Linear algebra cheat-sheet - Laurent Lessard

Linear algebra cheat-sheetLaurent LessardUniversity of Wisconsin MadisonLast updated: October 12, 2016 Matrix basicsA matrix is an array of Rm nmeans that:A= (mrows andncolumns)Two matrices can be multiplied if inner dimensions agree:C(m p)=A(m n)B(n p)wherecij=n k=1aikbkjTranspose: The transpose operatorATswaps rows andcolumns. IfA Rm nthenAT Rn mand (AT)ij=Aji. (AT)T=A. (AB)T= basics (cont d)Vector products. Ifx,y Rnare column vectors, Theinner productisxTy R( dot product) Theouter productisxyT Rn are just ordinary matrix multiplications!Inverse. LetA Rn n(square). If there existsB Rn nwithAB=IorBA=I(if one holds, then the other holds with thesameB) thenBis called theinverseofA, denotedB=A properties of the matrix inverse: A 1is unique if it exists.

Oct 12, 2016 · The singular value decomposition Every A 2Rm n can be factored as A (m n) = U 1 (m r) 1 (r r) VT 1 (n r)T (economy SVD) U 1 is orthogonal, its columns are the left singular vectors V 1 is orthogonal, its columns are the right singular vectors 1 is diagonal. ˙ 1 ˙ r >0 are the singular values Complete the orthogonal matrices so they become ...

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