# Tagged: limit

## Problem 744

A sequence of events $\{E_n\}_{n \geq 1}$ is said to be increasing if it satisfies the ascending condition
$E_1 \subset E_2 \subset \cdots \subset E_n \subset \cdots.$ Also, a sequence $\{E_n\}_{n \geq 1}$ is called decreasing if it satisfies the descending condition
$E_1 \supset E_2 \supset \cdots \supset E_n \supset \cdots.$

When $\{E_n\}_{n \geq 1}$ is an increasing sequence, we define a new event denoted by $\lim_{n \to \infty} E_n$ by
$\lim_{n \to \infty} E_n := \bigcup_{n=1}^{\infty} E_n.$

Also, when $\{E_n\}_{n \geq 1}$ is a decreasing sequence, we define a new event denoted by $\lim_{n \to \infty} E_n$ by
$\lim_{n \to \infty} E_n := \bigcap_{n=1}^{\infty} E_n.$

(1) Suppose that $\{E_n\}_{n \geq 1}$ is an increasing sequence of events. Then prove the equality of probabilities
$\lim_{n \to \infty} P(E_n) = P\left(\lim_{n \to \infty} E_n \right).$ Hence, the limit and the probability are interchangeable.

(2) Suppose that $\{E_n\}_{n \geq 1}$ is a decreasing sequence of events. Then prove the equality of probabilities
$\lim_{n \to \infty} P(E_n) = P\left(\lim_{n \to \infty} E_n \right).$

## Problem 403

Let $A$ be an $n\times n$ matrix. Suppose that $A$ has real eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$ with corresponding eigenvectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$.
Furthermore, suppose that
$|\lambda_1| > |\lambda_2| \geq \cdots \geq |\lambda_n|.$ Let
$\mathbf{x}_0=c_1\mathbf{u}_1+c_2\mathbf{u}_2+\cdots+c_n\mathbf{u}_n$ for some real numbers $c_1, c_2, \dots, c_n$ and $c_1\neq 0$.

Define
$\mathbf{x}_{k+1}=A\mathbf{x}_k \text{ for } k=0, 1, 2,\dots$ and let
$\beta_k=\frac{\mathbf{x}_k\cdot \mathbf{x}_{k+1}}{\mathbf{x}_k \cdot \mathbf{x}_k}=\frac{\mathbf{x}_k^{\trans} \mathbf{x}_{k+1}}{\mathbf{x}_k^{\trans} \mathbf{x}_k}.$

Prove that
$\lim_{k\to \infty} \beta_k=\lambda_1.$

## Problem 57

Let $A$ and $B$ be $n \times n$ matrices.

Prove that the characteristic polynomials for the matrices $AB$ and $BA$ are the same.

## Problem 50

Let
$A=\begin{bmatrix} \frac{1}{7} & \frac{3}{7} & \frac{3}{7} \\ \frac{3}{7} &\frac{1}{7} &\frac{3}{7} \\ \frac{3}{7} & \frac{3}{7} & \frac{1}{7} \end{bmatrix}$ be $3 \times 3$ matrix. Find

$\lim_{n \to \infty} A^n.$

(Nagoya University Linear Algebra Exam)