13 - Matrix inverses Vector norms errors Conditioning...

This preview shows pages 1–8. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern