机器学习数学基础1 青旅半醒 2024-04-20 09:14 21阅读 0赞 # 线性代数 # ## 矩阵 ## ### 矩阵的定义 ### ![1604514-20190721193349851-58793581.png][] * 由 m × n 个数 aij (i = 1,2,...,m; j = 1,2,...,n) 排成的 m 行 n 列的数表 A 就称为 m 行 n 列的**矩阵** * 这 m × n 个数称作矩阵 A 的**元素,**元素 aij 位于矩阵A 的第i 行第j 列 * m × n 矩阵 A 可以记作 Am×n,其中 m是行数,n是列数,m, n > 0 **特殊矩阵** ![1604514-20190721200935755-1883447346.png][] 对于Am×n,如果 m = n,即矩阵的**行数与列数相等**,那么称A为**方阵** **矩阵中的概念** **![1604514-20190721201148127-994156308.png][]** * 行数与列数都等于 n 的矩阵称为 n 阶矩阵,又称做 **n****阶方阵**,可以记作 An * 只有一行的矩阵 A1×n 称为行矩阵,又叫**行向量** * 同样,只有一列的矩阵 An×1 称为列矩阵,又叫**列向量** ![1604514-20190721201229710-2057567401.png][] * 对于方阵,从左上角到右下角的直线,叫做**主对角线**,主对角线上的元素称为**主对角** **线元素** **特殊矩阵** **![1604514-20190721201417233-905024153.png][]** 矩阵的元素全部为0,称为**零矩阵**,用 O 表示 ![1604514-20190721201450499-1114639441.png][] 对于方阵,如果只有对角线元素为1,其余元素都为0, 那么称为**单位矩阵**,一般用 I 或者 E 表示 ![1604514-20190721201537889-1859389215.png][] 对于方阵,不在对角线上的元素都为0, 称为**对角矩阵** ### 矩阵的加法 ### * 把矩阵的对应位元素相加 * 矩阵的形状必须一致,即必须是**同型矩阵** **![1604514-20190721201824320-2028767275.png][]** ### 矩阵的乘法 ### #### 1. 数与矩阵相乘 #### 数值与矩阵的每一个元素相乘 ![1604514-20190721202248785-920313770.png][] #### 2.矩阵与矩阵相乘 #### 左矩阵的每一行与右矩阵的每一行,对应每一行元素相乘 ![1604514-20190721202715868-676120950.png][] A × B,那么有 A 矩阵 m × n,B 矩阵 n × k,要求左侧矩阵的列数n,必须等于右侧矩阵的行数 n, 必须等于右侧矩阵的行数 n,结果矩阵 C 为 m × k 矩阵。 ![1604514-20190721202906160-893761557.png][] ![1604514-20190721203939153-1109265365.png][] ![1604514-20190721204012875-74543864.png][] 规则:一行乘一列,行定列移动,列尽下一行 ![1604514-20190721204156833-1801311081.png][] 如: ![1604514-20190721204317671-1634360249.png][] ### **矩阵的转置** ### **![1604514-20190721204413020-1702964596.png][]** * 把矩阵 A 的行换成相同序数的列,得到一个新矩阵,叫做 A 的**转置矩阵**,记作 AT * 行变列,列变行 * A 为 m × n 矩阵,转置之后为 n × m 矩阵 ### **矩阵的运算法则** ### **![1604514-20190721204647340-112011449.png][]** ### 矩阵的逆 ### ——对于 n 阶方阵 A,如果有一个 n 阶方阵 B,使得 AB = BA = E 就称矩阵 A 是可逆的,并把 B 称为 A 的逆矩阵 ![1604514-20190721205926182-29811376.png][] # 微积分基本知识 # ![1604514-20190721230655885-1874390570.png][] ## 导数 ## ![1604514-20190721234835684-163827446.png][] ![1604514-20190721235421708-2056079124.png][] ## 偏导数 ## ![1604514-20190721235500236-543050723.png][] ## 方向导数 ## ![1604514-20190721235538564-2120208632.png][] ## 梯度(Gradient) ## ![1604514-20190721235612311-1971034725.png][] ## 凸函数和凹函数 ## ![1604514-20190721235652140-1400489840.png][] # 概率统计基础知识 # ![1604514-20190722000003675-1090730985.png][] ## 常用统计变量 ## **• 样本均值** **![1604514-20190722000142389-687714817.png][]** **• 样本方差** ![1604514-20190722000209129-264066997.png][] **• 样本标准差** ![1604514-20190722000246967-611878523.png][] ## 常见概率分布 ## ![1604514-20190722000649131-1146411396.png][] ## 重要概率公式 ## ![1604514-20190722000843335-700387512.png][] 转载于:https://www.cnblogs.com/LXL616/p/11222709.html [1604514-20190721193349851-58793581.png]: https://image.dandelioncloud.cn/pgy_files/images/2024/04/18/03847bf7d6234712b3cdd41103da9db4.png [1604514-20190721200935755-1883447346.png]: 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