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Table of Contents Math Background Matrices Moore-Penrose Pseudo-Inverse Matrix | |
See also: matrix inversion |
The classical inverse of a matrix is restricted
to regular (square, non-singular)
matrices. This restriction has severe consequences regarding the type of
multivariate problems which can be solved by matrix algebra. In order to create
the necessary tools for extending the range of solvable problems, the idea of
inverting a given matrix has been extended to a more general level:
Moore-Penrose Pseudo-Inverse Matrix | Let A be an arbitrary matrix of order mn, and B be a matrix of order
nm. B is called the
"Moore-Penrose pseudo-inverse" of A, if
BAB = B AB is symmetric BA is symmetric |
What does this strange definition mean? Simply stated, the first two statements mean, that we neglect all non-invertible properties of a matrix while inverting the rest. The other two statements choose a suitable matrix B of all those matrices that satisfy the first two rules.
If a system of linear equations is not
solvable, one would like to know a good approximation of the solution anyway.
More exactly expressed, one would like to find a solution which minimizes the
error - and the Moore-Penrose pseudo-inverse really gives the best
approximation.
Last Update: 2005-Jän-25