# 0.2 General solutions of simultaneous equations  (Page 3/4)

 Page 3 / 4

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 ${\epsilon }^{\mathbf{T}*}\epsilon$ and the norm squared of the solution ${\mathbf{x}}^{\mathbf{T}*}\mathbf{x}$ are both minimized. This solution is denoted by

$\mathbf{x}={\mathbf{A}}^{+}\mathbf{b}$

where ${\mathbf{A}}^{+}$ is called the Moore-Penrose inverse [link] of $\mathbf{A}$ (and is also called the generalized inverse [link] and the pseudoinverse [link] )

Roger Penrose [link] showed that for all $\mathbf{A}$ , there exists a unique ${\mathbf{A}}^{+}$ satisfying the four conditions:

$\mathbf{A}{\mathbf{A}}^{+}\mathbf{A}=\mathbf{A}$
${\mathbf{A}}^{+}\mathbf{A}{\mathbf{A}}^{+}={\mathbf{A}}^{+}$
${\left[\mathbf{A}{\mathbf{A}}^{+}\right]}^{*}=\mathbf{A}{\mathbf{A}}^{+}$
${\left[{\mathbf{A}}^{+}\mathbf{A}\right]}^{*}={\mathbf{A}}^{+}\mathbf{A}$

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.

${\mathbf{A}}^{\mathbf{T}*}\mathbf{A}\mathbf{x}={\mathbf{A}}^{\mathbf{T}*}\mathbf{b}$

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

${\mathbf{A}}^{+}={\left[{\mathbf{A}}^{\mathbf{T}*}\mathbf{A}\right]}^{-\mathbf{1}}{\mathbf{A}}^{\mathbf{T}*}.$

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

${\mathbf{A}}^{+}={\left[{\mathbf{A}}^{\mathbf{T}*}{\mathbf{W}}^{\mathbf{T}*}\mathbf{W}\mathbf{A}\right]}^{-\mathbf{1}}{\mathbf{A}}^{\mathbf{T}*}{\mathbf{W}}^{\mathbf{T}*}\mathbf{W}.$

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

${\mathbf{A}}^{+}={\mathbf{A}}^{\mathbf{T}*}{\left[\mathbf{A}{\mathbf{A}}^{\mathbf{T}*}\right]}^{-\mathbf{1}}.$

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

${\mathbf{A}}^{+}={\left[{\mathbf{W}}^{\mathbf{T}}\mathbf{W}\right]}^{-\mathbf{1}}{\mathbf{A}}^{\mathbf{T}}{\left[\mathbf{A},{\left[{\mathbf{W}}^{\mathbf{T}}\mathbf{W}\right]}^{-\mathbf{1}},{\mathbf{A}}^{\mathbf{T}}\right]}^{-\mathbf{1}}.$

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

${\mathbf{A}}^{+}=\underset{\delta \to 0}{lim}{\left[{\mathbf{A}}^{\mathbf{T}*}\mathbf{A}+{\delta }^{\mathbf{2}}\mathbf{I}\right]}^{-\mathbf{1}}{\mathbf{A}}^{\mathbf{T}*}=\underset{\delta \to 0}{lim}{\mathbf{A}}^{\mathbf{T}*}{\left[\mathbf{A}{\mathbf{A}}^{\mathbf{T}*}+{\delta }^{\mathbf{2}}\mathbf{I}\right]}^{-\mathbf{1}}.$

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

• ${\left[{\mathbf{A}}^{+}\right]}^{+}=\mathbf{A}$
• ${\left[{\mathbf{A}}^{+}\right]}^{*}={\left[{\mathbf{A}}^{*}\right]}^{+}$
• ${\left[{\mathbf{A}}^{*}\mathbf{A}\right]}^{+}={\mathbf{A}}^{+}{\mathbf{A}}^{*+}$
• ${\lambda }^{+}=1/\lambda$ for $\lambda \ne 0$ else ${\lambda }^{+}=0$
• ${\mathbf{A}}^{+}={\left[{\mathbf{A}}^{*}\mathbf{A}\right]}^{+}{\mathbf{A}}^{*}={\mathbf{A}}^{*}{\left[\mathbf{A}{\mathbf{A}}^{*}\right]}^{+}$
• ${\mathbf{A}}^{*}={\mathbf{A}}^{*}\mathbf{A}{\mathbf{A}}^{+}={\mathbf{A}}^{+}\mathbf{A}{\mathbf{A}}^{*}$

It is informative to consider the range and null spaces [link] of $\mathbf{A}$ and ${\mathbf{A}}^{+}$

• $R\left(\mathbf{A}\right)=R\left(\mathbf{A}{\mathbf{A}}^{+}\right)=R\left(\mathbf{A}{\mathbf{A}}^{*}\right)$
• $R\left({\mathbf{A}}^{+}\right)=R\left({\mathbf{A}}^{*}\right)=R\left({\mathbf{A}}^{+}\mathbf{A}\right)=R\left({\mathbf{A}}^{*}\mathbf{A}\right)$
• $R\left(I-\mathbf{A}{\mathbf{A}}^{+}\right)=N\left(\mathbf{A}{\mathbf{A}}^{+}\right)=N\left({\mathbf{A}}^{*}\right)=N\left({\mathbf{A}}^{+}\right)=R{\left(\mathbf{A}\right)}^{\perp }$
• $R\left(I-{\mathbf{A}}^{+}\mathbf{A}\right)=N\left({\mathbf{A}}^{+}\mathbf{A}\right)=N\left(\mathbf{A}\right)=R{\left({\mathbf{A}}^{*}\right)}^{\perp }$

## 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).

what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
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
Got questions? Join the online conversation and get instant answers!