{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 求解线性方程组" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 逆矩阵" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "逆矩阵的几个属性:\n", "1. 若方阵 A 可逆，则 $A^{-1}A = I$，以及 $AA^{-1} = I$。\n", "2. 经过消除算法，A 不能存在全是 0 的行。pivots 不能有零。\n", "3. det(A) != 0\n", "4. $Ax = 0$: $x = 0$ 必须是唯一解。\n", "5. 若 A, B (同样行列) 都可逆，那么: $\\boxed{(AB)^{-1} = B^{-1}A^{-1}}$\n", "6. 一个 2x2 的矩阵，如果可逆，则 $ad - bc$ 不可为零。因为 \n", "$$\n", "\\begin{bmatrix}\n", "a & b \\\\\n", "c & d\n", "\\end{bmatrix}^{-1} = \\frac{1}{ad - bc}\n", "\\begin{bmatrix}\n", "d & -b \\\\\n", "-c & a\n", "\\end{bmatrix}\n", "$$\n", "7. 对角线矩阵(其余全为0):\n", "$$\n", "If \\quad A =\n", "\\begin{bmatrix}\n", "d_{1} \\\\\n", "& \\ddots \\\\\n", "& & d_{n}\n", "\\end{bmatrix}\n", "\\quad then \\quad\n", "A^{-1} = \n", "\\begin{bmatrix}\n", "1/d_{1} \\\\\n", "& \\ddots \\\\\n", "& & 1/d_{n}\n", "\\end{bmatrix}\n", "$$\n", "8. 高斯消元法求逆矩阵，求矩阵的秩: \n", "$$Multiply\\ \\begin{bmatrix} A & I \\end{bmatrix}\\ by \\ A^{-1}\\ to\\ get\\ \\begin{bmatrix} I & A^{-1} \\end{bmatrix}.$$\n", "9. 若 $K$ 沿主对角线对称，那么 $K^{-1}$ 也对称；求逆通常会让矩阵变得稠密；pivots 的乘积就是行列式的值。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**消元法 = 因式分解**： $A = LU$" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/plain": [ "3×3 Array{Int64,2}:\n", " 1 3 1\n", " 1 1 -1\n", " 3 11 6" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A = [1 3 1\n", " 1 1 -1\n", " 3 11 6]" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/plain": [ "3×3 Array{Float64,2}:\n", " -8.5 3.5 2.0\n", " 4.5 -1.5 -1.0\n", " -4.0 1.0 1.0" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "iA = inv(A)" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/plain": [ "3×3 Array{Float64,2}:\n", " 1.0 2.22045e-16 1.11022e-16\n", " 0.0 1.0 -1.11022e-16\n", " 0.0 -3.55271e-15 1.0 " ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A * iA" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "1.0000000000000009" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "det(A * iA)" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3×3 Array{Float64,2}:\n", " 1.0 0.0 0.0\n", " 1.0 1.0 0.0\n", " 3.0 -1.0 1.0" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "L, U = lu(A, Val{false})\n", "L" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3×3 Array{Float64,2}:\n", " 1.0 3.0 1.0\n", " 0.0 -2.0 -2.0\n", " 0.0 0.0 1.0" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "U" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3×3 Array{Float64,2}:\n", " 1.0 3.0 1.0\n", " 1.0 1.0 -1.0\n", " 3.0 11.0 6.0" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "L * U" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 矩阵的转置" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "1. $Ax$ 的转置是 $x^{T}A^{T}$, $AB$ 的转置是 $B^{T}A^{T}$, $A^{-1}$ 的转置是 $(A^{T})^{-1}$\n", "2. $x \\cdot y = x^{T}y$\n", "3. $Ax \\cdot y = x \\cdot A^{T}y$\n", "4. 对称矩阵的转置是其自身：$S^{T}=S$\n", "5. 正交矩阵的转置是其逆：$Q^{T} = Q^{-1}$" ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3×2 Array{Int64,2}:\n", " 1 4\n", " 2 5\n", " 3 6" ] }, "execution_count": 50, "metadata": {}, "output_type": "execute_result" } ], "source": [ "[1 2 3; 4 5 6]'" ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "true" ] }, "execution_count": 49, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Q = [1/3 2/3 2/3; 2/3 1/3 -2/3; 2/3 -2/3 1/3]\n", "isapprox(inv(Q), Q')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 如何理解置换矩阵\n", "\n", "置换矩阵就是每一行每一列都有一个 1. 记为 $P$, 而 $P^{T}$ 依然是置换矩阵。单位矩阵 $I$ 是最简单的置换矩阵。通过交换 $I$ 的行，可以得到全部可能的置换矩阵。A permutation matrix $P$ has the rows of the identity $I$ in any order.\n", "\n", "本质上，置换矩阵是对行交换的顺序的一种描述。 $PA$ 的含义就是，对 $A$ 进行 $P$ 所描述的行交换。" ] }, { "cell_type": "code", "execution_count": 51, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "5×5 Array{Int64,2}:\n", " 4 -2 -7 -4 -8\n", " 9 -6 -6 -1 -5\n", " -2 -9 3 -5 2\n", " 9 7 -9 5 -8\n", " -1 6 -3 9 6" ] }, "execution_count": 51, "metadata": {}, "output_type": "execute_result" } ], "source": [ "B = [4 -2 -7 -4 -8\n", " 9 -6 -6 -1 -5\n", " -2 -9 3 -5 2\n", " 9 7 -9 5 -8\n", " -1 6 -3 9 6]" ] }, { "cell_type": "code", "execution_count": 53, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "5×5 Array{Float64,2}:\n", " 1.0 0.0 0.0 0.0 0.0\n", " 1.0 1.0 0.0 0.0 0.0\n", " 0.444444 0.0512821 1.0 0.0 0.0\n", " -0.111111 0.410256 0.582822 1.0 0.0\n", " -0.222222 -0.794872 0.171779 0.0242696 1.0" ] }, "execution_count": 53, "metadata": {}, "output_type": "execute_result" } ], "source": [ "L, U, p = lu(B)\n", "L" ] }, { "cell_type": "code", "execution_count": 54, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "5×5 Array{Float64,2}:\n", " 9.0 -6.0 -6.0 -1.0 -5.0 \n", " 0.0 13.0 -3.0 6.0 -3.0 \n", " 0.0 0.0 -4.17949 -3.86325 -5.62393 \n", " 0.0 0.0 0.0 8.67894 9.95297 \n", " 0.0 0.0 0.0 0.0 -0.771206" ] }, "execution_count": 54, "metadata": {}, "output_type": "execute_result" } ], "source": [ "U" ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "5-element Array{Int64,1}:\n", " 2\n", " 4\n", " 1\n", " 5\n", " 3" ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p" ] } ], "metadata": { "kernelspec": { "display_name": "Julia 0.6.1", "language": "julia", "name": "julia-0.6" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "0.6.1" } }, "nbformat": 4, "nbformat_minor": 2 }