } \right)$ corresponding to $\lambda _ { 1 } , \cdots , \lambda _ { r }$, respectively.
III. Find an orthonormal basis, $\boldsymbol { b } _ { 1 } , \boldsymbol { b } _ { 2 } , \boldsymbol { b } _ { 3 }$, which belongs to either of $W \left( \lambda _ { 1 } \right) , \cdots , W \left( \lambda _ { r } \right)$, obtained in Question II.
IV. Find the spectral decomposition of $A$ :
$$A = \sum _ { i = 1 } ^ { r } \lambda _ { i } P _ { i }$$
where $\boldsymbol { P } _ { i }$ is the projection matrix onto $W \left( \lambda _ { i } \right)$.
V. Find $A ^ { n }$, where $n$ is any positive integer.

Problem 2

For a square matrix $\boldsymbol { A } , e ^ { \boldsymbol { A } }$ is defined as:
$$e ^ { A } = \boldsymbol { E } + \sum _ { k = 1 } ^ { \infty } \frac { 1 } { k ! } \boldsymbol { A } ^ { k } ,$$
where $\boldsymbol { E }$ is the identity matrix and $e$ is the base of natural logarithm.
I. Let $\boldsymbol { A }$ be a $3 \times 3$ square matrix which can be diagonalized by a regular matrix $\boldsymbol { P }$, i.e., $\boldsymbol { A } = \boldsymbol { P } \boldsymbol { D } \boldsymbol { P } ^ { - 1 }$, where $\boldsymbol { D }$ is a diagonal matrix:
$$\boldsymbol { D } = \left( \begin{array} { c c c } \lambda _ { 1 } & 0 & 0 \\ 0 & \lambda _ { 2 } & 0 \\ 0 & 0 & \lambda _ { 3 } \end{array} \right)$$
Here, $\lambda _ { 1 } , \lambda _ { 2 }$, and $\lambda _ { 3 }$ are complex numbers. Prove the following equation:
$$e ^ { \boldsymbol { A } } = \boldsymbol { P } \left( \begin{array} { c c c } e ^ { \lambda _ { 1 } } & 0 & 0 \\ 0 & e ^ { \lambda _ { 2 } } & 0 \\ 0 & 0 & e ^ { \lambda _ { 3 } } \end{array} \right) \boldsymbol { P } ^ { - 1 }$$
II. Let $\boldsymbol { A } = \left( \begin{array} { c c c } - 1 & 4 & 4 \\ - 5 & 8 & 10 \\ 3 & - 3 & - 5 \end{array} \right)$.

1. Find the regular matrix $\boldsymbol { P }$ and the diagonal matrix $\boldsymbol { D }$ such that $\boldsymbol { A } = \boldsymbol { P } \boldsymbol { D } \boldsymbol { P } ^ { - 1 }$.
2. Calculate $e ^ { \boldsymbol { A } }$.

III. Consider $\boldsymbol { A } = \left( \begin{array} { c c c } 0 & - x & 0 \\ x & 0 & 0 \\ 0 & 0 & 1 \end{array} \right) , \boldsymbol { B } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 1 \end{array} \right)$ and $\boldsymbol { a } = \left( \begin{array} { l } 1 \\ 1 \\ e \end{array} \right)$, where $x$ is a real number. In the following, the transpose of a vector $\boldsymbol { v }$ is denoted by $\boldsymbol { v } ^ { T }$.

1. Express the sum of the eigenvalues of $e ^ { \boldsymbol { A } }$ using $e$ and $x$.
2. Let $\boldsymbol { C } = \boldsymbol { B } e ^ { \boldsymbol { A } }$. Find the minimum and maximum values of $\frac { \boldsymbol { y } ^ { T } \boldsymbol { C } \boldsymbol { y } } { \boldsymbol { y } ^ { T } \boldsymbol { y } }$ for a real three-dimensional vector $\boldsymbol { y } ( \boldsymbol { y } \neq \mathbf { 0 } )$.
3. Let $f ( \boldsymbol { z } ) = \frac { 1 } { 2 } \boldsymbol { z } ^ { T } \boldsymbol { C } \boldsymbol { z } - \boldsymbol { a } ^ { T } \boldsymbol { z }$ for a real three-dimensional vector $\boldsymbol { z } = \left( \begin{array} { c } z _ { 1 } \\ z _ { 2 } \\ z _ { 3 } \end{array} \right)$ and $\boldsymbol { C }$ in III.2. Find $\sqrt { z _ { 1 } ^ { 2 } + z _ { 2 } ^ { 2 } + z _ { 3 } ^ { 2 } }$ for $\boldsymbol { z }$ such that $\frac { \partial f ( \boldsymbol { z } ) } { \partial z _ { 1 } } = \frac { \partial f ( \boldsymbol { z } ) } { \partial z _ { 2 } } = \frac { \partial f ( \boldsymbol { z } ) } { \partial z _ { 3 } } = 0$.

Let m be a positive real number and $u = \left[ \begin{array} { l l } x & y \end{array} \right]$. Given that
$$\mathrm { u } \cdot \left[ \begin{array} { l l } 1 & 2 \\ 2 & 1 \end{array} \right] = \mathrm { u } \cdot \left[ \begin{array} { c c } \mathrm { m } & 0 \\ 0 & \mathrm {~m} \end{array} \right]$$
the matrix equation has infinitely many solutions for u, what is m?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 2 } { 3 }$
D) 3
E) 4

Let $M = \left[ \begin{array} { r r } 1 & 1 \\ - 2 & 4 \end{array} \right]$ and $X = \left[ \begin{array} { l } 1 \\ 2 \end{array} \right]$ such that
$$\begin{aligned} & \mathrm { M } \cdot \mathrm { X } = \mathrm { aX } \\ & \mathrm { M } ^ { - 1 } \cdot \mathrm { X } = \mathrm { bX } \end{aligned}$$
For real numbers a and b satisfying these equalities, what is the sum $a + b$?
A) $\frac { 1 } { 3 }$
B) $\frac { 4 } { 3 }$
C) $\frac { 5 } { 3 }$
D) $\frac { 8 } { 3 }$
E) $\frac { 10 } { 3 }$