### 数学代写|matlab代写|Block Designs

MATLAB是一个编程和数值计算平台，被数百万工程师和科学家用来分析数据、开发算法和创建模型。

statistics-lab™ 为您的留学生涯保驾护航 在代写matlab方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写matlab代写方面经验极为丰富，各种代写matlab相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|matlab代写|Hadamard Matrices

In this section, we will show how Hadamard matrices can be used to construct block designs. An $n \times n$ matrix $H$ is called a Hadamard matrix if the entries in $H$ are all 1 or $-1$, and $H H^{T}=n I$, where $I$ is the $n \times n$ identity matrix.

For an $n \times n$ Hadamard matrix $H$, since $\frac{1}{n} H^{T}=H^{-1}$, then it is also true that $H^{T} H=n I$. As a result, since $H H^{T}=H^{T} H=n I$, we can see that the dot product of any row or column of $H$ with itself will be

equal to $n$, and the dot product of any two distinct rows or columns of $H$ will be equal to 0 . Thus, changing the sign of every entry in a row or column of $H$ will yield another Hadamard matrix. Also, a Hadamard matrix $H$ is said to be normalized if both the first row and first column of $H$ contain only positive ones. Every Hadamard matrix can be converted into a normalized Hadamard matrix by changing the sign of each of the entries in necessary rows and columns. Because both the first row and first column of a normalized Hadamard matrix $H$ will contain only positive ones, each of the other rows and columns of $H$ will contain the same number of positive and negative ones. Thus, for a Hadamard matrix $H$ of order $n$ (i.e., of size $n \times n$ ), if $n>1$, then $n$ must be even. In fact, if $n>2$, then $n$ must be a multiple of 4 . To see this, note that for $H=\left(h_{i j}\right)$, the following holds.
$$\sum_{j}\left(h_{1 j}+h_{2 j}\right)\left(h_{1 j}+h_{3 j}\right)=\sum_{j} h_{1 j}^{2}=n$$
Since $\left(h_{1 j}+h_{2 j}\right)\left(h_{1 j}+h_{3 j}\right)=0$ or 4 for each $j$, the result is apparent.
The only normalized Hadamard matrices of orders one and two are $H_{1}=[1]$ and $H_{2}=\left[\begin{array}{rr}1 & 1 \ 1 & -1\end{array}\right] .$ Also, $H_{4}=\left[\begin{array}{rr}H_{2} & H_{2} \ H_{2} & -H_{2}\end{array}\right]$ is a normalized Hadamard matrix of order four. This construction of $\mathrm{H}{4}$ from $\mathrm{H}{2}$ can be generalized. Specifically, if $H$ is a normalized Hadamard matrix, then so is $\left[\begin{array}{rr}H & H \ H & -H\end{array}\right]$. This shows that there are Hadamard matrices of order $2^{n}$ for every nonnegative integer $n$.

We are interested in Hadamard matrices because they provide us with a method for constructing block designs. For a normalized Hadamard matrix $H$ of order $4 t \geq 8$, if we delete both the first row and first column from $H$, and change all of the negative ones in $H$ into zeros, the resulting matrix will be an incidence matrix for a $(4 t-1,4 t-1,2 t-1,2 t-1, t-1)$ block design. We state this as the following theorem.

## 数学代写|matlab代写|Hadamard Matrices with Maple

In this section, we will show how Maple can be used to construct the Hadamard matrices $H_{2^{n}}$ and corresponding block designs. We will consider the design resulting from the incidence matrix in Example 2.3.

Because some of the functions that we will use are in the Maple LinearAlgebra package, we will begin by including this package.
$>$ with(LinearAlgebra):
Next, we will define the Hadamard matrix $H_{1}=[1]$.
$>H 1:=\operatorname{Matrix}([[1]])$;
$$H 1:=[1]$$

Recall that the Hadamard matrix $H_{2^{k}}$ can be constructed as a block matrix from the Hadamard matrix $H_{2^{k-1}}$. Thus, we can construct the Hadamard matrices $\mathrm{H}{2}, \mathrm{H}{4}$, and $\mathrm{H}{8}$ by using the Maple Matrix function as follows. $>$ H2 := Matrix $([[\mathrm{H} 1, \mathrm{H} 1],[\mathrm{H} 1,-\mathrm{H} 1]])$; $H 2:=\left[\begin{array}{rr}1 & 1 \ 1 & -1\end{array}\right]$ $>H 4:=\operatorname{Matrix}([\mathrm{H} 2, \mathrm{H} 2],[\mathrm{H} 2,-\mathrm{H} 2]]) ;$ $H{4}:=\left[\begin{array}{rrrrr}1 & 1 & 1 & 1 \ 1 & -1 & 1 & -1 \ 1 & 1 & -1 & -1 \ 1 & -1 & -1 & 1\end{array}\right]$
$>H 8:=\operatorname{Matrix}([[\mathrm{H} 4, \mathrm{H} 4],[\mathrm{H} 4,-\mathrm{H} 4]])$;
$H 8:=\left[\begin{array}{rrrrrrrr}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \ 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \ 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 \ 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 \ 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \ 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 \ 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \ 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1\end{array}\right]$
In the preceding three Matrix commands, the Hadamard matrices are constructed by listing their rows in order surrounded by brackets and separated by commas, with the individual blocks within each row also separated by commas. The normalized Hadamard matrices $H_{2^{k}}$ for $k \geq 4$ can be constructed similarly.

## 数学代写|matlab代写|Hadamard Matrices with MATLAB

In this section, we will show how MATLAB can be used to construct the Hadamard matrices $H_{2^{n}}$ and corresponding block designs. We will consider the design resulting from the incidence matrix in Example 2.3.
We will begin by defining the Hadamard matrix $H_{1}=[1]$.
Recall that the Hadamard matrix $H_{2^{k}}$ can be constructed as a block matrix from the Hadamard matrix $H_{2^{k-1}}$. Thus, we can construct the Hadamard matrices $\mathrm{H}{2}, \mathrm{H}{4}$, and $\mathrm{H}_{8}$ as follows.

$>>\mathrm{H} 2=\left[\begin{array}{llll}\mathrm{H} 1 & \mathrm{H} 1 ; & \mathrm{H} 1 & -\mathrm{H} 1\end{array}\right]$
$\mathrm{H} 2=$
$\begin{array}{rr}1 & 1 \ 1 & -1\end{array}$
$>\mathrm{H} 4=[\mathrm{H} 2 \mathrm{H} 2 ; \mathrm{H} 2-\mathrm{H} 2]$
$\mathrm{H} 4=$
$\begin{array}{rrrr}1 & 1 & 1 & 1 \ 1 & -1 & 1 & -1 \ 1 & 1 & -1 & -1 \ 1 & -1 & -1 & 1\end{array}$
$>\mathrm{HB}=[\mathrm{H} 4 \mathrm{H} 4 ; \mathrm{H} 4-\mathrm{H} 4]$
$H 8=$
$\begin{array}{rrrrrrrr}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \ 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \ 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 \ 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 \ 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \ 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 \ 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \ 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1\end{array}$
The parameters in the preceding three commands are an ordered list of the blocks that form the resulting matrices, with each row terminated by a semicolon. The normalized Hadamard matrices $H_{2^{k}}$ for $k \geq 4$ can be constructed similarly.

## 数学代写|matlab代写|Hadamard Matrices

∑j(H1j+H2j)(H1j+H3j)=∑jH1j2=n

## 数学代写|matlab代写|Hadamard Matrices with Maple

>with(LinearAlgebra)：

>H1:=矩阵⁡([[1]]);

H1:=[1]

>H8:=矩阵⁡([[H4,H4],[H4,−H4]]);
H8:=[11111111 1−11−11−11−1 11−1−111−1−1 1−1−111−1−11 1111−1−1−1−1 1−11−1−11−11 11−1−1−1−111 1−1−11−111−1]

## 数学代写|matlab代写|Hadamard Matrices with MATLAB

>>H2=[H1H1;H1−H1]
H2=
11 1−1
>H4=[H2H2;H2−H2]
H4=
1111 1−11−1 11−1−1 1−1−11
>H乙=[H4H4;H4−H4]
H8=
11111111 1−11−11−11−1 11−1−111−1−1 1−1−111−1−11 1111−1−1−1−1 1−11−1−11−11 11−1−1−1−111 1−1−11−111−1

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。