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 In a 2x2 matrix, the determinate is given by |A| = ad - bc

Determinate of a 3x3 matrix

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

Gauss-Jordan Elimination Using the Elementary Row Operations 1.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

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

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

Properties For A ∈ M(m,n;K) the pseudoinverse, A + ∈ M(n,m;K), satisfies these 4 properties 1. A A + A = A 2. A + A A + = A + 3. (AA + ) * = A A + 4. (A + A) * = A + A * = the conjugate transpose

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

Resources PenroseMatrixInverse.html PenroseMatrixInverse.html JordanElimination.html JordanElimination.html pdf pdf notes/Moore-Pinrose.pdf notes/Moore-Pinrose.pdf