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lecture2

# lecture2 - CSE6643/MATH6643:Numerical Linear Algebra Haesun...

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CSE6643/MATH6643:Numerical Linear Algebra Haesun Park [email protected] School of Computational Science and Engineering College of Computing Georgia Institute of Technology Atlanta, GA 30332, USA Lecture 2

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Introduction 1 Introduction 1.1 Topics 1. Linear system solving Ax = b 2. Least square problem Ax b min x bardbl Ax b bardbl 2 3. Eigenvalue problem Ax = λx, ( x negationslash = 0) 4. Linear programming min f ( x ) s.t. Ax = b where f ( x ) is a linear function. CSE6643/MATH6643:Numerical Linear Algebra – p.1/18
Introduction 1.2 Sherman-Morrison formula Sherman-Morrison formula: A is a nonsingular, and v T A 1 u negationslash = 1 then ( A + uv T ) 1 = A 1 A 1 uv T A 1 1 + v T A 1 u where we call uv T rank 1 update. Proof: ( A + uv T ) A 1 A - 1 uv T A - 1 1+ v T A - 1 u = I Iuv T A - 1 1+ v T A - 1 u + uv T A 1 uv T A - 1 uv T A - 1 1+ v T A - 1 u = I uv T A - 1 +( v T A - 1 u ) uv T A - 1 1+ v T A - 1 u + uv T A 1 = I CSE6643/MATH6643:Numerical Linear Algebra – p.2/18

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Introduction Sherman-Morrison-Woodbury formula: A is nonsingular and A R n × n . U R n × k , V R n × k and I k + V T A 1 U is nonsingular, then ( A + UV T ) 1 = A 1 A 1 U ( I k + V T A 1 U ) 1 V A 1 Example: A = 0 B @ 1 2 3 4 5 6 7 8 10 1 C A , b = 0 B @ 1 1 1 1 C A Assume A (1 , 1) changes to 3 , i.e. ˆ A = 0 B @ 3 2 3 4 5 6 7 8 10 1 C A = A + 0 B @ 2 0 0 1 C A 1 0 0 then ˆ x = ˆ A 1 b = A 1 A - 1 u T vA - 1 1+ v T Au b CSE6643/MATH6643:Numerical Linear Algebra – p.3/18
Introduction 1.3 Misc Outer product u · v T : n × n Inner product : u T · v , 1 × 1 Range ( A ) or Ran ( A ) Space spanned by columns of A rank ( A ) = dim( Range ( A )) orthogonal vectors u , v : v t u = bardbl v bardbl 2 bardbl u bardbl 2 cos θ = 0 orthonormal vectors u , v bardbl u bardbl 2 = 1 , bardbl v bardbl 2 = 1 orthogonal matrix A R n × n satisfies A T A = I n : A T A = 2 6 6 4 a T 1 . . . a T n 3 7 7 5 h a 1 · · · a n i = ( a T i a j ) = I n CSE6643/MATH6643:Numerical Linear Algebra – p.4/18

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Introduction Special matrix: diagonal matrix triangle matrix: upper and lower Hessenberg matrix: upper and lower
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lecture2 - CSE6643/MATH6643:Numerical Linear Algebra Haesun...

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