From 0d215e69dab9e93000a1524c6d78b25c56a49f12 Mon Sep 17 00:00:00 2001 From: MaoYZ Date: Wed, 4 Sep 2024 17:14:50 +0800 Subject: [PATCH] =?UTF-8?q?=E7=BA=BF=E6=80=A7=E4=BB=A3=E6=95=B0?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ...arestNeighbors_and_consine_simiarity.ipynb | 26 +++++- math/images/math_03.png | Bin 76343 -> 0 bytes math/images/math_04.png | Bin 69808 -> 0 bytes math/images/math_05.png | Bin 74950 -> 0 bytes math/images/math_06.png | Bin 68420 -> 0 bytes math/images/math_07.png | Bin 34103 -> 0 bytes math/images/math_08.png | Bin 75333 -> 0 bytes math/images/math_24.png | Bin 0 -> 17280 bytes math/images/math_25.png | Bin 0 -> 114687 bytes ...77\346\200\247\344\273\243\346\225\260.md" | 81 +++++++++++++++--- 10 files changed, 94 insertions(+), 13 deletions(-) delete mode 100644 math/images/math_03.png delete mode 100755 math/images/math_04.png delete mode 100644 math/images/math_05.png delete mode 100755 math/images/math_06.png delete mode 100755 math/images/math_07.png delete mode 100755 math/images/math_08.png create mode 100644 math/images/math_24.png create mode 100644 math/images/math_25.png diff --git a/machinelearning/recommand/03NearestNeighborsAndConsineSimiarity/NearestNeighbors_and_consine_simiarity.ipynb b/machinelearning/recommand/03NearestNeighborsAndConsineSimiarity/NearestNeighbors_and_consine_simiarity.ipynb index d167fbc..ae1cfff 100644 --- a/machinelearning/recommand/03NearestNeighborsAndConsineSimiarity/NearestNeighbors_and_consine_simiarity.ipynb +++ b/machinelearning/recommand/03NearestNeighborsAndConsineSimiarity/NearestNeighbors_and_consine_simiarity.ipynb @@ -73,13 +73,37 @@ ], "execution_count": 7 }, + { + "metadata": { + "ExecuteTime": { + "end_time": "2024-09-04T08:33:48.901702Z", + "start_time": "2024-09-04T08:33:48.896657Z" + } + }, + "cell_type": "code", + "source": "\n", + "id": "4c77487692826011", + "outputs": [ + { + "name": "stdout", + "output_type": 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z$E-nm(m8M5o$74l0(>S}v7hhL^WZW|!Oa2jojVOA%z(QdBUa;IQ;snOr@7;Wu3IE} z?H3Z#-18q2PoN4w(e9AoaZ~^J&hzuUuCN21-=1OGm^q-Q=5pY8c_&tK8d08~0BOiSqhEMk4?1-^I}B0AwcEvzx#Tm^b*ljrcOS f|DV!l68_LMgb&|u)>qs)1HP_lUe_o=TLt|ukw44| literal 0 HcmV?d00001 diff --git "a/math/\347\272\277\346\200\247\344\273\243\346\225\260.md" "b/math/\347\272\277\346\200\247\344\273\243\346\225\260.md" index d6ef887..ab3371a 100644 --- "a/math/\347\272\277\346\200\247\344\273\243\346\225\260.md" +++ "b/math/\347\272\277\346\200\247\344\273\243\346\225\260.md" @@ -27,11 +27,41 @@ ### 向量的运算 -![](images/math_03.png) - +向量有大小和方向 向量的加法:2维空间内,就是求给定2个向量所围成的平行四边形的对角线。 + + +向量基运算, 同位置+,-,乘 + +```text +(1,2,3) + (4,5,6) = (5,7,9) +(1,2,3) - (4,5,6) = (-3,-3,-3) +(1,2,3) * 3 = (3,6,9) +``` + +向量矩阵转置,行变列 +```text +(1, + 2, => (1,2,3) + 3) + +[ +[1,2], => [[1,3], +[3,4] [2,4]] +] + + +``` +```python +import numpy as np +d1 = np.array([[1,2],[3,4]]) +print(d1) +print(d1.T) +``` + + ### 基向量 向量 i, j 称为基向量,其他向量都可以通过对基向量缩放再相加的方法构造出来。基向量缩放的倍数对应向量的各个分量,即向量对应的坐标。 @@ -48,7 +78,23 @@ ### 范数 -![](images/math_04.png) +范数(norm) 是数学中的一种基本概念 +- L1范数 - 向量中各个元素绝对值之和 +- L2范数 - 向量的模长,每个元素平方求和,再开平方根 + - **X的转置.X = L2范数的平方** +- p范数, 向量中每个元素p幂次求和,再开p次根 + +```text +L1范数 +xT = (1,2,-3) ; ||x||1 = |1| +|2|+|3|=6 + +L2范数 +xT = (1,2,-3) ; ||x||2 = sqrt(1+4+9) = sqrt(12) + + +Lp范数 ||x||p = (|x1|^p + ... |xn|^p) ^ (1/p) +``` + @@ -59,7 +105,7 @@ ![](images/math_14.png) -当m=n时,我们称之为n阶矩阵或n阶方阵 +当m=n时,我们称之为n阶矩阵或n阶**方阵** 特殊情况: @@ -68,14 +114,9 @@ (2)列矩阵:当n=1时,即只有一列的矩阵! 称之为列矩阵或列向量 +**单位阵** 特殊的方阵,斜对角线都是1 -![](images/math_08.png) - -![](images/math_05.png) -![](images/math_06.png) - -![](images/math_07.png) ### 矩阵的加减法 @@ -104,7 +145,7 @@ print('水果价格浮动后价格为:\n',fruit_price*fudu) ### **矩阵与矩阵运算** - +#### 普通矩阵相乘 矩阵乘法公式: 设矩阵 A 是一个 m×n 的矩阵,矩阵 B 是一个 n×p 的矩阵,那么它们的乘积 C 是一个 m×p 的矩阵,其中 C 的第 i 行第 j 列元素为: @@ -112,12 +153,26 @@ print('水果价格浮动后价格为:\n',fruit_price*fudu) ![](images/math_13.png) -特别需要注意的就是:**前一个矩阵的列一定要和后一个矩阵的行大小相同**。 + +普通矩阵相乘: **前一个矩阵的列一定要和后一个矩阵的行大小相同**。 运算规则:需要注意进行矩阵乘法的两个矩阵要求前一个矩阵的列要和后一个矩阵的行的轴长相同,只有保证这个条件才能够进行矩阵乘法。也就是说前面的矩阵的列数和后面矩阵的行数一致两个矩阵才能点乘。 同时矩阵乘法是满足结合律并不满足交换律的,即 ABC 等于 A(BC),不等于 ACB。 我们的数据的维度往往是不符合要求的,这是就需要进行升维。 +#### 哈达玛积 + +矩阵大小一样, 同位置相乘 +![](images/math_24.png) + +#### 克罗内克积 + +### 矩阵的性质 +![](images/math_25.png) + +**A矩阵*I单位矩阵=A矩阵** + +**矩阵的逆A-¹ * 矩阵A = 单位矩阵I** @@ -141,6 +196,8 @@ A矩阵的行变成 同序号的列 得到的新矩阵称为 A矩阵的转置 下面是公式: ![](images/math_15.png) + + ### 矩阵的初等行变换 1. 换法变换:对换矩阵的两行。对换i,j两行,记作ri <->rj。

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