[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-348122-105":59,"doc-detail-348122-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","matrix-derivatives-crib-sheet","Matrix Derivatives - Crib Sheet","","This document provides a comprehensive and accessible reference guide for matrix and vector calculus, specifically focusing on derivative rules and manipulation techniques. It serves as a practical aid for students and practitioners who need to simplify algebraic expressions before performing differentiation. The content covers essential rules for matrix-vector multiplication, transposition, and distributivity, alongside a side-by-side comparison of common scalar and vector derivative identities to assist with conceptual retention and recall. By clarifying the non-commutative nature of matrix multiplication and providing standard derivative formulas for expressions such as quadratic forms, the guide helps users navigate complex linear algebra operations with greater efficiency and accuracy in their mathematical work.",{"@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":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/matrix-derivatives-crib-sheet/348122/",{"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-derivatives-crib-sheet/348122.png","ImageObject",300,407,{"name":92,"@type":93},"Kyle","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-08","2026-09-22",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",13,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What is the rule for the transpose of the product of two matrices?","Question",{"text":113,"@type":114},"The rule is (AB)^T = B^T A^T, which indicates that the order of the matrices is reversed when applying the transpose.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"Why is it important to remember that matrix multiplication is not commutative?",{"text":118,"@type":114},"Understanding that AB does not equal BA is crucial because it ensures that expressions are simplified or manipulated correctly without violating the fundamental algebraic properties of matrices.",{"name":120,"@type":111,"acceptedAnswer":121},"What is the vector derivative of the expression x^T Bx?",{"text":122,"@type":114},"The vector derivative of x^T Bx with respect to x is 2Bx, assuming B is a constant matrix.","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},348122,1790669081,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"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":8,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":81,"preview_page_count":4},3985741905716,"https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","Matrix derivatives cheat sheet  \nKirsty McNaught  \nOctober 2017  \n1 Matrix/vector manipulation  \nYou should be comfortable with these rules. They will come in handy when you want to simplify an expression before di􀀋erentiating. All bold capitals are matrices, bold lowercase are vectors.  \n\n| Rule | Comments |\n| --- | --- |\n| (AB)T = BT AT | order is reversed, everything is transposed |\n| (aT Bc)T = cT BTa | as above |\n| aTb = bTa | (the result is a scalar, and the transpose of a scalar is itself) |\n| (A + B)C = AC + BC | multiplication is distributive |\n| (a + b)TC = aTC + bTC | as above, with vectors |\n| AB  BA | multiplication is not commutative |\n\n2 Common vector derivatives  \nYou should know these by heart. They are presented alongside similar-looking scalar derivatives to help memory. This doesn't mean matrix derivatives always look just like scalar ones. In these examples, b is a constant scalar, and B is a constant matrix.  \n\n| Scalar derivative f (x) ! ~~ ~~ddfx | Vector derivative f (x) ! ~~ ~~ddfx |\n| --- | --- |\n\n\n| bx ! bbx ! bx2 ! 2xbx2 ! 2bx | xTB ! B\u003Cbr>xTb ! b\u003Cbr>xTx ! 2x\u003Cbr>xT Bx ! 2Bx |\n| --- | --- |\n\nFor a more comprehensive reference, see [https://www.math.uwaterloo.ca/](https://www.math.uwaterloo.ca/) ~hwolkowi/matrixcookbook. pdf","cbCaimQ9WJ3PKh72","https://ap.wps.com/l/cbCaimQ9WJ3PKh72","pdf",114016,"English","# Matrix/vector manipulation\n## Common vector derivatives","[{\"question\":\"What is the rule for the transpose of the product of two matrices?\",\"answer\":\"The rule is (AB)^T = B^T A^T, which indicates that the order of the matrices is reversed when applying the transpose.\"},{\"question\":\"Why is it important to remember that matrix multiplication is not commutative?\",\"answer\":\"Understanding that AB does not equal BA is crucial because it ensures that expressions are simplified or manipulated correctly without violating the fundamental algebraic properties of matrices.\"},{\"question\":\"What is the vector derivative of the expression x^T Bx?\",\"answer\":\"The vector derivative of x^T Bx with respect to x is 2Bx, assuming B is a constant matrix.\"}]","Matrix Derivatives - Crib Sheet | PDF",1790071222]