13 - Matrix inverses Vector norms errors Conditioning...

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Matrix inverses Vector norms & errors Conditioning Matrix Inverse and Condition Dhavide Aruliah UOIT MATH 2070U c D. Aruliah (UOIT) Matrix Inverse and Condition MATH 2070U 1 / 27
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Matrix inverses Vector norms & errors Conditioning Matrix Inverse and Condition 1 Matrix inverses 2 Vector norms and quantifying errors Vector norms Quantifying errors using norms 3 Conditioning of linear equations Matrix norms Condition numbers c D. Aruliah (UOIT) Matrix Inverse and Condition MATH 2070U 2 / 27
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Matrix inverses Vector norms & errors Conditioning The inverse of a matrix Matrix inverse Square matrix [ A ] R n × n is invertible (or regular or nonsingular ) if there exists [ B ] R n × n such that [ A ][ B ] = [ B ][ A ] = [ I ] Inverse of [ A ] is unique and denoted [ A ] - 1 ; [ A ] must be square e.g., - 2 - 2 4 1 - 3 0 - 4 4 1 - 1 = 1/8 - 3/4 - 1/2 1/24 - 7/12 - 1/6 1/3 - 2/3 - 1/3 M ATLAB has routine inv for computing matrix inverse format rat, A=[-2,-2,4;1,-3,0;-4,4,1], B=inv(A) c D. Aruliah (UOIT) Matrix Inverse and Condition MATH 2070U 4 / 27
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Matrix inverses Vector norms & errors Conditioning Interpretation of the inverse of a matrix Interpret linear system of equations [ A ] { x } = { b } as [ interactions ] { responses } = { stimuli } Let a - 1 i , j denote element ( i , j ) of [ A ] - 1 [ A ] { x } = { b } implies { x } = [ A ] - 1 { b } , e.g., x 1 = a - 1 11 b 1 + a - 1 12 b 2 + a - 1 13 b 3 x 2 = a - 1 21 b 1 + a - 1 22 b 2 + a - 1 23 b 3 x 3 = a - 1 31 b 1 + a - 1 32 b 2 + a - 1 33 b 3 Element a - 1 i , j of [ A ] - 1 denotes unit change in x i produced by unit change in b j when solving linear system [ A ] { x } = { b } c D. Aruliah (UOIT) Matrix Inverse and Condition MATH 2070U 5 / 27
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Matrix inverses Vector norms & errors Conditioning Computing the inverse of a matrix [ A ] - 1 computed by Gaussian elimination on [[ A ] | [ I ]] Computing [ A ] - 1 equivalent to n linear solves: e.g., solving a 11 a 12 a 13 1 0 0 a 21 a 22 a 23 0 1 0 a 31 a 32 a 33 0 0 1 is same as a 11 a 12 a 13 1 a 21 a 22 a 23 0 a 31 a 32 a 33 0 , a 11 a 12 a 13 0 a 21 a 22 a 23 1 a 31 a 32 a 33 0 , a 11 a 12 a 13 0 a 21 a 22 a 23 0 a 31 a 32 a 33 1 Each forward- & back-solve requires O ( n 2 ) flops Computing inverse requires O ( n 3 ) work on top of work required to compute LU factorisation c D. Aruliah (UOIT) Matrix Inverse and Condition MATH 2070U 6 / 27
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Matrix inverses Vector norms & errors Conditioning Motivation for vector norms Real numbers are ordered : given a , b R , either a < b , a > b , or a = b Vectors in R n are not ordered,e.g., expressions like 1 - 1 1 > - 1 1 - 1 or - 1 2 - 1 < 1 - 1 3 do not make sense (at least as scalars) Norms provide a way to order vectors, measure distance c D. Aruliah (UOIT) Matrix Inverse and Condition MATH 2070U 8 / 27
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Matrix inverses Vector norms & errors Conditioning Vector norms Vector norms Definition (Vector norm) Given a vector space V , a norm is a function k·k : V [ 0, ) satisfying three postulates: 1 k v k > 0 if v 6 = 0 for every v V 2 k λ v k = | λ | k v k for every λ R , v V 3 k u + v k ≤ k u k + k v k for every u , v V (triangle inequality) k x k provides notion of
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