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Linear algebra cheat-sheet Laurent Lessard University of Wisconsin–Madison Last updated: October 12, 2016 Matrix basics m×n A matrix is an array of numbers. A ∈ R means that: a . . . a 11 1n . . . A= . .. . (mrows and n columns) . . am1 ... amn Two matrices can be multiplied if inner dimensions agree: n C = A B where cij =Xaikbkj (m×p) (m×n)(n×p) k=1 T Transpose: The transpose operator A swaps rows and m×n T n×m T columns. If A ∈ R then A ∈ R and (A )ij = Aji. • T T (A ) =A. • T T T (AB) =B A . 2 Matrix basics (cont’d) Vector products. If x,y ∈ Rn are column vectors, • T The inner product is x y ∈ R (a.k.a. dot product) • T n×n The outer product is xy ∈R . These are just ordinary matrix multiplications! n×n n×n Inverse. Let A ∈ R (square). If there exists B ∈ R with AB =I or BA = I (if one holds, then the other holds with the −1 same B) then B is called the inverse of A, denoted B = A . Some properties of the matrix inverse: • −1 A is unique if it exists. • −1 −1 (A ) =A. • −1 T T −1 (A ) =(A ) . • −1 −1 −1 (AB) =B A . 3 Vector norms n A norm k·k : R → R is a function satisfying the properties: • kxk = 0 if and only if x = 0 (definiteness) • kcxk = |c|kxk for all c ∈ R (homogeneity) • kx +yk ≤ kxk+kyk (triangle inequality) Common examples of norms: 1 x2 • kxk =|x |+···+|x | (the 1-norm) 1 1 n • kxk =px2+···+x2 (the 2-norm) x1 2 1 n -1 1 • kxk =max |x | (max-norm) ∞ 1≤i≤n i -1 Properties of the 2-norm (Euclidean norm) norm ball: • If you see kxk, think kxk (it’s the default) {x | kxk = 1} 2 • xTx = kxk2 • xTy ≤ kxkkyk (Cauchy-Schwarz inequality) 4
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