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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

what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
Got questions? Join the online conversation and get instant answers!
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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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