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The second problem posed in the introduction is basically the solution of simultaneous linear equations [link] , [link] , [link] which is fundamental to linear algebra [link] , [link] , [link] and very important in diverse areas of applications in mathematics, numericalanalysis, physical and social sciences, engineering, and business. Since a system of linear equations may be over or under determined in a varietyof ways, or may be consistent but ill conditioned, a comprehensive theory turns out to be more complicated than it first appears. Indeed, there isa considerable literature on the subject of generalized inverses or pseudo-inverses . The careful statement and formulation of the general problem seems to have started with Moore [link] and Penrose [link] , [link] and developed by many others. Because the generalized solution of simultaneous equationsis often defined in terms of minimization of an equation error, the techniques are useful in a wide variety of approximation andoptimization problems [link] , [link] as well as signal processing.

The ideas are presented here in terms of finite dimensions using matrices. Many of the ideas extend to infinite dimensions using Banachand Hilbert spaces [link] , [link] , [link] in functional analysis.

The problem

Given an M by N real matrix A and an M by 1 vector b , find the N by 1 vector x when

a 11 a 12 a 13 a 1 N a 21 a 22 a 23 a 31 a 32 a 33 a M 1 a M N x 1 x 2 x 3 x N = b 1 b 2 b 3 b M

or, using matrix notation,

A x = b

If b does not lie in the range space of A (the space spanned by the columns of A ), there is no exact solution to [link] , therefore, an approximation problem can be posed by minimizing an equation error defined by

ε = A x - b .

A generalized solution (or an optimal approximate solution) to [link] is usually considered to be an x that minimizes some norm of ε . If that problem does not have a unique solution, further conditions, such as also minimizing the norm of x , are imposed. The l 2 or root-mean-squared error or Euclidean norm is ε T * ε and minimization sometimes has an analytical solution. Minimization of other norms such as l (Chebyshev) or l 1 require iterative solutions. The general l p norm is defined as q where

q = | | x | | p = ( n | x ( n ) | p ) 1 / p

for 1 < p < and a “pseudonorm" (not convex) for 0 < p < 1 . These can sometimes be evaluated using IRLS (iterative reweighted least squares) algorithms [link] , [link] , [link] , [link] , [link] .

If there is a non-zero solution of the homogeneous equation

A x = 0 ,

then [link] has infinitely many generalized solutions in the sense that any particular solution of [link] plus an arbitrary scalar times any non-zero solution of [link] will have the same error in [link] and, therefore, is also a generalized solution. The number of families of solutions is the dimensionof the null space of A .

This is analogous to the classical solution of linear, constant coefficient differential equationswhere the total solution consists of a particular solution plus arbitrary constants times the solutions to the homogeneous equation. The constants are determined from the initial(or other) conditions of the solution to the differential equation.

Ten cases to consider

Examination of the basic problem shows there are ten cases [link] listed in Figure 1 to be considered.These depend on the shape of the M by N real matrix A , the rank r of A , and whether b is in the span of the columns of A .

Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Basic vector space methods in signal and systems theory. OpenStax CNX. Dec 19, 2012 Download for free at http://cnx.org/content/col10636/1.5
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