{ "cells": [ { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "3ce13dbd-b25d-494c-a841-52b237d8240a" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Motivation\n", "\n", "Multivariate linear regression before linear algebra:\n", "\n", "$$\n", "\\begin{align*}\n", "y^{(1)} &\\approx \\theta_0 + \\theta_1 x_1^{(1)} + \\theta_2 x_2^{(1)} + \\ldots + \\theta_n x_n^{(1)} \\\\ \n", "y^{(2)} &\\approx \\theta_0 + \\theta_1 x_1^{(2)} + \\theta_2 x_2^{(2)} + \\ldots + \\theta_n x_n^{(2)} \\\\ \n", "\\ldots \\\\\n", "y^{(m)} &\\approx \\theta_0 + \\theta_1 x_1^{(m)} + \\theta_2 x_2^{(m)} + \\ldots + \\theta_n x_n^{(m)}\n", "\\end{align*}\n", "$$\n", "\n", "Multivariate linear regression after linear algebra\n", "$$\n", "\\mathbf{y} \\approx X \\boldsymbol{\\theta}\n", "$$\n" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "27119a93-4fda-404c-83bd-6e3386797809" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Linear Algebra in ML\n", "\n", "* Succinct notation for models and algorithms\n", "* Numerical tools (save coding!)\n", " $$\\boldsymbol{\\theta} = (X^{T}X)^{-1}X^T\\mathbf{y}$$\n", "* Inspiration for new models and problems: Netflix\n" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "94e2833b-1345-49a7-85e2-004c324144ae" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Netflix\n", "\n", "| user | Moonlight | The Shape of Water | Frozen | Moana |\n", "|-------|-----------|----------------------|--------|-----------| \n", "|Alice | 5 | 4 | 1 | |\n", "|Bob | | 5 | | 2 |\n", "|Carol | | | | 5 |\n", "|David | | | 5 | 5 |\n", "|Eve | 5 | 4 | | |\n", "\n", "\n", "Matrix completion problem, matrix factorization" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true, "nbpresent": { "id": "88982e7f-95ea-4222-ad68-0d7e43911762" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Topics\n", "\n", "* Matrices\n", "* Vectors\n", "* Matrix-Matrix multiplication (and special cases)\n", "* Tranpose\n", "* Inverse" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "aa0a3b44-b4ab-482c-a8cf-b58284779740" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Matrices\n", "\n", "A matrix is an rectangular array of numbers\n", "\n", "$$\n", "A = \n", "\\left[\n", "\\begin{array}{cc}\n", "101 & 10 \\\\\n", "54 & 13 \\\\ \n", "10 & 47\n", "\\end{array}\n", "\\right]\n", "$$\n", "\n", "When $A$ has $m$ rows and $n$ columns, we say that: \n", "* $A$ is an $m \\times n$ matrix\n", "* $A \\in \\mathbb{R}^{m \\times n}$\n", "\n", "The entry in row $i$ and column $j$ is denoted $A_{ij}$\n", "* sometimes $a_{ij}$ or $(A)_{ij}$" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "f6ebe35f-72f3-4f6f-aaae-cd1cb51ef087" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Example\n", "\n", "$$\n", "A = \n", "\\left[\n", "\\begin{array}{cc}\n", "101 & 10 \\\\\n", "54 & 13 \\\\ \n", "10 & 47\n", "\\end{array}\n", "\\right]\n", "$$\n", "\n", "* $A \\in \\mathbb{R}^{3 \\times 2}$\n", "* $A_{11}= 101$\n", "* $A_{32}=$ \n", "* $A_{22}=$\n", "* $A_{23}=$\n" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "d5d9c223-d434-40c0-b30d-17fe548f9f21" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Matrices in Python" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "nbpresent": { "id": "0bcd3d94-8e18-4bd3-8598-178ae406db64" }, "scrolled": true, "slideshow": { "slide_type": "-" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[101 10]\n", " [ 54 13]\n", " [ 10 47]]\n", "A has 3 rows and 2 columns\n" ] } ], "source": [ "import numpy as np\n", "\n", "# Pass a list of lists (rows) to the np.array constructor\n", "A = np.array([[101, 10], \n", " [54, 13], \n", " [10, 47]])\n", "\n", "print(A)\n", "\n", "m,n = A.shape\n", "\n", "print (\"A has %d rows and %d columns\" % (m, n))" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "45eef18f-3497-4513-98ed-f31ab0c8d828" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Matrix Indexing in Python\n", "\n", "Note that Python is zero-indexed" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "nbpresent": { "id": "2a9673a0-486d-4b32-b40c-ddce419b0890" }, "slideshow": { "slide_type": "-" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "101\n", "47\n", "13\n" ] }, { "ename": "IndexError", "evalue": "index 2 is out of bounds for axis 1 with size 2", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mIndexError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 7\u001b[0m \u001b[0mprint\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mA\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;31m# A_32 in math\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 8\u001b[0m \u001b[0mprint\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mA\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;31m# A_22 in math\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 9\u001b[0;31m \u001b[0mprint\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mA\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;31m# A_23 in math. ERROR!\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m", "\u001b[0;31mIndexError\u001b[0m: index 2 is out of bounds for axis 1 with size 2" ] } ], "source": [ "A = np.array([[101, 10], \n", " [54, 13], \n", " [10, 47]])\n", "\n", "\n", "print (A[0,0]) # A_11 in math\n", "print (A[2,1]) # A_32 in math\n", "print (A[1,1]) # A_22 in math\n", "print (A[1,2]) # A_23 in math. ERROR!" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "fc2fbc75-2132-4ebb-a9f2-9b46be06e418" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Vectors\n", "\n", "A vector is an $n \\times 1$ matrix:\n", "\n", "$$\n", "\\mathbf{x} = \n", "\\left[\n", "\\begin{array}{c}\n", "8 \\\\\n", "2.4 \\\\ \n", "1 \\\\\n", "-10 \n", "\\end{array}\n", "\\right]\n", "$$\n", "\n", "* We write $\\mathbf{x} \\in \\mathbb{R}^n$ (instead of $\\mathbf{x} \\in \\mathbb{R}^{n \\times 1}$)\n", "* The $i$th entry is $x_i$" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true, "nbpresent": { "id": "3ae48be8-32df-4c2d-bb66-019b61ae3ffb" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Indexing\n", "\n", "* Consider $\\mathbf{x} \\in \\mathbb{R}^4$ defined as $\\small\n", "\\mathbf{x} = \n", "\\left[\n", "\\begin{array}{c}\n", "8 \\\\\n", "2.4 \\\\ \n", "1 \\\\\n", "-10 \n", "\\end{array}\n", "\\right]\n", "$\n", "\n", " * $x_1 =$ \n", " * $x_4 =$ " ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "nbpresent": { "id": "eeb056fb-7c8c-43fb-9dec-2225b2444944" }, "slideshow": { "slide_type": "-" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "8.0\n", "-10.0\n" ] } ], "source": [ "x = np.array([8, 2.4, 1, -10]) # in numpy a vector is a 1d array\n", "print(x[0])\n", "print(x[3])" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "aa2b7f0b-ed9c-4f59-b320-c3debbecc0d8" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Addition\n", "\n", "If two matrices have the same size, we can add them by adding corresponding elements\n", "\n", "$$\n", "\\begin{bmatrix}\n", "1 & 2 \\\\\n", "3 & 4\n", "\\end{bmatrix} + \n", "\\begin{bmatrix}\n", "3 & 5 \\\\\n", "-1 & 0\n", "\\end{bmatrix} =\n", "\\begin{bmatrix}\n", "4 & 7 \\\\\n", "2 & 4\n", "\\end{bmatrix}\n", "$$\n", "\n", "* Subtraction is similar\n", "* Matrices of different sizes *cannot be added or subtracted*" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "nbpresent": { "id": "63ce192c-cb2c-47b7-b803-4f1ad79497b9" }, "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[4 7]\n", " [2 4]]\n" ] } ], "source": [ "A = np.array([[1, 2], \n", " [3, 4]])\n", "\n", "B = np.array([[3, 5],\n", " [-1, 0]])\n", "\n", "print(A + B)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "nbpresent": { "id": "b7b5a65e-65f6-4d96-983b-6fff6ec36289" }, "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "ename": "ValueError", "evalue": "operands could not be broadcast together with shapes (2,2) (1,3) ", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mValueError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 4\u001b[0m \u001b[0mC\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marray\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m3\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 5\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 6\u001b[0;31m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mA\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m", "\u001b[0;31mValueError\u001b[0m: operands could not be broadcast together with shapes (2,2) (1,3) " ] } ], "source": [ "A = np.array([[1, 2], \n", " [3, 4]])\n", "\n", "C = np.array([[1, 2, 3]])\n", "\n", "print(A + C)" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "nbpresent": { "id": "d26df4f9-5f14-4796-bd8a-ee88d48a4212" }, "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[11 22]\n", " [13 24]]\n" ] } ], "source": [ "# Beware: broadcasting. This will work, and is a nice feature, but\n", "# is *not* an accepted linear algebra operation\n", "\n", "A = np.array([[1, 2], \n", " [3, 4]])\n", "\n", "D = np.array([10, 20])\n", "print(A + D)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "nbpresent": { "id": "65b7d294-59a0-4693-8b1c-c84aa970b03f" }, "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[11 12]\n", " [23 24]]\n", "[[1, 3], array([-2, 0]), [1, 3], array([-2, 0])]\n" ] } ], "source": [ "# Do this to broadcast a column vector\n", "A = np.array([[1, 2], \n", " [3, 4]])\n", "\n", "D = np.array([[10], [20]]) # a 2x1 vector or \"column vector\"\n", "\n", "print(A + D)" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "9d088a56-fadd-4fba-b306-dd45fde526cd" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Scalar Multiplication\n", "\n", "A *scalar* $x \\in \\mathbb{R}$ is a real number (i.e., not a vector)\n", "\n", "$$\\text{e.g., } 2,\\, 3,\\, \\pi,\\, \\sqrt{2},\\, 1.843,\\, \\ldots$$\n", "\n", "Scalar times a matrix:\n", "\n", "$$\n", "2\n", "\\cdot\n", "\\begin{bmatrix}\n", "1 & 3 \\\\\n", "-2 & 0 \n", "\\end{bmatrix} \n", "= \n", "\\begin{bmatrix}\n", "2 & 6 \\\\\n", "-4 & 0 \n", "\\end{bmatrix} \n", "$$\n", "\n", "(multiply each entry by the scalar)" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "nbpresent": { "id": "c0be19b4-598e-49af-ba9b-29bfc7284b3c" }, "slideshow": { "slide_type": "-" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 2 6]\n", " [-4 0]]\n" ] } ], "source": [ "B = 2 * np.array([[1,3], [-2,0]])\n", "print(B)" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "58b19dac-35d9-4626-9264-a348b84aa711" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Dot Product\n", "\n", "* A type of multiplication between vectors\n", "* Let $\\mathbf{x}, \\mathbf{y}$ be vectors of same size ($\\mathbf{x}, \\mathbf{y} \\in \\mathbb{R}^n$). \n", "\n", "* Their dot product is\n", " $$\n", " \\begin{align*}\n", " \\mathbf{x}^T \\mathbf{y} \n", " &= \\sum_{i=1}^n x_i y_i \\\\\n", " &=\n", " \\begin{bmatrix}x_1 & x_2 & \\ldots & x_n \\end{bmatrix}\n", " \\begin{bmatrix}y_1 \\\\ y_2 \\\\ \\vdots \\\\ y_n \\end{bmatrix}\n", " \\end{align*}\n", " $$" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "nbpresent": { "id": "fa1ed065-713e-4b16-9bf2-058d20fb0222" }, "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "ename": "ValueError", "evalue": "shapes (3,) and (4,) not aligned: 3 (dim 0) != 4 (dim 0)", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mValueError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 2\u001b[0m \u001b[0my\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marray\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m4\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m5\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m6\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 3\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 4\u001b[0;31m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0my\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m", "\u001b[0;31mValueError\u001b[0m: shapes (3,) and (4,) not aligned: 3 (dim 0) != 4 (dim 0)" ] } ], "source": [ "x = np.array([1,2,3])\n", "y = np.array([2,4,5])\n", "\n", "print(np.dot(x, y))" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "1c85bcac-de56-40d3-af5e-1e7e027a93d6" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Matrix-Matrix Multiplication\n", "\n", "Can multiply two matrices **if their inner dimensions match**\n", "\n", "$$\n", " A \\in \\mathbb{R}^{m \\times n}, \n", " B \\in \\mathbb{R}^{n \\times p}\n", "$$\n", "\n", "$$\n", " C = AB \\quad \\in \\mathbb{R}^{m \\times p}\n", "$$\n", "\n", "The product has entries\n", "$$\n", " C_{ij} = \\sum_{k=1}^n A_{ik} B_{kj}\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "c79e9614-b6a2-43ff-8413-7599d92ebcd6" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Matrix-Matrix Multiplication\n", "\n", "$$\n", "C_{ij} = \\sum_{k=1}^n A_{ik} B_{kj}\n", "$$\n", "\n", "Dot product of $i$th row of $A$ and $j$th column of $B$.\n", "\n", "$\\newcommand{\\r}{\\mathbf}$\n", "\n", "$$\n", "\\begin{bmatrix}\n", "c_{11} & c_{12} & c_{13}\\\\\n", "c_{21} & c_{22} & c_{23}\\\\\n", "c_{31} & \\r{c_{32}} & c_{33}\\\\\n", "c_{41} & c_{42} & c_{43}\n", "\\end{bmatrix}\n", "=\n", "\\begin{bmatrix}\n", "a_{11} & a_{12} \\\\\n", "a_{21} & a_{22} \\\\\n", "\\r{a_{31}} & \\r{a_{32}} \\\\\n", "a_{41} & a_{42} \\\\\n", "\\end{bmatrix}\n", "\\begin{bmatrix}\n", "b_{11} & \\r{b_{12}} & b_{13} \\\\\n", "b_{21} & \\r{b_{22}} & b_{23} \n", "\\end{bmatrix}\n", "$$\n", "\n", "$$\n", "c_{32} = a_{31}b_{12} + a_{32}b_{22}\n", "$$\n" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "cd5ffdba-6e4d-4c09-95f3-abe15d379dd1" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Example\n", "\n", "$$\n", "A = \n", "\\begin{bmatrix}\n", "1 & -1 \\\\ \n", "0 & 3 \n", "\\end{bmatrix}, \\quad\n", "B = \n", "\\begin{bmatrix}\n", "3 & 2 \\\\\n", "-1 & 0\n", "\\end{bmatrix}\n", "$$\n", "\n", "$$\n", "AB = \\begin{bmatrix}\n", "1 & -1 \\\\ \n", "0 & 3 \n", "\\end{bmatrix}\n", "\\begin{bmatrix}\n", "3 & 2 \\\\\n", "-1 & 0\n", "\\end{bmatrix}\n", "= \n", "$$\n" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "nbpresent": { "id": "0d5c6005-cfeb-4892-bb07-92f2de2cb98b" }, "slideshow": { "slide_type": "-" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 3 -2]\n", " [ 0 0]]\n" ] } ], "source": [ "A = np.array([[1, -1], [0, 3]])\n", "B = np.array([[3, 2], [-1, 0]])\n", "print(A * B) # NOT matrix multiplication" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "nbpresent": { "id": "0349337c-7d36-4c9c-9b79-7136ae5add95" }, "slideshow": { "slide_type": "-" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 4 2]\n", " [-3 0]]\n", "[[ 4 2]\n", " [-3 0]]\n" ] } ], "source": [ "# OH NO! A*B gives elementwise multiplication!\n", "\n", "# Use np.dot for matrix multiplication\n", "print(np.dot(A, B)) \n", "\n", "print(A.dot(B))" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "15ec1f58-35b2-4f51-a433-715b0944f1f9" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Multiplication Properties\n", "\n", "* Associative\n", " $$\n", " (AB)C = A(BC)\n", " $$\n", "* Distributive\n", " $$ A(B+C) = AB + AC $$\n", " $$ (B+C)D = BD + CD $$\n", "* **Not commutative**\n", " $$\\r{AB \\neq BA}$$\n" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "b14e5fe8-84d2-423b-a98b-b8d2f4d4387f" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Matrix-Vector Multiplication\n", "\n", "A (worthy) special case of matrix-matrix multiplication:\n", "\n", "$$\n", "A \\in \\mathbb{R}^{m \\times n}, \\quad \\mathbf{x} \\in \\mathbb{R}^n\n", "$$\n", "\n", "$$\n", "\\mathbf{y} = A\\mathbf{x} \\in \\mathbb{R}^{m}\n", "$$\n", "\n", "Definition\n", "$$\n", "y_i = \\sum_{j=1}^n A_{ij} x_j\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "2bca12c8-4b46-47ed-9887-cf232cf24b8d" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Matrix-Vector Multiplication\n", "\n", "$$\n", "y_i = \\sum_{j=1}^n A_{ij} x_j\n", "$$\n", "\n", "\n", "$$\n", "\\begin{bmatrix}\n", "y_1 \\\\ \n", "y_2 \\\\\n", "\\r{y_3} \\\\\n", "y_4\n", "\\end{bmatrix}\n", " =\n", " \\begin{bmatrix}\n", " a_{11} & a_{12} \\\\\n", " a_{21} & a_{22} \\\\\n", " \\r{a_{31}} & \\r{a_{32}} \\\\\n", " a_{41} & a_{42}\n", " \\end{bmatrix}\n", " \\begin{bmatrix}\n", " \\r{x_1} \\\\\n", " \\r{x_2}\n", " \\end{bmatrix}\n", "$$\n", "\n", "$$\n", "y_3 = a_{31} x_1 + a_{32} x_2\n", "$$\n" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "712f78bf-e8b4-4b88-a1fc-daf51a09b60c" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Example\n", "\n", "$$\n", "A = \n", "\\begin{bmatrix}\n", " 1 & -1 \\\\ \n", " 0 & 3 \n", "\\end{bmatrix}, \n", "\\quad \n", "\\mathbf{x} = \\begin{bmatrix}1 \\\\ -1\\end{bmatrix}\n", " \\quad\n", " \\mathbf{z} = \\begin{bmatrix}8 \\\\ 1.5\\end{bmatrix}\n", "$$\n", "\n", "\n", "$$\n", "A \\mathbf{x} = \n", " \\begin{bmatrix}1 & -1 \\\\ 0 & 3 \\end{bmatrix}\n", " \\begin{bmatrix}1 \\\\ -1\\end{bmatrix}\n", " = \n", "$$\n", "\n", "$$\n", "A \\mathbf{z} = \\qquad \\qquad \\qquad \\quad\n", "$$" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "nbpresent": { "id": "13e0c2ea-bde5-483d-b50f-62312aff62a6" }, "slideshow": { "slide_type": "-" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 2 -3]\n", "[6.5 4.5]\n" ] } ], "source": [ "A = np.array([[1, -1], [0, 3]])\n", "\n", "x = np.array([1, -1])\n", "z = np.array([8, 1.5])\n", "\n", "print(np.dot(A, x))\n", "print(np.dot(A, z))" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[ 1 -4]\n" ] } ], "source": [ "A = np.array([[1, -1], [0, 3]])\n", "x = np.array([1, -1])\n", "\n", "print(np.dot(x,A))" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "nbpresent": { "id": "13e0c2ea-bde5-483d-b50f-62312aff62a6" }, "slideshow": { "slide_type": "-" } }, "outputs": [ { "ename": "ValueError", "evalue": "shapes (2,2) and (1,2) not aligned: 2 (dim 1) != 1 (dim 0)", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mValueError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 6\u001b[0m \u001b[0mx_colvec\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marray\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 7\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 8\u001b[0;31m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mA\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx_rowvec\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;31m# error\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 9\u001b[0m \u001b[0;31m#print(np.dot(A, x_colvec))\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 10\u001b[0m \u001b[0;31m#print(np.dot(x_rowvec, A))\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;31mValueError\u001b[0m: shapes (2,2) and (1,2) not aligned: 2 (dim 1) != 1 (dim 0)" ] } ], "source": [ "A = np.array([[1, -1], [0, 3]])\n", "\n", "# A vector can also be represented as a 2D array. In this case \n", "# you must be careful about whether it is a row or column vector\n", "x_rowvec = np.array([[1, -1]])\n", "x_colvec = np.array([[1], [-1]])\n", "\n", "print(np.dot(A, x_rowvec)) # error\n", "#print(np.dot(A, x_colvec))\n", "#print(np.dot(x_rowvec, A))" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "df982e8e-91da-4290-9346-f0877ab6a042" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Transpose\n", "\n", "Transposition of a matrix swaps the rows and columns\n", "$$\n", "A = \\begin{bmatrix}1 & -1 \\\\ 0 & 3 \\end{bmatrix}, \n", "\\quad\n", "A^T = \\begin{bmatrix}1 & 0 \\\\ -1 & 3 \\end{bmatrix}.\n", "$$\n", "\n", "**Definition**:\n", "\n", "* Let $A \\in \\mathbb{R}^{m \\times n}$\n", "* The **transpose** $A^T \\in \\mathbb{R}^{n \\times m}$ has entries\n", "\n", "$$\n", "(A^T)_{ij} = A_{ji}.\n", "$$\n" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "54ee526c-4d39-4b1a-b75f-f61be931c900" }, "slideshow": { "slide_type": "subslide" } }, "source": [ "# Example\n", "\n", "$\\newcommand{\\b}{\\mathbf}$\n", "\n", "$$\n", "A = \\begin{bmatrix} 3 & 2 \\\\ -1 & 0 \\\\ 1 & 4 \\end{bmatrix}\\qquad\n", "A^T = \n", "\\begin{bmatrix} 3 & -1 & 1 \\\\ 2 & 0 & 4 \\end{bmatrix}\n", "$$\n", "\n", "\n", "\n", "$$\n", "\\b{x} = \\begin{bmatrix} 1 \\\\ -3 \\\\ 2\\end{bmatrix}\n", "\\qquad\n", "\\b{x}^T = \n", "\\begin{bmatrix} 1 & -3 & 2\\end{bmatrix}\n", "$$" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "nbpresent": { "id": "056c0cfd-c621-4803-bfc2-4fbf10bededf" }, "slideshow": { "slide_type": "-" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 3 -1 1]\n", " [ 2 0 4]]\n" ] } ], "source": [ "A = np.array([[3, 2], [-1, 0], [1, 4]])\n", "\n", "print(A.T) # numpy array has a transpose property" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "7fb2b059-98bf-48b8-b40d-4f2330e74a80" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Vector Norm\n", "\n", "The *norm* of a vector is\n", " $$\n", " \\begin{align*}\n", " \\| \\b{x} \\| &= \\sqrt{x_1^2 + x_2^2 + \\ldots + x_n^2} \\\\\n", " &= \\sqrt{\\b{x}^T \\b{x}}\n", " \\end{align*}\n", " $$\n", "\n", "Geometric interpretation: length of the vector \n" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "ed1f4a24-b662-4e0e-a450-1b243e7595a6" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Transpose Properties\n", "\n", "* Transpose of transpose\n", "\n", " $$ (A^T)^T = A $$\n", "\n", "* Transpose of sum\n", "\n", " $$ (A+B)^T = A^T + B^T $$\n", "\n", "* Transpose of product\n", "\n", " $$ (AB)^T = B^T A^T $$\n", "\n" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "6435e1f9-aca2-4265-8ae0-b90ab9d4302a" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Identity\n", "\n", "The identity matrix $I \\in \\mathbb{R}^{n \\times n}$ has entries\n", "\n", "$$\n", "I_{ij} = \\begin{cases}1 & i=j \\\\ 0 & i \\neq j\\end{cases},\n", "$$\n", "\n", "$$\n", "I_{1 \\times 1} = [1], \\qquad\n", " I_{2 \\times 2} = \\begin{bmatrix}1 & 0 \\\\ 0 & 1\\end{bmatrix},\n", " \\qquad \n", " I_{3 \\times 3} = \\begin{bmatrix}1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1\\end{bmatrix}.\n", "$$\n", "\n", "For any $A, B$ of appropriate dimensions\n", "$$\n", "\\begin{align*}\n", " IA &= A \\\\\n", " BI &= B\n", " \\end{align*}\n", " $$" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "nbpresent": { "id": "c7e8b795-a3a2-4232-bcb7-4b4737e02aa4" }, "slideshow": { "slide_type": "subslide" } }, "outputs": [], "source": [ "# Use numpy.eye to create identity matrices of different dimensions\n", "\n", "I = np.eye(1)\n", "print(I)\n", "\n", "I = np.eye(2)\n", "print(I)\n", "\n", "I = np.eye(3)\n", "print(I)" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "nbpresent": { "id": "c12378cf-3605-4a0c-894e-6017e648727c" }, "slideshow": { "slide_type": "subslide" } }, "outputs": [], "source": [ "I = np.eye(2)\n", "A = np.array([[1,2], [3,4]])\n", "\n", "print(A)\n", "print(np.dot(A, I))\n", "print(np.dot(I, A))" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "ffb6f363-6fa0-4bee-92bd-db142814787a" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Inverse\n", "\n", "The inverse $A^{-1} \\in \\mathbb{R}^{n \\times n}$ of a ***square*** matrix $A \\in \\mathbb{R}^{n \\times n}$ satisfies\n", "$$\n", "AA^{-1} = I = A^{-1}A\n", "$$\n", "\n", "Compare to division of scalars\n", "$$\n", "x x^{-1} = 1 = x^{-1} x\n", "$$\n", "\n", "**Not all matrices are invertible**\n", "\n", "* E.g., $A$ not square, $A = [0]$, $A = \\begin{bmatrix}0 & 0\\\\ 0 & 0\\end{bmatrix}$, many more" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "7b7cf2c5-4712-4a1c-b916-e78ee3f15440" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Inverse\n", "\n", "$$\n", "A = \n", "\\begin{bmatrix}1 & 0 \\\\ 0 & 2\\end{bmatrix}, \\qquad\n", "B = \\begin{bmatrix} 1 & 0 \\\\ 0 & \\frac{1}{2}\\end{bmatrix}\n", "$$\n", "\n", "Is $B$ the inverse of $A$?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "$$\n", "A = \\begin{bmatrix}1 & -1 \\\\ 0 & 3 \\end{bmatrix} \\qquad \n", "A^{-1} = \\begin{bmatrix}1 & \\frac{1}{3} \\\\ 0 & \\frac{1}{3}\\end{bmatrix}\n", "$$\n", "Verify on your own.\n" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "nbpresent": { "id": "ea7f0954-eabf-4fa5-aa61-ac11fa5dbd43" }, "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[1. 1.]\n", " [0. 1.]]\n" ] }, { "ename": "ZeroDivisionError", "evalue": "division by zero", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mZeroDivisionError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 6\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlinalg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0minv\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mA\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 7\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 8\u001b[0;31m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m/\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m", "\u001b[0;31mZeroDivisionError\u001b[0m: division by zero" ] } ], "source": [ "import numpy as np\n", "\n", "A = np.array([[1, -1], [0, 1]])\n", "\n", "# Use numpy.linalg.inv to invert a matrix\n", "print(np.linalg.inv(A))\n", "\n", "print(1/0)" ] }, { "cell_type": "markdown", "metadata": { "nbpresent": { "id": "aae603e1-6e1a-45ae-bac3-59bbf3acea7d" }, "slideshow": { "slide_type": "slide" } }, "source": [ "# Inverse Properties\n", "\n", "* Inverse of inverse\n", "\n", " $$ (A^{-1})^{-1} = A $$\n", " \n", "* Inverse of product\n", "\n", " $$(AB)^{-1} = B^{-1}A^{-1} $$\n", "\n", "* Inverse of transpose \n", " $$ (A^{-1})^T = (A^T)^{-1} := A^{-T} $$\n" ] } ], "metadata": { "anaconda-cloud": {}, "celltoolbar": "Slideshow", "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.1" }, "nbpresent": { "slides": { "075ca6a0-0a22-4236-b1d2-499bed6095d9": { "id": "075ca6a0-0a22-4236-b1d2-499bed6095d9", "prev": "c16392bb-f5fe-4417-b6fc-273e88d42cc5", "regions": { "c6a1861c-0c36-4d8d-b83e-135e4edee00e": { "attrs": { "height": 0.8, "width": 0.8, "x": 0.1, "y": 0.1 }, "content": { "cell": 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