As in the third part, we assume that $B = A + \mathbf{u u}^T$ with $A \in \mathcal{S}_n(\mathbb{R})$ a symmetric matrix, and $\mathbf{u} \in \mathbb{R}^n$ a vector such that $\|\mathbf{u}\| = 1$. We denote by $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$ the eigenvalues of $A$ and $\mu_1 \leqslant \mu_2 \leqslant \cdots \leqslant \mu_n$ those of $B$. We admit that $$\lambda_1 \leqslant \mu_1 \leqslant \lambda_2 \leqslant \mu_2 \leqslant \cdots \leqslant \lambda_n \leqslant \mu_n.$$ We further assume that there exists an integer $m \in \{1,2,\ldots,n-1\}$ such that the eigenvalues of $A$ satisfy $$0 = \lambda_1 = \lambda_2 = \cdots = \lambda_m < \lambda_{m+1} \leqslant \cdots \leqslant \lambda_n.$$ Let $\varepsilon \in ]0, \lambda_{m+1}[$. We suppose that $\left\langle \mathbf{u}, \left(A - \varepsilon \mathbb{I}_n\right)^{-1} \mathbf{u} \right\rangle < -1$. Show that $(B - \varepsilon \mathbb{I}_n)$ is invertible.

As in the third part, we suppose that $B = A + \mathbf{u}\mathbf{u}^T$ with $A \in \mathcal{S}_n(\mathbb{R})$ a symmetric matrix, and $\mathbf{u} \in \mathbb{R}^n$ a vector such that $\|\mathbf{u}\| = 1$. We denote by $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$ the eigenvalues of $A$ and $\mu_1 \leqslant \mu_2 \leqslant \cdots \leqslant \mu_n$ those of $B$. We admit that $$\lambda_1 \leqslant \mu_1 \leqslant \lambda_2 \leqslant \mu_2 \leqslant \cdots \leqslant \lambda_n \leqslant \mu_n.$$ We further suppose that there exists an integer $m \in \{1, 2, \ldots, n-1\}$ such that the eigenvalues of $A$ satisfy $$0 = \lambda_1 = \lambda_2 = \cdots = \lambda_m < \lambda_{m+1} \leqslant \cdots \leqslant \lambda_n.$$ Let $\varepsilon \in ]0, \lambda_{m+1}[$. We suppose that $\left\langle \mathbf{u}, \left(A - \varepsilon \mathbb{I}_n\right)^{-1}\mathbf{u}\right\rangle < -1$.
Show that $\operatorname{Tr}\left(\left(B - \varepsilon \mathbb{I}_n\right)^{-1}\right) > \operatorname{Tr}\left(\left(A - \varepsilon \mathbb{I}_n\right)^{-1}\right)$.

As in the third part, we assume that $B = A + \mathbf{u u}^T$ with $A \in \mathcal{S}_n(\mathbb{R})$ a symmetric matrix, and $\mathbf{u} \in \mathbb{R}^n$ a vector such that $\|\mathbf{u}\| = 1$. We denote by $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$ the eigenvalues of $A$ and $\mu_1 \leqslant \mu_2 \leqslant \cdots \leqslant \mu_n$ those of $B$. We admit that $$\lambda_1 \leqslant \mu_1 \leqslant \lambda_2 \leqslant \mu_2 \leqslant \cdots \leqslant \lambda_n \leqslant \mu_n.$$ We further assume that there exists an integer $m \in \{1,2,\ldots,n-1\}$ such that the eigenvalues of $A$ satisfy $$0 = \lambda_1 = \lambda_2 = \cdots = \lambda_m < \lambda_{m+1} \leqslant \cdots \leqslant \lambda_n.$$ Let $\varepsilon \in ]0, \lambda_{m+1}[$. We suppose that $\left\langle \mathbf{u}, \left(A - \varepsilon \mathbb{I}_n\right)^{-1} \mathbf{u} \right\rangle < -1$. Show that $\operatorname{Tr}\left(\left(B - \varepsilon \mathbb{I}_n\right)^{-1}\right) > \operatorname{Tr}\left(\left(A - \varepsilon \mathbb{I}_n\right)^{-1}\right)$.

As in the third part, we suppose that $B = A + \mathbf{u}\mathbf{u}^T$ with $A \in \mathcal{S}_n(\mathbb{R})$ a symmetric matrix, and $\mathbf{u} \in \mathbb{R}^n$ a vector such that $\|\mathbf{u}\| = 1$. We denote by $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$ the eigenvalues of $A$ and $\mu_1 \leqslant \mu_2 \leqslant \cdots \leqslant \mu_n$ those of $B$. We admit that $$\lambda_1 \leqslant \mu_1 \leqslant \lambda_2 \leqslant \mu_2 \leqslant \cdots \leqslant \lambda_n \leqslant \mu_n.$$ We further suppose that there exists an integer $m \in \{1, 2, \ldots, n-1\}$ such that the eigenvalues of $A$ satisfy $$0 = \lambda_1 = \lambda_2 = \cdots = \lambda_m < \lambda_{m+1} \leqslant \cdots \leqslant \lambda_n.$$ Let $\varepsilon \in ]0, \lambda_{m+1}[$. We suppose that $\left\langle \mathbf{u}, \left(A - \varepsilon \mathbb{I}_n\right)^{-1}\mathbf{u}\right\rangle < -1$.
Show that $\mu_m > \varepsilon$.

As in the third part, we assume that $B = A + \mathbf{u u}^T$ with $A \in \mathcal{S}_n(\mathbb{R})$ a symmetric matrix, and $\mathbf{u} \in \mathbb{R}^n$ a vector such that $\|\mathbf{u}\| = 1$. We denote by $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$ the eigenvalues of $A$ and $\mu_1 \leqslant \mu_2 \leqslant \cdots \leqslant \mu_n$ those of $B$. We admit that $$\lambda_1 \leqslant \mu_1 \leqslant \lambda_2 \leqslant \mu_2 \leqslant \cdots \leqslant \lambda_n \leqslant \mu_n.$$ We further assume that there exists an integer $m \in \{1,2,\ldots,n-1\}$ such that the eigenvalues of $A$ satisfy $$0 = \lambda_1 = \lambda_2 = \cdots = \lambda_m < \lambda_{m+1} \leqslant \cdots \leqslant \lambda_n.$$ Let $\varepsilon \in ]0, \lambda_{m+1}[$. We suppose that $\left\langle \mathbf{u}, \left(A - \varepsilon \mathbb{I}_n\right)^{-1} \mathbf{u} \right\rangle < -1$. Show that $\mu_m > \varepsilon$.

Let $I = \left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right)$ and $P = \left( \begin{array} { l l } 2 & 0 \\ 0 & 3 \end{array} \right)$. Let $Q = \left( \begin{array} { l l } x & y \\ z & 4 \end{array} \right)$ for some non-zero real numbers $x , y$, and $z$, for which there is a $2 \times 2$ matrix $R$ with all entries being non-zero real numbers, such that $Q R = R P$.
Then which of the following statements is (are) TRUE?

(A)	The determinant of $Q - 2 I$ is zero
(B)	The determinant of $Q - 6 I$ is 12
(C)	The determinant of $Q - 3 I$ is 15
(D)	$y z = 2$

The set of all values of $\lambda$ for which the system of linear equations: $2x_1 - 2x_2 + x_3 = \lambda x_1$ $2x_1 - 3x_2 + 2x_3 = \lambda x_2$ $-x_1 + 2x_2 = \lambda x_3$ has a non-trivial solution:
(1) is an empty set
(2) is a singleton
(3) contains two elements
(4) contains more than two elements

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.

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 $z _ { 1 } + z _ { 2 } + z _ { 3 } = 0$.

Answer the following questions concerning the matrix $\boldsymbol{A}$ given by
$$A = \left( \begin{array}{ccc} 0 & 3 & 0 \\ -3 & 0 & 4 \\ 0 & -4 & 0 \end{array} \right)$$
In the following, $\boldsymbol{I}$ is the $3 \times 3$ identity matrix, $\boldsymbol{O}$ is the $3 \times 3$ zero matrix, $n$ is an integer greater than or equal to 0 and $t$ is a real number.

1. Obtain all eigenvalues of the matrix $\boldsymbol{A}$.
2. Find coefficients $a, b$ and $c$ of the following equation satisfied by $\boldsymbol{A}$: $$\boldsymbol{A}^3 + a\boldsymbol{A}^2 + b\boldsymbol{A} + c\boldsymbol{I} = \boldsymbol{O}$$
3. Obtain $\boldsymbol{A}^{2n+1}$.
4. Since Equation (2) is satisfied, the following equation holds: $$\exp(t\boldsymbol{A}) = p\boldsymbol{A}^2 + q\boldsymbol{A} + r\boldsymbol{I}$$ Express coefficients $p, q$ and $r$ in terms of $t$ without using the imaginary unit.

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.

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) $\frac { 10 } { 3 }$