[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572367-105":53,"doc-detail-572367-en":126},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","18085-linear-algebra-cheat-sheet","18.085 - Linear Algebra - Cheat Sheet","","18.085 Linear Algebra cheat sheet (Spring 2014) covering core results for rectangular and square matrices. Includes transpose rules, conditions for positive semi-definite and positive definite forms, and key definitions and formulas for eigenvalues, eigenvectors, trace and determinant relationships. Summarizes LU factorization and invertibility criteria, plus inverse identities and symmetric, skew-symmetric spectral properties. Covers orthonormal diagonalization, LDLT and Cholesky decompositions, and tests for positive definiteness or semi-definiteness.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":51,"@type":70,"position":76},"https://docshare.wps.com/template/general/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/18085-linear-algebra-cheat-sheet/572367/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/18085-linear-algebra-cheat-sheet/572367.png","ImageObject",442,249,{"name":88,"@type":89},"Putri","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-19","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":79},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"How do transposes interact with matrix products, and what are the key properties of ATA type matrices?","Question",{"text":108,"@type":109},"Use (AB)T = BTAT. The matrix ATA is always square, symmetric, and positive semi-definite; with linearly independent columns of A it becomes positive definite. Similar statements hold for ATCA with C diagonal positive.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"What characterizes eigenvalues and eigenvectors for square matrices, and how do trace and determinant relate to them?",{"text":113,"@type":109},"Eigenvalues and eigenvectors are defined for square matrices via Av = λv, where eigenvalues may be complex and a matrix can be defective with fewer than n eigenvectors. If eigenvalues are counted with multiplicity, then tr(A) equals the sum of eigenvalues and det(A) equals the product of eigenvalues.",{"name":115,"@type":106,"acceptedAnswer":116},"What are the main criteria for invertibility and how can inverses be computed using LU or spectral information?",{"text":117,"@type":109},"A is invertible if it has n nonzero pivots, or equivalently det(A) ≠ 0, or all eigenvalues are nonzero, or its columns are linearly independent, or Au = 0 implies u = 0. Conversely, violating these makes A singular. In general Au = f has the unique solution u = A^{-1}f, and if A = SΛS^{-1} then A^{-1} uses the same eigenvectors with inverse eigenvalues.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},572367,1792088758,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":79,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":73,"language":135,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":61,"update_tm":139,"read_time":9},962085571259,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","18.085 :: Linear algebra cheat sheet :: Spring 2014  \n1. Rectangular m-by-n matrices  \n􀀏 Rule for transposes:  \n(AB)T = BT AT  \n􀀏 The matrix ATA is always square, symmetric, and positive semi-definite. If in addition A has linearly independent columns, then ATA is positive definite.  \n􀀏 Take C a diagonal matrix with positive elements. Then AT CA is always square, symmetric, and positive semi-definite. If in addition A has linearly independent columns, then AT CA is positive definite.  \n2. Square n-by-n matrices  \n􀀏 Eigenvalues and eigenvectors are defined only for square matrices:  \nAv = 􀀕v  \nThe eigenvalues may be complex. There may be fewer than n eigenvectors (in that case we say the matrix is defective) .  \n􀀏 In matrix form, we can always write  \nAS = S􀀃 ;  \nwhere S has the eigenvectors in the columns (S could be a rectangular matrix), and 􀀃 is diagonal with the eigenvalues on the diagonal.  \n􀀏 If A has n eigenvectors (a full set), then S is square and invertible, and A = S􀀃S􀀀1:  \n􀀏 If eigenvalues are counted with their multiplicity, then  \ntr (A) = A 11 + : : : + Ann = 􀀕 1 + : : : + 􀀕n:  \n􀀏 If eigenvalues are counted with their multiplicity, then  \ndet (A) = 􀀕 1 􀀂 : : : 􀀂 􀀕n:  \n􀀏 Addition of a multiple of the identity shifts the eigenvalues:  \n􀀕j (A + cI) = 􀀕j (A) + c:  \nNo such rule exists in general when adding A + B.  \n􀀏 Gaussian elimination gives  \nA = LU:  \nL is lower triangular with ones on the diagonal, and with the elimination multipliers below the diagonal. U is upper triangular with the pivots on the diagonal.  \n􀀏 A is invertible if either of the following criteria is satisfied: there are n nonzero pivots; the determinant is not zero; the eigenvalues are all nonzero; the columns are linearly independent; the only way to have Au = 0 is if u = 0 .  \n􀀏 Conversely, if either of these criteria is violated, then the matrix is singular (not invertible) .  \n3. Invertible matrices.  \n􀀏 The system Au = f has a unique solution u = A 􀀀1f.  \n􀀏 (AB)􀀀1 = B 􀀀1A 􀀀1 .  \n􀀏 (AT ) 􀀀1 = (A􀀀1)T:  \n􀀏 If A = S􀀃S􀀀1, then A 􀀀1 = S􀀃􀀀1S 􀀀1 (same eigenvectors, inverse eigenvalues) .  \n4. Symmetric matrices (AT = A.)  \n􀀏 Eigenvalues are real, eigenvectors are orthogonal, and there are always n eigenvectors (full set) .(Precision concerning eigenvectors: accidentally, they may not come as orthogonal. But that case always corresponds to a multiple eigenvalue. There exists another choice of eigenvectors such that they are orthogonal.)  \n􀀏 Can write  \nA = Q􀀃QT ;  \nwhere Q􀀀1 = QT (orthonormal matrix) .  \n􀀏 Can modify A = LU into A = LDLT with diagonal D , and the pivots on the diagonal of D.  \n􀀏 Can define positive definite matrices only in the symmetric case. A symmetric matrix is positive definite if either of the following criteria holds: all the pivots are positive; all the eigenvalues are positive; all the upper-left determinants are positive; or xTAx > 0 unless x = 0 .  \n􀀏 Similar definition for positive semi-definite. The creiteria are: all the pivots are nonnegative; all the eigenvalues are nonnegative; all the upper-left determinants are nonnegative; or xTAx 􀀕 0.  \n􀀏 If A is positive definite, we have the Cholesky decomposition  \nA = RTR;  \nwhich corresponds to the choice R = pDLT .  \n5. Skew-symmetric matrices (AT = 􀀀A.)  \n􀀏 Eigenvalues are purely imaginary, eigenvectors are orthogonal, and there are always n eigenvectors (full set) .","cbCaicGwGJZArf2L","https://ap.wps.com/l/cbCaicGwGJZArf2L","pdf",71457,"English","# Rectangular m-by-n matrices\n## Rules for transposes and positive (semi-)definite forms\n# Square n-by-n matrices\n## Eigenvalues, eigenvectors, trace/determinant, and shifts\n## LU factorization and invertibility criteria\n# Invertible matrices\n## Solving Au=f and inverse identities\n# Symmetric matrices\n## Orthogonality, diagonalization, LDLT, and Cholesky\n## Positive definite and positive semi-definite tests\n# Skew-symmetric matrices\n## Spectral properties","[{\"question\":\"How do transposes interact with matrix products, and what are the key properties of ATA type matrices?\",\"answer\":\"Use (AB)T = BTAT. The matrix ATA is always square, symmetric, and positive semi-definite; with linearly independent columns of A it becomes positive definite. Similar statements hold for ATCA with C diagonal positive.\"},{\"question\":\"What characterizes eigenvalues and eigenvectors for square matrices, and how do trace and determinant relate to them?\",\"answer\":\"Eigenvalues and eigenvectors are defined for square matrices via Av = λv, where eigenvalues may be complex and a matrix can be defective with fewer than n eigenvectors. If eigenvalues are counted with multiplicity, then tr(A) equals the sum of eigenvalues and det(A) equals the product of eigenvalues.\"},{\"question\":\"What are the main criteria for invertibility and how can inverses be computed using LU or spectral information?\",\"answer\":\"A is invertible if it has n nonzero pivots, or equivalently det(A) ≠ 0, or all eigenvalues are nonzero, or its columns are linearly independent, or Au = 0 implies u = 0. Conversely, violating these makes A singular. In general Au = f has the unique solution u = A^{-1}f, and if A = SΛS^{-1} then A^{-1} uses the same eigenvectors with inverse eigenvalues.\"}]","18.085 - Linear Algebra - Cheat Sheet | PDF",1791926986]