{ "metadata": { "language_info": { "name": "python", "version": "3.8.10", "mimetype": "text/x-python", "codemirror_mode": { "name": "ipython", "version": 3 }, "pygments_lexer": "ipython3", "nbconvert_exporter": "python", "file_extension": ".py" }, "orig_nbformat": 4, "kernelspec": { "name": "python3", "display_name": "Python 3.8.10 64-bit ('penter': conda)" }, "interpreter": { "hash": "66a9b9246fc3be4ca617a120a8795ea86c1f91adc8c4462264e6f2ff62103632" } }, "nbformat": 4, "nbformat_minor": 2, "cells": [ { "cell_type": "code", "execution_count": 1, "source": [ "%matplotlib inline\r\n", "# 如果对带有一个变量或是未赋值语句的cell执行操作，Jupyter将会自动打印该变量而无需一个输出语句。\r\n", "from IPython.core.interactiveshell import InteractiveShell\r\n", "InteractiveShell.ast_node_interactivity = \"all\" #默认为'last'\r\n", "# display(Math(latex_s))和display(Latex(latex_s))输出的是latex类型， \r\n", "# display(Markdown(latex_s))输出的是markdown\r\n", "# 推荐markdown和Latex；而Math只支持纯latex\r\n", "from IPython.display import display, Latex, Math, Markdown\r\n", "import numpy as np\r\n", "import matplotlib.pyplot as plt\r\n", "from matplotlib.patches import Polygon\r\n", "from matplotlib.collections import PatchCollection\r\n", "import seaborn as sns\r\n", "plt.rcParams['text.usetex'] = True\r\n", "plt.rcParams['text.latex.preamble'] = r'\\usepackage{amsmath} \\usepackage{amsfonts}'\r\n", "# plt.rcParams.update({\r\n", "# #'font.size': 8,\r\n", "# #'text.usetex': True,\r\n", "# 'text.latex.preamble': [r'\\usepackage{amsmath}', #for \\text command\r\n", "# r'\\usepackage{amsfonts}', # \\mathbb is provided by the LaTeX package amsfonts\r\n", "# ]\r\n", "# })\r\n", "# 参数'text.latex.preamble' or 'pgf.preamble'（这两个不是同一个参数） 以前是用数组，现在用字符串\r\n", "# 罗马体 operatorname -> mbox or mathrm\r\n", "\r\n", "# https://matplotlib.org/stable/tutorials/text/mathtext.html\r\n", "\r\n", "# matplotlib.rcParams['text.latex.unicode'] = True\r\n", "# matplotlib.rcParams['text.latex.preamble'] = [\r\n", "# '\\\\usepackage{CJK}',\r\n", "# r'\\AtBeginDocument{\\begin{CJK}{UTF8}{gbsn}}',\r\n", "# r'\\AtEndDocument{\\end{CJK}}',\r\n", "# ]\r\n", "\r\n", "# Matplotlib中文显示和Latex\r\n", "#import matplotlib.font_manager as mf # 导入字体管理器\r\n", "#my_font= mf.FontProperties(fname='C://Windows//Fonts/simsun.ttc') # 加载字体" ], "outputs": [], "metadata": {} }, { "cell_type": "code", "execution_count": 2, "source": [ "\r\n", "\r\n", "def show_latex(eqn:int, latex_s:str , validated:bool):\r\n", " fig, ax = plt.subplots(figsize=(20, 0.7))\r\n", " #latex_s = r\"$\\alpha _ { 1 } ^ { r } \\gamma _ { 1 } + \\dots + \\alpha _ { N } ^ { r } \\gamma _ { N } = 0 \\quad ( r = 1 , . . . , R ) ,$\"\r\n", " plt.text(0.01, 0.5, \"({})\".format(eqn), ha='left', va='center', fontsize=20)\r\n", " #水平和垂直方向居中对齐\r\n", " plt.text(0.5, 0.5, latex_s, ha='center', va='center', fontsize=20)\r\n", " if validated:\r\n", " plt.text(0.97, 0.5, r\"ok\", ha='right', va='center', fontsize=20, color = \"g\")\r\n", " else:\r\n", " plt.text(0.97, 0.5, r\"error\", ha='right', va='center', fontsize=20, color = \"r\")\r\n", " # 隐藏框线\r\n", " ax.spines['top'].set_visible(False)\r\n", " ax.spines['right'].set_visible(False)\r\n", " ax.spines['bottom'].set_visible(False)\r\n", " ax.spines['left'].set_visible(False)\r\n", " # 隐藏坐标轴的刻度信息\r\n", " plt.xticks([])\r\n", " plt.yticks([])\r\n", "\r\n", "def is_numpy(value):\r\n", " return hasattr(value, 'dtype')\r\n", "\r\n", "def validate(left, right, eqn, latex_s = None): \r\n", " # np.allclose(1, 1) 判断两个向量是否相近https://numpy.org/doc/stable/reference/generated/numpy.allclose.html#numpy.allclose \r\n", " validated = np.allclose(left, right) # (left == right).all()\r\n", " #print(\"{} \\n \".format((left == right)))\r\n", " #print(\"方程({})的结果: {}\".format(eqn, validated))\r\n", " if not validated:\r\n", " print(\"({}) {} = {}\".format(eqn, left, right))\r\n", " if latex_s:\r\n", " display(Markdown(latex_s)) \r\n", " show_latex(eqn, latex_s, validated)\r\n" ], "outputs": [], "metadata": {} }, { "cell_type": "markdown", "source": [ "# The Matrix Cookbook\r\n", "[原文](https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf)\r\n", "## 符号和命名\r\n", "$\\mathbf{A}\\rightarrow\\;$ Matrix\r\n", "\r\n", "$\\mathbf{A}_{ij}\\rightarrow\\;$ Matrix indexed for some purpose\r\n", "\r\n", "$\\mathbf{A}_{i}\\rightarrow\\;$ Matrix indexed for some purpose\r\n", "\r\n", "$\\mathbf{A}^{ij}\\rightarrow\\;$ Matrix indexed for some purpose\r\n", "\r\n", "$\\mathbf{A}^{n}\\rightarrow\\;$ Matrix indexed for some purpose or the the $n^{th}$ power of a square matrix\r\n", "\r\n", "$\\mathbf{A}^{-1}\\rightarrow\\;$ The inverse matrix of matrix $\\mathbf{A}$\r\n", "\r\n", "$\\mathbf{A}^{\\dagger}\\rightarrow\\;$ The pseudo inverse matrix of the matrix $\\mathbf{A}$\r\n", "\r\n", "$\\mathbf{A}^{1/2}\\rightarrow\\;$ The square root of a matrix (if unique), not elementwise\r\n", "\r\n", "$(\\mathbf{A})_{i,j}\\rightarrow\\;$ The $(i,j)^{th}$ entry of the matrix $\\mathbf{A}$\r\n", "\r\n", "$\\mathbf{A}_{i,j}\\rightarrow\\;$ The $(i,j)^{th}$ entry of the matrix $\\mathbf{A}$\r\n", "\r\n", "$[\\mathbf{A}]_{i,j}\\rightarrow\\;$ The $ij$-suubmatrix, i.e. $\\mathbf{A}$ with $i^{th}$ row and $j^{th}$ column deleted\r\n", "\r\n", "$\\vec{a}\\rightarrow\\;$ Vector (column-vector)\r\n", "\r\n", "$\\vec{a}_{i}\\rightarrow\\;$ Vector indexed for some purpose\r\n", "\r\n", "$\\alpha_{i}\\rightarrow\\;$ The $i^{th}$ element of vector $\\vec{a}$\r\n", "\r\n", "$\\alpha\\rightarrow\\;$ Scalar\r\n", "\r\n", "---\r\n", "\r\n", "$\\Re z\\rightarrow\\;$ Real part of a scalar\r\n", "\r\n", "$\\Re \\mathbf{z}\\rightarrow\\;$ Real part of a vectorS\r\n", "\r\n", "$\\Re \\mathbf{Z}\\rightarrow\\;$ Real part of a matrix\r\n", "\r\n", "$\\Im z\\rightarrow\\;$ Imaginary part of a scalar\r\n", "\r\n", "$\\Im \\mathbf{z}\\rightarrow\\;$ Imaginary part of a vectorS\r\n", "\r\n", "$\\Im \\mathbf{Z}\\rightarrow\\;$ Imaginary part of a matrix\r\n", "\r\n", "---\r\n", "\r\n", "$det(\\mathbf{A})\\rightarrow\\;$ Determinant of $\\mathbf{A}$\r\n", "\r\n", "$Tr(\\mathbf{A})\\rightarrow\\;$ Trace of the matrix $\\mathbf{A}$\r\n", "\r\n", "$diag(\\mathbf{A})\\rightarrow\\;$ Diagonal matrix of the matrix $\\mathbf{A}$, i.e. $(diag(\\mathbf{A}))_{ij} = \\delta_{ij}\\mathbf{A}_{ij}$\r\n", "\r\n", "$eig(\\mathbf{A})\\rightarrow\\;$ Eigenvalues of the matrix $\\mathbf{A}$\r\n", "\r\n", "$vec(\\mathbf{A})\\rightarrow\\;$ The vector-version of the matrix $\\mathbf{A}$\r\n", "\r\n", "$sup\\rightarrow\\;$ Supremum of a set\r\n", "\r\n", "$\\Vert\\mathbf{A}\\Vert\\rightarrow\\;$ Matrix norm (subscript if any denotes what norm)\r\n", "\r\n", "$\\mathbf{A}^{T}\\rightarrow\\;$ Transposed matrix\r\n", "\r\n", "$\\mathbf{A}^{-T}\\rightarrow\\;$ The inverse of the transposed and vice versa, $\\mathbf{A}^{-T} = (\\mathbf{A}^{-1})^{T} = (\\mathbf{A}^{T})^{-1}$\r\n", "\r\n", "$\\mathbf{A}^{\\star}\\rightarrow\\;$ Complex conjugated matrix\r\n", "\r\n", "$\\mathbf{A}^{H}\\rightarrow\\;$ Transposed and complex conjugated matrix (Hermitian)\r\n", "\r\n", "---\r\n", "\r\n", "$\\mathbf{A}\\circ\\mathbf{B}$ Hadamard (elementwise) product\r\n", "\r\n", "$\\mathbf{A}\\otimes\\mathbf{B}$ Kronecker product\r\n", "\r\n", "---\r\n", "\r\n", "$\\mathbf{0}\\rightarrow\\;$ The null matrix. Zero in all entries.\r\n", "\r\n", "$\\mathbf{I}\\rightarrow\\;$ The idnentity matrix\r\n", "\r\n", "$\\mathbf{J}^{ij}\\rightarrow\\;$ The single entry matrix, 1 at (i,j) and zero elsewhere\r\n", "\r\n", "$\\mathbf{\\Sigma}\\rightarrow\\;$ A posisitive definite matrix\r\n", "\r\n", "$\\mathbf{\\Lambda}\\rightarrow\\;$ A diagonal matrix" ], "metadata": {} }, { "cell_type": "markdown", "source": [ "## 1 Basics\r\n", "$$\\begin{align} (\\mathbf{A}\\mathbf{B})^{-1} &= \\mathbf{B}^{-1}\\mathbf{A}^{-1} \\\\ (\\mathbf{ABC\\ldots})^{-1} &= \\ldots\\mathbf{C}^{-1}\\mathbf{B}^{-1}\\mathbf{A}^{-1} \\\\ (\\mathbf{A}^{T})^{-1} &= (\\mathbf{A}^{-1})^{T} \\\\ (\\mathbf{A} + \\mathbf{B})^{T} &= \\mathbf{A}^{T} + \\mathbf{B}^{T} \\\\ (\\mathbf{AB})^{T} &= \\mathbf{B}^{T}\\mathbf{A}^{T} \\\\ (\\mathbf{ABC\\ldots})^{T} &= \\ldots\\mathbf{C}^{T}\\mathbf{B}^{T}\\mathbf{A}^{T} \\\\ (\\mathbf{A}^{H})^{-1} &= (\\mathbf{A}^{-1})^{H} \\\\ (\\mathbf{A} + \\mathbf{B})^{H} &= \\mathbf{B}^{H} + \\mathbf{A}^{H} \\\\ (\\mathbf{AB})^{H} &= \\mathbf{B}^{H}\\mathbf{A}^{H} \\\\ (\\mathbf{ABC\\ldots})^{H} &= \\ldots\\mathbf{C}^{H}\\mathbf{B}^{H}\\mathbf{A}^{H} \\\\ \\end{align}$$" ], "metadata": {} }, { "cell_type": "markdown", "source": [ "### 1 Basics Proofs\r\n", "- equation (1)):$(\\mathbf{A}\\mathbf{B})^{-1} = \\mathbf{B}^{-1}\\mathbf{A}^{-1}$\r\n", "Multiply both side with $AB \\Rightarrow (\\mathbf{A}\\mathbf{B})^{-1}\\mathbf{AB} = \\mathbf{B}^{-1}\\mathbf{A}^{-1}\\mathbf{AB}$\r\n", "Now, $LHS = (\\mathbf{A}\\mathbf{B})^{-1}\\mathbf{AB}=\\mathbf{I} \\text{ and } RHS = \\mathbf{B}^{-1}\\mathbf{A}^{-1}\\mathbf{AB}=\\mathbf{B}^{-1}\\mathbf{I}\\mathbf{B} =\\mathbf{I} = LHS$\r\n", "\r\n", "---\r\n", "\r\n", "- equation (2):$(\\mathbf{ABC\\ldots})^{-1} = \\ldots\\mathbf{C}^{-1}\\mathbf{B}^{-1}\\mathbf{A}^{-1}$\r\n", "This is generalized case of equation (1) and can be proved similarly.\r\n", "> generalized 广义\r\n", "\r\n", "---\r\n", "\r\n", "- equation (3):$(\\mathbf{A}^{T})^{-1} = (\\mathbf{A}^{-1})^{T}$\r\n", "$$\\begin{aligned} RHS &= (\\mathbf{A}^{-1})^{T} \\\\&= (\\mathbf{A}^{-1})^{T}\\mathbf{A}^T(\\mathbf{A}^T)^{-1} \\quad \\because \\mathbf{A}^T(\\mathbf{A}^T)^{-1}=\\mathbf{I} \\\\&=(\\mathbf{AA^{-1}})^T(\\mathbf{A^T})^{-1} \\quad \\because \\mathbf{B^T}\\mathbf{A^T}=(\\mathbf{AB})^T \\quad \\text{See proof of equation (5) for this} \\\\&=\\mathbf{I^T(A^T)^{-1}}\\\\&=\\mathbf{(A^T)^{-1}}\\\\&=LHS\\end{aligned}$$\r\n", "\r\n", "---\r\n", "\r\n", "- equation (4):$(\\mathbf{A} + \\mathbf{B})^{T} = \\mathbf{A}^{T} + \\mathbf{B}^{T}$\r\n", "The $(i,j)^{th}$ element of $(\\mathbf A^T+\\mathbf B^T)$ is the sum of $(i,j)^{th}$ elements of $\\mathbf A^T$ and $\\mathbf B^T$, which are $(j,i)^{th}$ element of $\\mathbf A$ and $\\mathbf B$, respectively. Thus the $(i,j)^{th}$ element of $\\mathbf A^T+\\mathbf B^T$ is the $(j,i)^{th}$ element of the sum of $\\mathbf A$ and $\\mathbf B$, which is equal to the $(i,j)^{th}$ element of the transpose $(\\mathbf {A+B})^T$.\r\n", "\r\n", "---\r\n", "\r\n", "- equation (5):$(\\mathbf{AB})^{T} = \\mathbf{B}^{T}\\mathbf{A}^{T}$\r\n", "$$(ab)^T_{ki} = (ab)_{ik} = \\sum_{j=1}^na_{ij}b_{jk}$$\r\n", "$$(a^Tb^T)_{ki} = \\sum_{j=1}^nb^T_{kj}a^T_{ji} = \\sum_{j=1}^nb_{jk}a_{ij} =\\sum_{j=1}^na_{ij}b_{jk} = (ab)^T_{ki}$$\r\n", "\r\n", "---\r\n", "\r\n", "- equation (6):$(\\mathbf{ABC\\ldots})^{T} = \\ldots\\mathbf{C}^{T}\\mathbf{B}^{T}\\mathbf{A}^{T}$\r\n", "This is generalized form of above equation (5). To extend it to more than two matrices, use induction:\r\n", "Suppose that for some $n$, we have\r\n", "$$\\begin{aligned}\\mathbf{(A_1A_2\\cdots A_n)^T} = \\mathbf{A^T_n \\cdots A^T_2A^T_1} \\tag{proof 1-6-1}\\end{aligned}$$\r\n", "Note that we have already derived (5) for $n=2$.\r\n", "Then, using the two matrix result and (proof 1-6-1), we have\r\n", "$$\\begin{aligned}\\mathbf{(A_1A_2\\cdots A_nA_{n+1})^T} &= \\mathbf{((A_1A_2\\cdots A_n)A_{n+1})^T}\\\\&=\\mathbf{A_{n+1}^T(A_1A_2\\cdots A_n)^T}\\\\&=\\mathbf{A_{n+1}^TA_{n}^T\\cdots A_{2}^TA_{1}^T}\\end{aligned}$$\r\n", "\r\n", "---\r\n", "\r\n", "- equation (7):$(\\mathbf{A}^{H})^{-1} = (\\mathbf{A}^{-1})^{H}$\r\n", "$$\\begin{aligned}RHS &= \\mathbf{(A^{-1})^H} \\\\&= \\mathbf{(A^{-1})^H A^H (A^H)^{-1}} \\\\&= \\mathbf{(AA^{-1})^H (A^T)^{-1}} \\quad \\because \\mathbf{B^H A^H = (AB)^H} \\text{ See proof of equation (9) fro this} \\\\&= \\mathbf{I^H (A^H)^{-1}} \\\\&= \\mathbf{(A^H)^{-1}} \\\\&= LHS\\end{aligned}$$\r\n", "---\r\n", "\r\n", "- equation (8):$(\\mathbf{A} + \\mathbf{B})^{H} = \\mathbf{B}^{H} + \\mathbf{A}^{H}$\r\n", "\r\n", "---\r\n", "\r\n", "- equation (9):$(\\mathbf{AB})^{H} = \\mathbf{B}^{H}\\mathbf{A}^{H}$\r\n", "\r\n", "---\r\n", "\r\n", "- equation (10):$(\\mathbf{ABC\\ldots})^{H} = \\ldots\\mathbf{C}^{H}\\mathbf{B}^{H}\\mathbf{A}^{H}$\r\n" ], "metadata": {} }, { "cell_type": "markdown", "source": [ "> Left Hand Side (LHS) 左手边;Right Hand Side (RHS) 右手边" ], "metadata": {} }, { "cell_type": "markdown", "source": [ "### 1 Basics Validate" ], "metadata": {} }, { "cell_type": "code", "execution_count": 3, "source": [ "# lets play with the above equations and equatities to see for ouselves\r\n", "# caution the above probably work only for square matrices\r\n", "width = 3\r\n", "height = 3\r\n", "A = np.random.random([width, height])\r\n", "B = np.random.random([width, height])\r\n", "C = np.random.random([width, height])\r\n", "\r\n", "I = np.eye(width)\r\n", "\r\n", "\r\n", "A2 = np.random.random([2, 2])\r\n", "B2 = np.random.random([2, 2])\r\n", "C2 = np.random.random([2, 2])\r\n", "I2 = np.eye(2)\r\n", "\r\n", "A4 = np.random.random([4, 4])\r\n", "B4 = np.random.random([4, 4])\r\n", "C4 = np.random.random([4, 4])\r\n", "I4 = np.eye(4)\r\n", "\r\n", "a = np.random.random([width, 1])\r\n", "\r\n", "u = np.random.random([width, 1])\r\n", "v = np.random.random([width, 1])\r\n" ], "outputs": [], "metadata": {} }, { "cell_type": "code", "execution_count": 4, "source": [ "\r\n", "# equation 1:\r\n", "lft_hand = np.linalg.inv(A.dot(B))\r\n", "rgt_hand = np.dot(np.linalg.inv(B), np.linalg.inv(A))\r\n", "validate(lft_hand, rgt_hand, 1, r\"$(\\mathbf{A}\\mathbf{B})^{-1} = \\mathbf{B}^{-1}\\mathbf{A}^{-1}$\")\r\n", "\r\n", "# equation 2:\r\n", "lft_hand = np.linalg.inv(A.dot(B).dot(C))\r\n", "rgt_hand = np.dot(np.linalg.inv(C), np.linalg.inv(B)).dot(np.linalg.inv(A))\r\n", "validate(lft_hand, rgt_hand, 2, r\"$(\\mathbf{ABC\\ldots})^{-1} = \\ldots\\mathbf{C}^{-1}\\mathbf{B}^{-1}\\mathbf{A}^{-1}$\")\r\n", "\r\n", "# equation 3:\r\n", "lft_hand = np.linalg.inv(A.T)\r\n", "rgt_hand = np.transpose(np.linalg.inv(A))\r\n", "validate(lft_hand, rgt_hand, 3, r\"$(\\mathbf{A}^{T})^{-1} = (\\mathbf{A}^{-1})^{T}$\")\r\n", "\r\n", "# equation 4:\r\n", "lft_hand = (A+B).T\r\n", "rgt_hand = A.T+B.T\r\n", "validate(lft_hand, rgt_hand, 4, r\"$(\\mathbf{A} + \\mathbf{B})^{T} = \\mathbf{A}^{T} + \\mathbf{B}^{T}$\")\r\n", "\r\n", "# equation 5:\r\n", "lft_hand = A.dot(B).T\r\n", "rgt_hand = B.T.dot(A.T)\r\n", "validate(lft_hand, rgt_hand, 5, r\"$(\\mathbf{AB})^{T} = \\mathbf{B}^{T}\\mathbf{A}^{T}$\")\r\n", "\r\n", "# equation 7:\r\n", "lft_hand = np.linalg.inv(np.conjugate(A))\r\n", "rgt_hand = np.conjugate(np.linalg.inv(A))\r\n", "validate(lft_hand, rgt_hand, 7, r\"$(\\mathbf{A}^{H})^{-1} = (\\mathbf{A}^{-1})^{H}$\")\r\n", "\r\n", "# equation 8:\r\n", "lft_hand = np.conjugate(A+B)\r\n", "rgt_hand = np.conjugate(A) + np.conjugate(B)\r\n", "validate(lft_hand, rgt_hand, 8, r\"$(\\mathbf{A} + \\mathbf{B})^{H} = \\mathbf{B}^{H} + \\mathbf{A}^{H}$\")\r\n", "\r\n", "# equation 9:\r\n", "lft_hand = np.conjugate(A*B)\r\n", "rgt_hand = np.conjugate(B)*np.conjugate(A)\r\n", "validate(lft_hand, rgt_hand, 9, r\"$(\\mathbf{A \\star B})^{H} = \\mathbf{B}^{H} \\star \\mathbf{A}^{H} $\")" ], "outputs": [ { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$(\\mathbf{A}\\mathbf{B})^{-1} = \\mathbf{B}^{-1}\\mathbf{A}^{-1}$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$(\\mathbf{ABC\\ldots})^{-1} = \\ldots\\mathbf{C}^{-1}\\mathbf{B}^{-1}\\mathbf{A}^{-1}$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$(\\mathbf{A}^{T})^{-1} = (\\mathbf{A}^{-1})^{T}$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$(\\mathbf{A} + \\mathbf{B})^{T} = \\mathbf{A}^{T} + \\mathbf{B}^{T}$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$(\\mathbf{AB})^{T} = \\mathbf{B}^{T}\\mathbf{A}^{T}$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$(\\mathbf{A}^{H})^{-1} = (\\mathbf{A}^{-1})^{H}$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$(\\mathbf{A} + \\mathbf{B})^{H} = \\mathbf{B}^{H} + \\mathbf{A}^{H}$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$(\\mathbf{A \\star B})^{H} = \\mathbf{B}^{H} \\star \\mathbf{A}^{H} $" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "
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}, "metadata": {} } ], "metadata": {} }, { "cell_type": "markdown", "source": [ "### 1.1 Trace\r\n", "$$\\begin{align} Tr(\\mathbf{A}) &= \\sum_{i} \\mathbf{A}_{ii} \\\\ Tr(\\mathbf{A}) &= \\sum_{i}\\lambda_{i}, \\;\\;\\lambda_{i} = eig(\\mathbf{A}) \\\\ Tr(\\mathbf{A}) &= Tr(\\mathbf{A}^{T}) \\\\ Tr(\\mathbf{AB}) &= Tr(\\mathbf{BA}) \\\\ Tr(\\mathbf{A}+\\mathbf{B}) &= Tr(\\mathbf{A}) + Tr(\\mathbf{B}) \\\\ Tr(\\mathbf{ABC}) &= Tr(\\mathbf{BCA}) = Tr(\\mathbf{CAB}) \\\\ \\vec{a}^{T}\\vec{a} &= Tr(\\vec{a}\\vec{a}^{T}) \\end{align}$$" ], "metadata": {} }, { "cell_type": "markdown", "source": [ "#### 1.1 Trace Proofs\r\n", "\r\n", "- equation (11):$Tr(\\mathbf{A}) = \\sum_{i} \\mathbf{A}_{ii}$\r\n", "By definition of trace of the matrix, which is the sum of elements on the main diagonal of $\\mathbf{A}$.\r\n", "\r\n", "---\r\n", "\r\n", "- equation (12):$Tr(\\mathbf{A}) = \\sum_{i}\\lambda_{i}, \\;\\;\\lambda_{i} = eig(\\mathbf{A})$\r\n", "By definition, the characteristic polynomial of an $n \\times n$ matrix $\\mathbf{A}$ is given by\r\n", "$$p(t) = \\operatorname{det}(\\mathbf{A-tI}) = (-1)^n \\Big( t^n - (\\operatorname{tr}\\mathbf A)t^{n-1} + \\cdots + (-1)^n \\operatorname{det}\\mathbf{A} \\Big)$$\r\n", "On the other hand, $p(t)=(−1)^n(t−\\lambda_1)\\cdots(t−\\lambda_n)$, where the $\\lambda_j$ are the eigenvalues of $\\mathbf A$. So, comparing coefficients, we have $\\operatorname{tr}\\mathbf A=\\lambda_1+\\cdots+\\lambda_n$.\r\n", "\r\n", "---\r\n", "\r\n", "- equation (13):$Tr(\\mathbf{A}) = Tr(\\mathbf{A}^{T})$\r\n", "Elements of any matrix $\\mathbf{A}$ can be represented by $\\mathbf{A}_{ij}$. After transpose, the corresponding becomes $\\mathbf{A}_{ji}$. But for diagonal elements, $i=j$, therefore, from definition of trace of matrix $\\operatorname{Tr}(\\mathbf A)=\\operatorname{Tr}(\\mathbf A^T)= \\sum_i \\mathbf A_{ii}$\r\n", "\r\n", "---\r\n", "\r\n", "- equation (14):$Tr(\\mathbf{AB}) = Tr(\\mathbf{BA})$\r\n", "Let $\\mathbf{A}$ be a $n \\times m$ and \\mathbf{B} be a $m \\times n$ matrix, we have\r\n", "$$\\begin{aligned}\\operatorname{Tr}(\\mathbf {AB}) &= \\sum_{i=1}^n(\\mathbf{AB})_{ii} \\\\&=\\sum_{i=1}^n\\sum_{j=1}^m \\mathbf {A}_{ij} \\mathbf {B}_{ji} \\\\&= \\sum_{j=1}^m\\sum_{i=1}^n \\mathbf {B}_{ji} \\mathbf {A}_{ij} \\\\&= \\sum_{j=1}^m(\\mathbf{BA})_{jj} \\\\&= \\operatorname{Tr}(\\mathbf{BA})\\end{aligned}$$\r\n", "\r\n", "---\r\n", "\r\n", "- equation (15):$Tr(\\mathbf{A}+\\mathbf{B}) = Tr(\\mathbf{A}) + Tr(\\mathbf{B})$\r\n", "$$\\begin{aligned}RHS &= \\operatorname{Tr}(\\mathbf A) + \\operatorname{Tr}(\\mathbf B) \\\\&=\\sum_{k=1}^na_{kk} + \\sum_{k=1}^nb_{kk} \\\\&=\\sum_{k=1}^n(a_{kk} + b_{kk}) \\\\&= \\operatorname{Tr}(\\mathbf{A}+\\mathbf{B}) \\\\&=LHS \\end{aligned}$$\r\n", "\r\n", "---\r\n", "\r\n", "- equation (16):$Tr(\\mathbf{ABC}) = Tr(\\mathbf{BCA}) = Tr(\\mathbf{CAB})$\r\n", "More general form of (proof 14).\r\n", "\r\n", "---\r\n", "\r\n", "- equation (17):$\\mathbf{a^Ta} = Tr(\\mathbf{aa^T})$" ], "metadata": {} }, { "cell_type": "markdown", "source": [ "#### 1.1 Trace Validate" ], "metadata": {} }, { "cell_type": "code", "execution_count": 5, "source": [ "\r\n", "# equation 11:\r\n", "lft_hand = np.trace(A)\r\n", "rgt_hand = np.sum(np.diag(A))\r\n", "validate(lft_hand, rgt_hand, 11, r\"$Tr(\\mathbf{A}) = \\sum_{i} \\mathbf{A}_{ii}$\")\r\n", "\r\n", "# equation 12:\r\n", "lft_hand = np.trace(A)\r\n", "rgt_hand = np.sum(np.linalg.eigvals(A))\r\n", "validate(lft_hand, rgt_hand, 12, r\"$Tr(\\mathbf{A}) = \\sum_{i}\\lambda_{i}, \\;\\;\\lambda_{i} = eig(\\mathbf{A})$\")\r\n", "\r\n", "# equation 13:\r\n", "lft_hand = np.trace(A)\r\n", "rgt_hand = np.trace(A.T)\r\n", "validate(lft_hand, rgt_hand, 13, r\"$Tr(\\mathbf{A}) = Tr(\\mathbf{A}^{T})$\")\r\n", "\r\n", "# equation 14:\r\n", "lft_hand = np.trace(A.dot(B))\r\n", "rgt_hand = np.trace(B.dot(A))\r\n", "validate(lft_hand, rgt_hand, 14, r\"$Tr(\\mathbf{AB}) = Tr(\\mathbf{BA})$\")\r\n", "\r\n", "# equation 15:\r\n", "lft_hand = np.trace(A+B)\r\n", "rgt_hand = np.trace(A) + np.trace(B)\r\n", "validate(lft_hand, rgt_hand, 15, r\"$Tr(\\mathbf{A}+\\mathbf{B}) = Tr(\\mathbf{A}) + Tr(\\mathbf{B})$\")\r\n", "\r\n", "# equation 16:\r\n", "lft_hand = np.trace(np.dot(A.dot(B), C))\r\n", "rgt_hand = np.trace(np.dot(B.dot(C), A))\r\n", "validate(lft_hand, rgt_hand, 16, r\"$Tr(\\mathbf{ABC}) = Tr(\\mathbf{BCA}) = Tr(\\mathbf{CAB})$\")\r\n", "\r\n", "# equation 17:\r\n", "lft_hand = np.dot(a.T, a)\r\n", "rgt_hand = np.trace(np.dot(a, a.T))\r\n", "validate(lft_hand, rgt_hand, 17, r\"$\\vec{a}^{T}\\vec{a} = Tr(\\vec{a}\\vec{a}^{T})$\")" ], "outputs": [ { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$Tr(\\mathbf{A}) = \\sum_{i} \\mathbf{A}_{ii}$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$Tr(\\mathbf{A}) = \\sum_{i}\\lambda_{i}, \\;\\;\\lambda_{i} = eig(\\mathbf{A})$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$Tr(\\mathbf{A}) = Tr(\\mathbf{A}^{T})$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$Tr(\\mathbf{AB}) = Tr(\\mathbf{BA})$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$Tr(\\mathbf{A}+\\mathbf{B}) = Tr(\\mathbf{A}) + Tr(\\mathbf{B})$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$Tr(\\mathbf{ABC}) = Tr(\\mathbf{BCA}) = Tr(\\mathbf{CAB})$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$\\vec{a}^{T}\\vec{a} = Tr(\\vec{a}\\vec{a}^{T})$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "
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}, "metadata": {} } ], "metadata": {} }, { "cell_type": "markdown", "source": [ "### 1.2 Determinant\r\n", "Let $A$ be an $n \\times n$ matrix.\r\n", "$$\\begin{align} \\mathrm{det}(\\mathbf{A}) &= \\prod_i{\\lambda_i} \\quad \\lambda_i=\\mathrm{eig}(\\mathbf{A})\\\\ \\mathrm{det}(c\\mathbf{A}) &=c^n \\mathrm{det}(\\mathbf{A}), \\quad \\text{if } \\mathbf{A} \\in \\mathbb{R}^{n \\times n} \\\\ \\mathrm{det}(\\mathbf{A}^T) &=\\mathrm{det}(\\mathbf{A}) \\\\ \\mathrm{det}(\\mathbf{AB}) &=\\mathrm{det}(\\mathbf{A})\\mathrm{det}(\\mathbf{B}) \\\\ \\mathrm{det}(\\mathbf{A}^{-1}) &= 1/\\mathrm{det}(\\mathbf{A}) \\\\ \\mathrm{det}(\\mathbf{A}^{n}) &= \\mathrm{det}(\\mathbf{A})^n \\\\ \\mathrm{det}(\\mathbf{I+uv^T}) &= 1+\\mathbf{u^Tv} \\\\ \\end{align}$$\r\n", "\r\n", "For $n=2$:\r\n", "\r\n", "$$\\begin{align} \\mathrm{det}(\\mathbf{I+A}) = 1+\\mathrm{det}(\\mathbf{A})+\\mathrm{Tr}(\\mathbf{A}) \\end{align}$$\r\n", "\r\n", "For $n=3$:\r\n", "\r\n", "$$\\begin{align} \\mathrm{det}(\\mathbf{I+A}) = 1+\\mathrm{det}(\\mathbf{A})+\\mathrm{Tr}(\\mathbf{A})+\\frac{1}{2}\\mathrm{Tr}(\\mathbf{A})^2-\\frac{1}{2}\\mathrm{Tr}(\\mathbf{A}^2) \\end{align}$$\r\n", "\r\n", "For $n=4$:\r\n", "\r\n", "$$\\begin{equation}\\begin{aligned} \\mathrm{det}(\\mathbf{I+A}) =\\; &1+\\mathrm{det}(\\mathbf{A})+\\mathrm{Tr}(\\mathbf{A})\\\\ &+\\frac{1}{2}\\mathrm{Tr}(\\mathbf{A})^2-\\frac{1}{2}\\mathrm{Tr}(\\mathbf{A}^2)\\\\ &+\\frac{1}{6}\\mathrm{Tr}(\\mathbf{A})^3-\\frac{1}{2}\\mathrm{Tr}(\\mathbf{A})\\mathrm{Tr}(\\mathbf{A}^2)+\\frac{1}{3}\\mathrm{Tr}(\\mathbf{A}^3) \\\\ \\end{aligned}\\end{equation}$$\r\n", "\r\n", "For small $\\varepsilon$, the following approximation holds\r\n", "对于小$\\varepsilon$，以下近似成立\r\n", "\r\n", "$$\\begin{align} \\mathrm{det}(\\mathbf{I+\\varepsilon A}) \\cong 1+\\mathrm{det}(\\mathbf{A})+\\varepsilon\\mathrm{Tr}(\\mathbf{A})+\\frac{1}{2}\\varepsilon^2\\mathrm{Tr}(\\mathbf{A})^2-\\frac{1}{2}\\varepsilon^2\\mathrm{Tr}(\\mathbf{A}^2) \\end{align}$$\r\n" ], "metadata": {} }, { "cell_type": "markdown", "source": [ "#### 1.2 Determinant Validate" ], "metadata": {} }, { "cell_type": "code", "execution_count": 6, "source": [ "\r\n", "# equation 18:\r\n", "lft_hand = np.linalg.det(A)\r\n", "rgt_hand = np.prod(np.linalg.eigvals(A))\r\n", "validate(lft_hand, rgt_hand, 18, r\"$\\mathrm{det}(\\mathbf{A}) = \\prod_i{\\lambda_i} \\quad \\lambda_i=\\mathrm{eig}(\\mathbf{A})$\")\r\n", "\r\n", "# equation 19:\r\n", "c = 1.23\r\n", "lft_hand = np.linalg.det(c*A)\r\n", "rgt_hand =np.power(c,A.shape[0])*np.linalg.det(A)\r\n", "validate(lft_hand, rgt_hand, 19, r\"$\\mathrm{det}(c\\mathbf{A}) = c^n \\mathrm{det}(\\mathbf{A}), \\quad \\text{if } \\mathbf{A} \\in \\mathbb{R}^{n \\times n}$\")\r\n", "\r\n", "# equation 20:\r\n", "lft_hand = np.linalg.det(A.T)\r\n", "rgt_hand = np.linalg.det(A)\r\n", "validate(lft_hand, rgt_hand, 20, r\"$\\mathrm{det}(\\mathbf{A}^T) = \\mathrm{det}(\\mathbf{A})$\")\r\n", "\r\n", "# equation 21:\r\n", "lft_hand = np.linalg.det(A.dot(B))\r\n", "rgt_hand = np.linalg.det(A) * np.linalg.det(B)\r\n", "validate(lft_hand, rgt_hand, 21, r\"$\\mathrm{det}(\\mathbf{AB}) =\\mathrm{det}(\\mathbf{A})\\mathrm{det}(\\mathbf{B})$\")\r\n", "\r\n", "# equation 22:\r\n", "lft_hand = np.linalg.det(np.linalg.inv(A))\r\n", "rgt_hand = 1.0/np.linalg.det(A)\r\n", "validate(lft_hand, rgt_hand, 22, r\"$\\mathrm{det}(\\mathbf{A}^{-1}) = 1/\\mathrm{det}(\\mathbf{A})$\")\r\n", "\r\n", "\r\n", "# equation 23:\r\n", "n = 5\r\n", "lft_hand = np.linalg.det(np.linalg.matrix_power(A, n))\r\n", "rgt_hand = np.power(np.linalg.det(A), n) \r\n", "validate(lft_hand, rgt_hand, 23, r\"$\\mathrm{det}(\\mathbf{A}^{n}) = \\mathrm{det}(\\mathbf{A})^n$\")\r\n", "\r\n", "\r\n", "# equation 24:\r\n", "lft_hand = np.linalg.det(I + u.dot(v.T))\r\n", "rgt_hand = 1 + u.T.dot(v)\r\n", "validate(lft_hand, rgt_hand, 24, r\"$\\mathrm{det}(\\mathbf{I+uv^T}) = 1+\\mathbf{u^Tv}$\")\r\n", "\r\n", "# equation 25:\r\n", "lft_hand = np.linalg.det(I2 + A2)\r\n", "rgt_hand = 1 + np.linalg.det(A2) + np.trace(A2)\r\n", "validate(lft_hand, rgt_hand, 25, r\"For $n=2$: $\\mathrm{det}(\\mathbf{I+A}) = 1+\\mathrm{det}(\\mathbf{A})+\\mathrm{Tr}(\\mathbf{A})$\")\r\n", "\r\n", "# equation 26:\r\n", "lft_hand = np.linalg.det(I + A)\r\n", "rgt_hand = 1 + np.linalg.det(A) + np.trace(A) + 0.5*np.power(np.trace(A),2) - 0.5*np.trace(np.linalg.matrix_power(A,2))\r\n", "validate(lft_hand, rgt_hand, 26, r\"For $n=3$: $\\mathrm{det}(\\mathbf{I+A}) = 1+\\mathrm{det}(\\mathbf{A})+\\mathrm{Tr}(\\mathbf{A})+\\frac{1}{2}\\mathrm{Tr}(\\mathbf{A})^2-\\frac{1}{2}\\mathrm{Tr}(\\mathbf{A}^2)$\")\r\n", "\r\n", "# equation 27:\r\n", "lft_hand = np.linalg.det(I4 + A4)\r\n", "rgt_hand = 1 + np.linalg.det(A4) + np.trace(A4) + 0.5*np.power(np.trace(A4),2) - 0.5*np.trace(np.linalg.matrix_power(A4,2)) +1/6 * np.power(np.trace(A4),3) - 0.5*np.trace(A4)*np.trace(np.linalg.matrix_power(A4,2)) +1/3*np.trace(np.linalg.matrix_power(A4,3))\r\n", "validate(lft_hand, rgt_hand, 27, r\"For $n=4$: $\\mathrm{det}(\\mathbf{I+A}) = 1+\\mathrm{det}(\\mathbf{A})+\\mathrm{Tr}(\\mathbf{A}) +\\frac{1}{2}\\mathrm{Tr}(\\mathbf{A})^2-\\frac{1}{2}\\mathrm{Tr}(\\mathbf{A}^2) +\\frac{1}{6}\\mathrm{Tr}(\\mathbf{A})^3-\\frac{1}{2}\\mathrm{Tr}(\\mathbf{A})\\mathrm{Tr}(\\mathbf{A}^2)+\\frac{1}{3}\\mathrm{Tr}(\\mathbf{A}^3)$\")\r\n", "\r\n", "# equation 28:\r\n", "varepsilon = 1e-8\r\n", "lft_hand = np.linalg.det(I + varepsilon*A)\r\n", "rgt_hand = 1 + np.linalg.det(A) + varepsilon*np.trace(A) + 0.5*np.power(varepsilon,2)*np.power(np.trace(A),2) - 0.5*np.power(varepsilon,2)*np.trace(np.linalg.matrix_power(A,2))\r\n", "validate(lft_hand, rgt_hand, 28, r\"$\\mathrm{det}(\\mathbf{I+\\varepsilon A}) \\cong 1+\\mathrm{det}(\\mathbf{A})+\\varepsilon\\mathrm{Tr}(\\mathbf{A})+\\frac{1}{2}\\varepsilon^2\\mathrm{Tr}(\\mathbf{A})^2-\\frac{1}{2}\\varepsilon^2\\mathrm{Tr}(\\mathbf{A}^2)$\")\r\n" ], "outputs": [ { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$\\mathrm{det}(\\mathbf{A}) = \\prod_i{\\lambda_i} \\quad \\lambda_i=\\mathrm{eig}(\\mathbf{A})$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$\\mathrm{det}(c\\mathbf{A}) = c^n \\mathrm{det}(\\mathbf{A}), \\quad \\text{if } \\mathbf{A} \\in \\mathbb{R}^{n \\times n}$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$\\mathrm{det}(\\mathbf{A}^T) = \\mathrm{det}(\\mathbf{A})$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$\\mathrm{det}(\\mathbf{AB}) =\\mathrm{det}(\\mathbf{A})\\mathrm{det}(\\mathbf{B})$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$\\mathrm{det}(\\mathbf{A}^{-1}) = 1/\\mathrm{det}(\\mathbf{A})$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$\\mathrm{det}(\\mathbf{A}^{n}) = \\mathrm{det}(\\mathbf{A})^n$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$\\mathrm{det}(\\mathbf{I+uv^T}) = 1+\\mathbf{u^Tv}$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "For $n=2$: $\\mathrm{det}(\\mathbf{I+A}) = 1+\\mathrm{det}(\\mathbf{A})+\\mathrm{Tr}(\\mathbf{A})$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "For $n=3$: $\\mathrm{det}(\\mathbf{I+A}) = 1+\\mathrm{det}(\\mathbf{A})+\\mathrm{Tr}(\\mathbf{A})+\\frac{1}{2}\\mathrm{Tr}(\\mathbf{A})^2-\\frac{1}{2}\\mathrm{Tr}(\\mathbf{A}^2)$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "For $n=4$: $\\mathrm{det}(\\mathbf{I+A}) = 1+\\mathrm{det}(\\mathbf{A})+\\mathrm{Tr}(\\mathbf{A}) +\\frac{1}{2}\\mathrm{Tr}(\\mathbf{A})^2-\\frac{1}{2}\\mathrm{Tr}(\\mathbf{A}^2) +\\frac{1}{6}\\mathrm{Tr}(\\mathbf{A})^3-\\frac{1}{2}\\mathrm{Tr}(\\mathbf{A})\\mathrm{Tr}(\\mathbf{A}^2)+\\frac{1}{3}\\mathrm{Tr}(\\mathbf{A}^3)$" ] }, "metadata": {} }, { "output_type": "stream", "name": "stdout", "text": [ "(28) 1.0000000230135486 = 0.8651635248813441\n" ] }, { "output_type": "display_data", "data": { "text/plain": [ "" ], "text/markdown": [ "$\\mathrm{det}(\\mathbf{I+\\varepsilon A}) \\cong 1+\\mathrm{det}(\\mathbf{A})+\\varepsilon\\mathrm{Tr}(\\mathbf{A})+\\frac{1}{2}\\varepsilon^2\\mathrm{Tr}(\\mathbf{A})^2-\\frac{1}{2}\\varepsilon^2\\mathrm{Tr}(\\mathbf{A}^2)$" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/plain": [ "
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}, "metadata": {} } ], "metadata": {} }, { "cell_type": "markdown", "source": [ "### 1.3 The Special Case 2x2 (特殊情况)\r\n", "Consider the matrix $\\mathbf{A}$\r\n", "\r\n", "$$\\mathbf{A} = \\begin{bmatrix} \\mathbf{A}_{11} & \\mathbf{A}_{12} \\\\ \\mathbf{A}_{21} & \\mathbf{A}_{22} \\end{bmatrix}$$\r\n", "\r\n", "Determinant and trace\r\n", "\r\n", "$$\\begin{align} \\mathrm{det}(\\mathbf{A}) &= \\mathbf{A}_{11}\\mathbf{A}_{22} - \\mathbf{A}_{12}\\mathbf{A}_{21} \\\\ \\mathrm{Tr}(\\mathbf{A}) &= \\mathbf{A}_{11}+\\mathbf{A}_{22}\\end{align}$$\r\n", "\r\n", "Eigenvalues\r\n", "\r\n", "$$\\lambda^2 - \\lambda \\cdot \\mathrm{Tr}(\\mathbf{A}) + \\mathrm{det}(\\mathbf{A}) = 0 \\\\ \\lambda_1 = \\frac{\\mathrm{Tr}(\\mathbf{A}) + \\sqrt{\\mathrm{Tr}(\\mathbf{A})^2 - 4\\mathrm{det}(\\mathbf{A})}}{2} \\quad \\lambda_2 = \\frac{\\mathrm{Tr}(\\mathbf{A}) - \\sqrt{\\mathrm{Tr}(\\mathbf{A})^2 - 4\\mathrm{det}(\\mathbf{A})}}{2} \\\\ \\lambda_1 + \\lambda_2 = \\mathrm{Tr}(\\mathbf{A}) \\quad \\lambda_1 \\lambda_2 =\\mathrm{det}(\\mathbf{A})$$\r\n", "\r\n", "Eigenvectors\r\n", "\r\n", "$$\\mathbf{v}_1 \\propto \\begin{bmatrix} \\mathbf{A}_{12} \\\\ \\lambda_1-\\mathbf{A}_{11} \\end{bmatrix} \\quad \\mathbf{v}_2 \\propto \\begin{bmatrix} \\mathbf{A}_{12} \\\\ \\lambda_2-\\mathbf{A}_{11} \\end{bmatrix}$$\r\n", "\r\n", "Inverse\r\n", "\r\n", "$$\\begin{align} \\mathbf{A}^{-1} = \\frac{1}{\\mathrm{det}(\\mathbf{A})} \\begin{bmatrix} \\mathbf{A}_{22} & -\\mathbf{A}_{12} \\\\ -\\mathbf{A}_{21} & \\mathbf{A}_{11} \\end{bmatrix} \\end{align}$$" ], "metadata": {} }, { "cell_type": "markdown", "source": [ "## 2 Derivatives" ], "metadata": {} }, { "cell_type": "code", "execution_count": 7, "source": [ "from sympy import MatrixSymbol, Inverse, symbols, Determinant, Trace, Derivative\r\n", "from sympy import MatAdd, Identity, MatMul" ], "outputs": [], "metadata": {} }, { "cell_type": "code", "execution_count": 8, "source": [ "# k = symbols(\"k\")\r\n", "\r\n", "# X = MatrixSymbol(\"X\", k, k)\r\n", "# x = MatrixSymbol(\"x\", k, 1)\r\n", "\r\n", "# A = MatrixSymbol(\"A\", k, k)\r\n", "# B = MatrixSymbol(\"B\", k, k)\r\n", "# C = MatrixSymbol(\"C\", k, k)\r\n", "# D = MatrixSymbol(\"D\", k, k)\r\n", "\r\n", "# a = MatrixSymbol(\"a\", k, 1)\r\n", "# b = MatrixSymbol(\"b\", k, 1)\r\n", "# c = MatrixSymbol(\"c\", k, 1)\r\n", "# d = MatrixSymbol(\"d\", k, 1)" ], "outputs": [], "metadata": {} }, { "cell_type": "code", "execution_count": 9, "source": [ "# 把assert 都去掉了\r\n", "# A.diff(A) == Derivative(A, A)\r\n", "\r\n", "# A.T.diff(A) == Derivative(A.T, A)\r\n", "\r\n", "# (2*A).diff(A) == Derivative(2*A, A)\r\n", "\r\n", "# MatAdd(A, A).diff(A) == Derivative(MatAdd(A, A), A)\r\n", "\r\n", "# (A + B).diff(A) == Derivative(A + B, A)" ], "outputs": [], "metadata": {} }, { "cell_type": "code", "execution_count": null, "source": [ "# #------------matrix_derivative_trivial\r\n", "# # equation 33:\r\n", "# rgt_hand = 0\r\n", "# lft_hand = X.diff(A)\r\n", "\r\n", "# #------------matrix_derivative_vectors_and_scalars\r\n", "# # Cookbook example 69:\r\n", "# expr = x.T*a\r\n", "# expr.diff(x) == a\r\n", "# expr = a.T*x\r\n", "# expr.diff(x) == a\r\n", "\r\n", "## Cookbook example 70:\r\n", "# expr = a.T*X*b\r\n", "# expr.diff(X) == a*b.T\r\n", "\r\n", "# # Cookbook example 71:\r\n", "# expr = a.T*X.T*b\r\n", "# expr.diff(X) == b*a.T\r\n", "\r\n", "# # Cookbook example 77:\r\n", "# expr = b.T*X.T*X*c\r\n", "# expr.diff(X) == X*b*c.T + X*c*b.T\r\n", "\r\n", "# # Cookbook example 78:\r\n", "# expr = (B*x + b).T*C*(D*x + d)\r\n", "# expr.diff(x) == B.T*C*(D*x + d) + D.T*C.T*(B*x + b)\r\n", "\r\n", "# # Cookbook example 81:\r\n", "# expr = x.T*B*x\r\n", "# expr.diff(x) == B*x + B.T*x\r\n", "\r\n", "# # Cookbook example 82:\r\n", "# expr = b.T*X.T*D*X*c\r\n", "# expr.diff(X) == D.T*X*b*c.T + D*X*c*b.T\r\n", "\r\n", "# # Cookbook example 83:\r\n", "# expr = (X*b + c).T*D*(X*b + c)\r\n", "# expr.diff(X) == D*(X*b + c)*b.T + D.T*(X*b + c)*b.T\r\n", "\r\n", "\r\n", "# #-----------matrix_derivative_with_inverse\r\n", "# # Cookbook example 61:\r\n", "# expr = a.T*Inverse(X)*b\r\n", "# expr.diff(X) == -Inverse(X).T*a*b.T*Inverse(X).T\r\n", "\r\n", "# # Cookbook example 63:\r\n", "# expr = Trace(A*Inverse(X)*B)\r\n", "# expr.diff(X) == -(X**(-1)*B*A*X**(-1)).T\r\n", "\r\n", "# # Cookbook example 64:\r\n", "# expr = Trace(Inverse(X + A))\r\n", "# expr.diff(X) == -(Inverse(X + A)).T**2" ], "outputs": [], "metadata": {} }, { "cell_type": "code", "execution_count": null, "source": [ "# #---------------matrix_derivatives_of_traces\r\n", "# # First order:\r\n", "\r\n", "# # Cookbook example 99:\r\n", "# expr = Trace(X)\r\n", "# expr.diff(X) == Identity(k)\r\n", "\r\n", "# # Cookbook example 100:\r\n", "# expr = Trace(X*A)\r\n", "# expr.diff(X) == A.T\r\n", "\r\n", "# # Cookbook example 101:\r\n", "# expr = Trace(A*X*B)\r\n", "# expr.diff(X) == A.T*B.T\r\n", "\r\n", "# # Cookbook example 102:\r\n", "# expr = Trace(A*X.T*B)\r\n", "# expr.diff(X) == B*A\r\n", "\r\n", "# # Cookbook example 103:\r\n", "# expr = Trace(X.T*A)\r\n", "# expr.diff(X) == A\r\n", "\r\n", "# # Cookbook example 104:\r\n", "# expr = Trace(A*X.T)\r\n", "# expr.diff(X) == A\r\n", "\r\n", "# # Cookbook example 105:\r\n", "# # TODO: TensorProduct is not supported\r\n", "# #expr = Trace(TensorProduct(A, X))\r\n", "# #expr.diff(X) == Trace(A)*Identity(k)\r\n", "\r\n", "# ## Second order:\r\n", "\r\n", "# # Cookbook example 106:\r\n", "# expr = Trace(X**2)\r\n", "# expr.diff(X) == 2*X.T\r\n", "\r\n", "# # Cookbook example 107:\r\n", "# expr = Trace(X**2*B)\r\n", "# # TODO: wrong result\r\n", "# #expr.diff(X) == (X*B + B*X).T\r\n", "# expr = Trace(MatMul(X, X, B))\r\n", "# expr.diff(X) == (X*B + B*X).T\r\n", "\r\n", "# # Cookbook example 108:\r\n", "# expr = Trace(X.T*B*X)\r\n", "# expr.diff(X) == B*X + B.T*X\r\n", "\r\n", "# # Cookbook example 109:\r\n", "# expr = Trace(B*X*X.T)\r\n", "# expr.diff(X) == B*X + B.T*X\r\n", "\r\n", "# # Cookbook example 110:\r\n", "# expr = Trace(X*X.T*B)\r\n", "# expr.diff(X) == B*X + B.T*X\r\n", "\r\n", "# # Cookbook example 111:\r\n", "# expr = Trace(X*B*X.T)\r\n", "# expr.diff(X) == X*B.T + X*B\r\n", "\r\n", "# # Cookbook example 112:\r\n", "# expr = Trace(B*X.T*X)\r\n", "# expr.diff(X) == X*B.T + X*B\r\n", "\r\n", "# # Cookbook example 113:\r\n", "# expr = Trace(X.T*X*B)\r\n", "# expr.diff(X) == X*B.T + X*B\r\n", "\r\n", "# # Cookbook example 114:\r\n", "# expr = Trace(A*X*B*X)\r\n", "# expr.diff(X) == A.T*X.T*B.T + B.T*X.T*A.T\r\n", "\r\n", "# # Cookbook example 115:\r\n", "# expr = Trace(X.T*X)\r\n", "# expr.diff(X) == 2*X\r\n", "# expr = Trace(X*X.T)\r\n", "# expr.diff(X) == 2*X\r\n", "\r\n", "# # Cookbook example 116:\r\n", "# expr = Trace(B.T*X.T*C*X*B)\r\n", "# expr.diff(X) == C.T*X*B*B.T + C*X*B*B.T\r\n", "\r\n", "# # Cookbook example 117:\r\n", "# expr = Trace(X.T*B*X*C)\r\n", "# expr.diff(X) == B*X*C + B.T*X*C.T\r\n", "\r\n", "# # Cookbook example 118:\r\n", "# expr = Trace(A*X*B*X.T*C)\r\n", "# expr.diff(X) == A.T*C.T*X*B.T + C*A*X*B\r\n", "\r\n", "# # Cookbook example 119:\r\n", "# expr = Trace((A*X*B + C)*(A*X*B + C).T)\r\n", "# expr.diff(X) == 2*A.T*(A*X*B + C)*B.T\r\n", "\r\n", "# # Cookbook example 120:\r\n", "# # TODO: no support for TensorProduct.\r\n", "# # expr = Trace(TensorProduct(X, X))\r\n", "# # expr = Trace(X)*Trace(X)\r\n", "# # expr.diff(X) == 2*Trace(X)*Identity(k)" ], "outputs": [], "metadata": {} } ] }