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In using sparsity in posing a signal processing problem (e.g. compressive sensing), an l 1 norm can be used (or even an l 0 “pseudo norm”) to obtain solutions with zero components if possible [link] , [link] .

In addition to using side conditions to achieve a unique solution, side conditions are sometimes part of the original problem. One interesting caserequires that certain of the equations be satisfied with no error and the approximation be achieved with the remaining equations.

Moore-penrose pseudo-inverse

If the l 2 norm is used, a unique generalized solution to [link] always exists such that the norm squared of the equation error ε T * ε and the norm squared of the solution x T * x are both minimized. This solution is denoted by

x = A + b

where A + is called the Moore-Penrose inverse [link] of A (and is also called the generalized inverse [link] and the pseudoinverse [link] )

Roger Penrose [link] showed that for all A , there exists a unique A + satisfying the four conditions:

A A + A = A
A + A A + = A +
[ A A + ] * = A A +
[ A + A ] * = A + A

There is a large literature on this problem. Five useful books are [link] , [link] , [link] , [link] , [link] . The Moore-Penrose pseudo-inverse can be calculated in Matlab [link] by the pinv(A,tol) function which uses a singular value decomposition (SVD) to calculate it. There are a variety of other numerical methodsgiven in the above references where each has some advantages and some disadvantages.


For cases 2a and 2b in Figure 1, the following N by N system of equations called the normal equations [link] , [link] have a unique minimum squared equation error solution (minimum ϵ T ϵ ). Here we have the over specified case with more equations than unknowns.A derivation is outlined in "Derivations" , equation [link] below.

A T * A x = A T * b

The solution to this equation is often used in least squares approximation problems. For these two cases A T A is non-singular and the N by M pseudo-inverse is simply,

A + = [ A T * A ] - 1 A T * .

A more general problem can be solved by minimizing the weighted equation error, ϵ T W T W ϵ where W is a positive semi-definite diagonal matrix of the error weights. The solution to that problem [link] is

A + = [ A T * W T * W A ] - 1 A T * W T * W .

For the case 3a in Figure 1 with more unknowns than equations, A A T is non-singular and has a unique minimum norm solution, | | x | | . The N by M pseudoinverse is simply,

A + = A T * [ A A T * ] - 1 .

with the formula for the minimum weighted solution norm | | x | | is

A + = [ W T W ] - 1 A T A [ W T W ] - 1 A T - 1 .

For these three cases, either [link] or [link] can be directly calculated, but not both. However, they are equal so you simply use the one with the non-singularmatrix to be inverted. The equality can be shown from an equivalent definition [link] of the pseudo-inverse given in terms of a limit by

A + = lim δ 0 [ A T * A + δ 2 I ] - 1 A T * = lim δ 0 A T * [ A A T * + δ 2 I ] - 1 .

For the other 6 cases, SVD or other approaches must be used. Some properties [link] , [link] are:

  • [ A + ] + = A
  • [ A + ] * = [ A * ] +
  • [ A * A ] + = A + A * +
  • λ + = 1 / λ for λ 0 else λ + = 0
  • A + = [ A * A ] + A * = A * [ A A * ] +
  • A * = A * A A + = A + A A *

It is informative to consider the range and null spaces [link] of A and A +

  • R ( A ) = R ( A A + ) = R ( A A * )
  • R ( A + ) = R ( A * ) = R ( A + A ) = R ( A * A )
  • R ( I - A A + ) = N ( A A + ) = N ( A * ) = N ( A + ) = R ( A )
  • R ( I - A + A ) = N ( A + A ) = N ( A ) = R ( A * )

The cases with analytical soluctions

The four Penrose equations in [link] are remarkable in defining a unique pseudoinverse for any A with any shape, any rank, for any of the ten cases listed in Figure 1.However, only four cases of the ten have analytical solutions (actually, all do if you use SVD).

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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what is biological synthesis of nanoparticles
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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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