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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.

Properties

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

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
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or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
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ramon Reply
Kristine 2*2*2=8
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Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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J, combine like terms 7x-4y
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what is the problem that i will help you to self with?
Asali
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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China
Cied
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what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
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what is nano technology
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what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
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Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
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Azam
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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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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