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Includes trace, norms, linear independence, span and rank, inverse, orthogonal and unitary matrices, column space and null space, projections, determinants, quadratic forms, and eigenvalues/eigenvectors. Also expands into matrix factorizations like eigendecomposition, Cholesky, and singular value decomposition, then covers matrix calculus with gradients, Hessians, convexity, Taylor expansions, least squares, and optimization perspectives.",{"@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/linear-algebra-review-comprehensive-matrix-and-vector-reference/348119/",{"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/linear-algebra-review-comprehensive-matrix-and-vector-reference/348119.png","ImageObject",300,407,{"name":92,"@type":93},"WPS_1786070896","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-04","2026-09-22",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",14,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What topics are covered in the basics and notation section?","Question",{"text":113,"@type":114},"It introduces linear systems via Ax=b, defines matrices and vectors with dimension conventions, and sets notation for matrix entries, rows, columns, and special vectors like all-ones and all-zeros.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How does the document explain different forms of matrix multiplication?",{"text":118,"@type":114},"It derives the product rules for vector-vector inner/outer products, matrix-vector products, and general matrix-matrix multiplication using entries Cij = Σk AikBkj.",{"name":120,"@type":111,"acceptedAnswer":121},"Which matrix factorizations and calculus tools are included for deeper analysis?",{"text":122,"@type":114},"It covers eigendecomposition, Cholesky decomposition, and singular value decomposition, then introduces gradient and Hessian concepts with convexity and Taylor expansions, followed by least squares and eigenvalues as optimization objects.","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},348119,1790528214,{"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":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":145,"read_time":146,"preview_page_count":4},549768072016,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Linear Algebra Review  \nConner DiPaolo  \nContents  \n1 Basics 2  \n1.1 Notation ................................................ 2  \n2 Matrix Multiplication 3  \n2.1 Vector-Vector Products ....................................... 3  \n2.2 Matrix-Vector Products ....................................... 3  \n2.3 Matrix-Matrix Multiplication .................................... 3  \n2.4 Properties of Matrix Multiplication ................................. 4  \n3 Operations and Properties 4  \n3.1 The Identity Matrix and Diagonal Matrices ............................ 4  \n3.2 The Transpose: A> .......................................... 4  \n3.3 Symmetric Matrices ......................................... 4  \n3.4 The Conjugate Transpose: A􀀃 .................................... 4  \n3.5 Hermitian Matrices .......................................... 5  \n3.6 The Trace tr(A) ........................................... 5  \n3.7 Norms ................................................. 5  \n3.8 Linear Independence, Span and Rank ............................... 6  \n3.9 The Inverse A 􀀀1 ........................................... 7  \n3.10 Orthogonal Matrices ......................................... 7  \n3.11 Unitary Matrices ........................................... 7  \n3.12 Columnspace and Nullspace of a Matrix .............................. 8  \n3.13 Projections .............................................. 8  \n3.14 The Determinant ........................................... 8  \n3.15 Quadratic Forms and Positive Semide􀀌nite Matrices ....................... 9  \n3.16 Eigenvalues and Eigenvectors: Ax = 􀀕x .............................. 9  \n3.17 Eigenvalues and Eigenvectors of Symmetric Matrices ....................... 10  \n4 Matrix Factorizations 10  \n4.1 Eigendecomposition: A = X􀀃X􀀀1 ................................. 10  \n4.2 Cholesky Decomposition: A = LL> ................................. 10  \n4.3 Singular Value Decomposition: A = U􀀆V􀀃 ............................ 11  \n5 Matrix Calculus 11  \n5.1 The Gradient: rf .......................................... 12  \n5.2 The Hessian: r2 f .......................................... 12  \n5.2.1 Convexity ........................................... 12  \n5.3 Gradients and Hessians of Quadratic and Linear Functions ................... 12  \n5.3.1 Taylor Expansions: f (x) 􀀙 f (a) + rf(a)>(x 􀀀 a) + ~~1~~2 (x 􀀀 a)>r2 f (x 􀀀 a) ....... 14  \n5.4 Least Squares ............................................. 14  \n5.5 Eigenvalues as Optimization ..................................... 14  \nThis is an adaptation of Stanford's CS229 Linear Algebra Review and Reference 1 , written by Zico Kolterand updated by Chuong Do, and some extra material that will be helpful in the course.  \n1[http://cs229.stanford.edu/section/cs229-linalg.pdf](http://cs229.stanford.edu/section/cs229-linalg.pdf)  \nTo the Reader: If you want a brief and very comprehensive reference/resource of anything Linear Algebra related (including a ton of matrix derivatives at the beginning) check out the Matrix Cookbook 2 . It has over 800 citations in the literature!  \n1 Basics  \nLinear Algebra allows us to interact with linear operators (or systems of linear equations) in a more powerful manner than dealing with equations. For example, consider the linear system  \nx 1 + 2x2 = 5  \n3x1 + 4x2 = 1 :  \nThis can be solved for x 1 and x2 using substitution, but it is convenient (for many reasons) to investigate this system more compactly, as a matrix-vector product. Namely,  \nAx = b  \nwhere  \nA = 􀀔13 24􀀕 and b = 􀀔51􀀕 :  \nHere, A is a matrix and b is a vector. Matrices are linear operators. That is, they obey the following property, where A is a matrix in Rm 􀀂n , x and y are vectors in Rn , and c and d are scalars:  \nA (cx + dy) = cAx + dAy:  \nNote that this implies that  \nA0 = 0 :  \n1.1 Notation  \n- We denote an m 􀀂 n matrix of real numbers A as A 2 Rm 􀀂n (\\A is in R m by n\") . Similarly, to declare am 􀀂 n matrix of complex numbers we say B 2 Cm 􀀂n.  \n- We denote a vector x w","cbCaieUqXuxW8PCA","https://ap.wps.com/l/cbCaieUqXuxW8PCA","pdf",337688,15,"English","# Contents\n## Basics\n## Matrix Multiplication\n## Operations and Properties\n## Matrix Factorizations\n## Matrix Calculus","[{\"question\":\"What topics are covered in the basics and notation section?\",\"answer\":\"It introduces linear systems via Ax=b, defines matrices and vectors with dimension conventions, and sets notation for matrix entries, rows, columns, and special vectors like all-ones and all-zeros.\"},{\"question\":\"How does the document explain different forms of matrix multiplication?\",\"answer\":\"It derives the product rules for vector-vector inner/outer products, matrix-vector products, and general matrix-matrix multiplication using entries Cij = Σk AikBkj.\"},{\"question\":\"Which matrix factorizations and calculus tools are included for deeper analysis?\",\"answer\":\"It covers eigendecomposition, Cholesky decomposition, and singular value decomposition, then introduces gradient and Hessian concepts with convexity and Taylor expansions, followed by least squares and eigenvalues as optimization objects.\"}]","Linear Algebra Review - Comprehensive Matrix and Vector Reference | PDF",1790071216,38]