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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 .

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Moore-Penrose Pseudoinverse & Generalized Inverse

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  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

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