1 / 11

Moore-Penrose Pseudoinverse & Generalized Inverse

Moore-Penrose Pseudoinverse & Generalized Inverse. Matt Connor Fall 2013. Inverse- when A is combined with its inverse you get the identity (I) Identity (I) - when combined with any other element X it will produce X ex: B*I = B. Determinate. Denoted |A| General form of a 2x2 is .

ron
Télécharger la présentation

Moore-Penrose Pseudoinverse & Generalized Inverse

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Moore-Penrose Pseudoinverse & Generalized Inverse • Matt Connor • Fall 2013

  2. Inverse- when A is combined with its inverse you get the identity (I) • Identity (I) - when combined with any other element X it will produce X • ex: B*I = B

  3. Determinate • Denoted |A| • General form of a 2x2 is • In a 2x2 matrix, the determinate is given by |A| = ad - bc

  4. Determinate of a 3x3 matrix

  5. If A is an nxn matrix, and |A|≠0 then we call it nonsingular • nonsingular matrices are invertible • Some methods are Gauss-Jordan Elimination, Gaussian Elimination, and LU Decomposition

  6. Gauss-Jordan Elimination • Using the Elementary Row Operations • Interchanging two rows or columns • 2. Adding a multiple of one row or column to another • 3. Multiplying any row or column by a nonzero element

  7. Moore-Penrose Pseudoinverse • A generalization of the inverse matrix. • Discovered by Moore in 1920, Penrose in 1955 independently • Does not have to be nxn matrix • Found using Singular Value Decomposition • Common cases are over real and complex numbers • can be used for matrices over a commutative ring

  8. Uses • Compute a best fit solution to a system of linear equations that does not have a unique solution • Find the minimum solution to a linear system with multiple solutions • Finding the condition number • measures how sensitive a function is to a change in the input

  9. Properties • For A∈M(m,n;K) the pseudoinverse , A+∈M(n,m;K), satisfies these 4 properties • A A+A = A • A+A A+ = A+ • (AA+)* = A A+ • (A+ A)* = A+A • * = the conjugate transpose

  10. For any matrix A, there is exactly one matrix A+, that satisfies the four properties of the Moore-Penrose Pseudoinverse • A matrix that satisfies the first two conditions is called a Generalized inverse • These always exist, but do not imply uniqueness, uniqueness is established by the last two conditions

  11. Resources • http://arxiv.org/pdf/1110.6882.pdf • http://mathworld.wolfram.com/Moore-PenroseMatrixInverse.html • http://mathworld.wolfram.com/MatrixInverse.html • http://mathworld.wolfram.com/Gauss-JordanElimination.html • http://www.math.wustl.edu/~sawyer/handouts/GenrlInv.pdf • http://faculty.kfupm.edu.sa/MATH/jaafarm/lec-notes/Moore-Pinrose.pdf

More Related