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
• Some
matrices are invertible
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
• http://arxiv.org/pdf/1110.6882.pdf
•
http://mathworld.wolfram.com/MoorePenroseMatrixInverse.html
•
http://mathworld.wolfram.com/MatrixInverse.html
•
http://mathworld.wolfram.com/GaussJordanElimination.html
•
http://www.math.wustl.edu/~sawyer/handouts/GenrlInv.
pdf
•
http://faculty.kfupm.edu.sa/MATH/jaafarm/lecnotes/Moore-Pinrose.pdf