# 物理代写|量子力学代写quantum mechanics代考|PHYS3040

#### Doug I. Jones

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## 物理代写|量子力学代写quantum mechanics代考|Laplace expansion and determinant of a matrix

The Laplace expansion is a method to obtain the determinant of a matrix based on cofactors minor matrix. The cofactors, $\mathbf{C}{\mathrm{ij}}$, of the square matrix $\mathbf{A}$ are $(-1)^{\mathrm{i}+\mathrm{j}}$ times the determinant of the submatrix $\mathbf{A}{\mathrm{ij}}, \mathrm{D}\left(\mathbf{A}{\mathrm{ij}}\right)$, obtained from $\mathbf{A}$ by deleting $\mathrm{i}^{\text {th }}$ rows and $j^{\text {th }}$ columns of $A$. The $D\left(\mathbf{A}{\mathrm{ij}}\right)$ is called minor $\mathrm{M}{\mathrm{ij}}$ of element $\mathrm{a}{\mathrm{ij}}$ of a determinant $\mathrm{D}$ obtained by deleting row $i$ and column $\mathrm{j}$. The cofactors of $\mathbf{A}$ form a new matrix called cofactor matrix, $\mathrm{C}$, whose elements are:
\begin{aligned} &C_{i j}=(-1)^{i+j} \cdot D\left(\mathbf{A}{i j}\right) \ &C{i j}=(-1)^{i+j} \cdot M_{i j} \end{aligned}
The determinant, $\mathrm{D}$, of a square matrix of order $\mathrm{n}$ can be obtained by the expansion along any row $\mathrm{i}$ or by the expansion of any column $\mathrm{j}$ according to Laplace expansion.
\begin{aligned} &D=a_{i 1} C_{i 1}+a_{i 2} C_{i 2}+\ldots+a_{i n} C_{i n}=\sum_{j=1}^{n} a_{i j} C_{i j} \ &D=a_{1 j} C_{1 j}+a_{2 j} C_{2 j}+\ldots+a_{i j} C_{n j}=\sum_{i=1}^{i n} a_{i j} C_{i j} \end{aligned}
Let us find the determinant of the matrix $\mathbf{A}$ below:
\begin{aligned} &\mathbf{A}=\left[\begin{array}{ccc} 1 & 2 & 3 \ -2 & 1 & 2 \ 3 & -1 & -1 \end{array}\right] \ &\operatorname{det}(\mathbf{A})=1 \times\left|\begin{array}{cc} 1 & 2 \ -1 & -1 \end{array}\right|-2 \times\left|\begin{array}{cc} -2 & 2 \ 3 & -1 \end{array}\right|+3 \times\left|\begin{array}{cc} -2 & 1 \ 3 & -1 \end{array}\right| \ &\operatorname{det}(\mathbf{A})=1 \times(-1+2)-2 \times(2-6)+3 \times(2-3)=6 \end{aligned}

## 物理代写|量子力学代写quantum mechanics代考|Antisymmetry in matrices and permutation matrix

A permutation is an operation that changes two rows or two columns of a matrix which gives the minus determinant of the former matrix. Supposing the $2 \times 2$ matrix $\mathbf{A}$ has determinant $\mathbf{D}$, if two rows or two columns of a matrix $\mathbf{A}$ are interchanged, then the determinant of the second matrix is $-\mathrm{D}$. This is the antisymmetry property of the determinants. The transformation of matrix $\mathbf{A}$ into matrix $\mathbf{B}$ occurs by the permutation matrix, $\mathbf{P}$.
$\mathbf{A}=\left[\begin{array}{ll}a_{11} & a_{12} \ a_{21} & a_{22}\end{array}\right], \quad \operatorname{det} \mathbf{A}=D$
If $\mathbf{B}=\left[\begin{array}{ll}a_{21} & a_{22} \ a_{11} & a_{12}\end{array}\right]$, then $\operatorname{det} \mathbf{B}=-D$
$\operatorname{det} \mathbf{A}=a_{11} a_{22}-a_{21} a_{12}$
$\operatorname{det} \mathbf{B}=a_{21} a_{12}-a_{11} a_{22}$
The permutation matrix is derived from an identity matrix where the unit diagonal elements of the former are reordered in the latter. The multiplication of the permutation matrix over matrix $\mathbf{A}$ gives the matrix $\mathbf{B}$ (where the rows are interchanged).
\begin{aligned} &\mathbf{I}{2}=\left[\begin{array}{ll} 1 & 0 \ 0 & 1 \end{array}\right], \quad \mathbf{P}=\left[\begin{array}{ll} 0 & 1 \ 1 & 0 \end{array}\right] \ &{\left[\begin{array}{ll} 0 & 1 \ 1 & 0 \end{array}\right] \cdot\left[\begin{array}{ll} a{11} & a_{12} \ a_{21} & a_{22} \end{array}\right]=\left[\begin{array}{ll} a_{21} & a_{22} \ a_{11} & a_{12} \end{array}\right]} \end{aligned}

# 量子力学代考

## 物理代写|量子力学代写quantum mechanics代考|Laplace expansion and determinant of a matrix

$D(\mathbf{A i j})$ 被称为次要 Mij元表的aij行列式的 D通过册除行获得 $i$ 和列j. 的辅因子 $\mathbf{A}$ 形成一个 称为辅因子矩阵的新矩阵， $\mathrm{C}$ ，其元牱为:
$$C_{i j}=(-1)^{i+j} \cdot D(\mathbf{A} i j) \quad C i j=(-1)^{i+j} \cdot M_{i j}$$

$$D=a_{i 1} C_{i 1}+a_{i 2} C_{i 2}+\ldots+a_{i n} C_{i n}=\sum_{j=1}^{n} a_{i j} C_{i j} \quad D=a_{1 j} C_{1 j}+a_{2 j} C_{2 j}+$$

$$\mathbf{A}=\left[\begin{array}{llllllll} 1 & 2 & 3 & -2 & 1 & 23 & -1 & -1 \end{array}\right] \quad \operatorname{det}(\mathbf{A})=1 \times\left|\begin{array}{llll} 1 & 2 & -1 & -1 \end{array}\right|$$

## 物理代写|量子力学代写quantum mechanics代考|Antisymmetry in matrices and permutation matrix

$\mathbf{A}=\left[\begin{array}{llll}a_{11} & a_{12} & a_{21} & a_{22}\end{array}\right], \quad \operatorname{det} \mathbf{A}=D$

$\operatorname{det} \mathbf{A}=a_{11} a_{22}-a_{21} a_{12}$
$\operatorname{det} \mathbf{B}=a_{21} a_{12}-a_{11} a_{22}$

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## 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。

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