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Unformatted text preview: 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 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 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 MATLAB 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 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 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 a 31 a 32 a 33 , a 11 a 12 a 13 a 21 a 22 a 23 1 a 31 a 32 a 33 , a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 1 Each forward & backsolve 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 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 Matrix inverses Vector norms & errors Conditioning Vector norms...
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This note was uploaded on 06/20/2011 for the course MATH 2070 taught by Professor Aruliahdhavidhe during the Winter '10 term at UOIT.
 Winter '10
 aruliahdhavidhe
 Math

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