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The material develops Jacobians of matrix functions, Kronecker-product representations, and practical finite-difference approximations with accuracy and roundoff analysis. It then connects optimization, nonlinear root-finding, and adjoint (reverse-mode) differentiation, culminating in forward/reverse automatic differentiation, ODE sensitivities, calculus of variations, and derivatives for random functions with stochastic methods and the reparameterization trick.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/matrix-calculus-for-machine-learning-and-beyond-lecture-notes/290834/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/matrix-calculus-for-machine-learning-and-beyond-lecture-notes/290834.png","ImageObject",300,407,{"name":92,"@type":93},"Skyler","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-02","2026-09-17",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",17,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"How do the notes interpret derivatives for matrix and vector functions?","Question",{"text":113,"@type":114},"They treat derivatives as linear operators and build rules for matrix and vector product differentiation, extending scalar-variable calculus to vector-valued settings.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What is covered about Jacobians in this text?",{"text":118,"@type":114},"The notes derive Jacobians of matrix functions, include worked examples such as a two-by-two matrix-square case, and explain Kronecker-product notation for expressing the Jacobian and its computational implications.",{"name":120,"@type":111,"acceptedAnswer":121},"How are automatic differentiation and adjoint differentiation related?",{"text":122,"@type":114},"They introduce reverse-mode “adjoint” differentiation and then connect it to forward- and reverse-mode automatic differentiation, including how derivatives propagate through computational graphs.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},290834,1789642709,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":130,"read_time":145,"preview_page_count":4},2336464648746,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Matrix Calculus  \n(for Machine Learning and Beyond)  \nLecturers: Alan Edelman and Steven G. Johnson  \nNotes by Paige Bright, Alan Edelman, and Steven G. Johnson  \narXiv :2501 . 14787v1 [math .HO] 7 Jan 2025  \nBased on MIT course 18.S096 (now 18.063) in IAP 2023  \nContents  \nIntroduction 4  \n1 Overview and Motivation 5  \n1.1 Applications .................................................. 5  \n1.2 First Derivatives ............................................... 6  \n1.3 Intro: Matrix and Vector Product Rule .................................. 8  \n2 Derivatives as Linear Operators 9  \n2.1 Revisiting single-variable calculus ...................................... 9  \n2.2 Linear operators ............................................... 10  \n2.2.1 Directional derivatives ........................................ 12  \n2.3 Revisiting multivariable calculus, Part 1: Scalar-valued functions .................... 12  \n2.4 Revisiting multivariable calculus, Part 2: Vector-valued functions .................... 14  \n2.5 The Chain Rule ................................................ 16  \n2.5.1 Cost of Matrix Multiplication .................................... 17  \n2.6 Beyond Multi-Variable Derivatives ..................................... 18  \n3 Jacobians of Matrix Functions 20  \n3.1 Derivatives of matrix functions: Linear operators ............................. 20  \n3.2 A simple example: The two-by-two matrix-square function ........................ 21  \n3.2.1 The matrix-squaring four-by-four Jacobian matrix ........................ 23  \n3.3 Kronecker Products ............................................. 23  \n3.3.1 Key Kronecker-product identity ................................... 25  \n3.3.2 The Jacobian in Kronecker-product notation ........................... 26  \n3.3.3 The computational cost of Kronecker products .......................... 27  \n4 Finite-Difference Approximations 29  \n4.1 Why compute derivatives approximately instead of exactly? ....................... 29  \n4.2 Finite-Difference Approximations: Easy Version .............................. 29  \n4.3 Example: Matrix squaring .......................................... 30  \n4.4 Accuracy of Finite Differences ........................................ 31  \n4.5 Order of accuracy ............................................... 32  \n4.6 Roundoff error ................................................ 32  \n4.7 Other finite-difference methods ....................................... 32  \n5 Derivatives in General Vector Spaces 34  \n5.1 A Simple Matrix Dot Product and Norm .................................. 34  \n5.2 Derivatives, Norms, and Banach spaces ................................... 37  \n6 Nonlinear Root-Finding, Optimization, and Adjoint Differentiation 39  \n6.1 Newton’s Method ............................................... 39  \n6.1.1 Scalar Functions ........................................... 39  \n6.1.2 Multidimensional Functions ..................................... 39  \n6.2 Optimization ................................................. 40  \n6.2.1 Nonlinear Optimization ....................................... 40  \n6.2.2 Engineering/Physical Optimization ................................. 42  \n6.3 Reverse-mode “Adjoint” Differentiation ................................... 42  \n6.3.1 Nonlinear equations ......................................... 44  \n6.3.2 Adjoint methods and AD ...................................... 44  \n6.3.3 Adjoint-method example ....................................... 44  \n7 Derivative of Matrix Determinant and Inverse 47  \n7.1 Two Derivations ............................................... 47  \n7.2 Applications .................................................. 48  \n7.2.1 Characteristic Polynomial ...................................... 48  \n7.2.2 The Logarithmic Derivative ..................................... 48  \n7.3 Jacobian of the Inverse ............................................ 49  \n8 Forward and Reverse-Mode Automatic Differentiation 50  \n8.1 Automatic ","cbCaisdZdPL03tIg","https://ap.wps.com/l/cbCaisdZdPL03tIg","pdf",1884010,101,"English","# Introduction\n## Overview and Motivation\n## Applications\n## First Derivatives\n## Derivatives as Linear Operators\n## Jacobians of Matrix Functions\n## Finite-Difference Approximations\n## Derivatives in General Vector Spaces\n## Nonlinear Root-Finding, Optimization, and Adjoint Differentiation\n## Derivative of Matrix Determinant and Inverse\n## Forward and Reverse-Mode Automatic Differentiation\n## Differentiating ODE solutions\n## Calculus of Variations\n## Derivatives of Random Functions","[{\"question\":\"How do the notes interpret derivatives for matrix and vector functions?\",\"answer\":\"They treat derivatives as linear operators and build rules for matrix and vector product differentiation, extending scalar-variable calculus to vector-valued settings.\"},{\"question\":\"What is covered about Jacobians in this text?\",\"answer\":\"The notes derive Jacobians of matrix functions, include worked examples such as a two-by-two matrix-square case, and explain Kronecker-product notation for expressing the Jacobian and its computational implications.\"},{\"question\":\"How are automatic differentiation and adjoint differentiation related?\",\"answer\":\"They introduce reverse-mode “adjoint” differentiation and then connect it to forward- and reverse-mode automatic differentiation, including how derivatives propagate through computational graphs.\"}]","Matrix Calculus - (for Machine Learning and Beyond) - Lecture Notes | PDF",255]