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

 Page 4 / 4
• If $\mathbf{A}$ is case 1a, (square and nonsingular), then
${\mathbf{A}}^{+}={\mathbf{A}}^{-\mathbf{1}}$
• If $\mathbf{A}$ is case 2a or 2b, (over specified) then
${\mathbf{A}}^{+}={\left[{\mathbf{A}}^{\mathbf{T}}\mathbf{A}\right]}^{-\mathbf{1}}{\mathbf{A}}^{\mathbf{T}}$
• If $\mathbf{A}$ is case 3a, (under specified) then
${\mathbf{A}}^{+}={\mathbf{A}}^{\mathbf{T}}{\left[\mathbf{A}{\mathbf{A}}^{\mathbf{T}}\right]}^{-\mathbf{1}}$

Figure 2. Four Cases with Analytical Solutions

Fortunately, most practical cases are one of these four but even then, it is generally faster and less error prone to use special techniques on the normal equationsrather than directly calculating the inverse matrix. Note the matrices to be inverted above are all $r$ by $r$ ( $r$ is the rank) and nonsingular. In the other six cases from the ten in Figure 1, these would be singular, so alternate methods suchas SVD must be used [link] , [link] , [link] .

In addition to these four cases with “analytical” solutions, we can pose a more general problem by asking for an optimal approximation with a weighted norm [link] to emphasize or de-emphasize certain components or range of equations.

• If $\mathbf{A}$ is case 2a or 2b, (over specified) then the weighted error pseudoinverse is
${\mathbf{A}}^{+}={\left[{\mathbf{A}}^{\mathbf{T}*}{\mathbf{W}}^{\mathbf{T}*}\mathbf{W}\mathbf{A}\right]}^{-\mathbf{1}}{\mathbf{A}}^{\mathbf{T}*}{\mathbf{W}}^{\mathbf{T}*}\mathbf{W}$
• If $\mathbf{A}$ is case 3a, (under specified) then the weighted norm pseudoinverse is
${\mathbf{A}}^{+}={\left[{\mathbf{W}}^{\mathbf{T}}\mathbf{W}\right]}^{-\mathbf{1}}{\mathbf{A}}^{\mathbf{T}}{\left[\mathbf{A},{\left[{\mathbf{W}}^{\mathbf{T}}\mathbf{W}\right]}^{-\mathbf{1}},{\mathbf{A}}^{\mathbf{T}}\right]}^{-\mathbf{1}}$

Figure 3. Three Cases with Analytical Solutions and Weights

These solutions to the weighted approxomation problem are useful in their own right but also serve as the foundation to the Iterative Reweighted Least Squares (IRLS)algorithm developed in the next chapter.

## Geometric interpretation and least squares approximation

A particularly useful application of the pseudo-inverse of a matrix is to various least squared error approximations [link] , [link] . A geometric view of the derivation of the normal equations can be helpful. If $\mathbf{b}$ does not lie in the range space of $\mathbf{A}$ , an error vector is defined as the difference between $\mathbf{A}\mathbf{x}$ and $\mathbf{b}$ . A geometric picture of this vector makes it clear that for the length of $\epsilon$ to be minimum, it must be orthogonal to the space spanned by the columns of $\mathbf{A}$ . This means that ${\mathbf{A}}^{*}\epsilon =\mathbf{0}$ . If both sides of [link] are multiplied by ${\mathbf{A}}^{*}$ , it is easy to see that the normal equations of [link] result in the error being orthogonal to the columns of $\mathbf{A}$ and, therefore its being minimal length. If $\mathbf{b}$ does lie in the range space of $\mathbf{A}$ , the solution of the normal equations gives the exact solution of [link] with no error.

For cases 1b, 1c, 2c, 2d, 3a, 3b, and 3c, the homogeneous equation [link] has non-zero solutions. Any vector in the space spanned by these solutions (the null space of $\mathbf{A}$ ) does not contribute to the equation error $\epsilon$ defined in [link] and, therefore, can be added to any particular generalized solution of [link] to give a family of solutions with the same approximation error. If the dimension of the nullspace of $\mathbf{A}$ is $d$ , it is possible to find a unique generalized solution of [link] with $d$ zero elements. The non-unique solution for these seven cases can be written in the form [link] .

$\mathbf{x}={\mathbf{A}}^{+}\mathbf{b}+\left[\mathbf{I}-{\mathbf{A}}^{+}\mathbf{A}\right]\mathbf{y}$

where $\mathbf{y}$ is an arbitrary vector. The first term is the minimum norm solution given by the Moore-Penrose pseudo-inverse ${\mathbf{A}}^{+}$ and the second is a contribution in the null space of $\mathbf{A}$ . For the minimum $||x||$ , the vector $\mathbf{y}=0$ .

## Derivations

To derive the necessary conditions for minimizing $q$ in the overspecified case, we differentiate $q={ϵ}^{\mathbf{T}}ϵ$ with respect to $\mathbf{x}$ and set that to zero. Starting with the error

$q={ϵ}^{\mathbf{T}}ϵ={\left[\mathbf{Ax}-\mathbf{b}\right]}^{\mathbf{T}}\left[\mathbf{Ax}-\mathbf{b}\right]={\mathbf{x}}^{\mathbf{T}}{\mathbf{A}}^{\mathbf{T}}\mathbf{A}\mathbf{x}-{\mathbf{x}}^{\mathbf{T}}{\mathbf{A}}^{\mathbf{T}}\mathbf{b}-{\mathbf{b}}^{\mathbf{T}}\mathbf{A}\mathbf{x}+{\mathbf{b}}^{\mathbf{T}}\mathbf{b}$
$q={\mathbf{x}}^{\mathbf{T}}{\mathbf{A}}^{\mathbf{T}}\mathbf{A}\mathbf{x}-\mathbf{2}{\mathbf{x}}^{\mathbf{T}}{\mathbf{A}}^{\mathbf{T}}\mathbf{b}+{\mathbf{b}}^{\mathbf{T}}\mathbf{b}$

and taking the gradient or derivative gives

${\nabla }_{\mathbf{x}}q=2{\mathbf{A}}^{\mathbf{T}}\mathbf{A}\mathbf{x}-2{\mathbf{A}}^{\mathbf{T}}\mathbf{b}=\mathbf{0}$

$q={ϵ}^{\mathbf{T}}{\mathbf{W}}^{\mathbf{T}}\mathbf{W}ϵ={\left[\mathbf{Ax}-\mathbf{b}\right]}^{\mathbf{T}}{\mathbf{W}}^{\mathbf{T}}\mathbf{W}\left[\mathbf{Ax}-\mathbf{b}\right]$

using the same steps as before gives the normal equations for the minimum weighted squared error as

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

and the pseudoinverse as

$\mathbf{x}={\left[{\mathbf{A}}^{\mathbf{T}}{\mathbf{W}}^{\mathbf{T}}\mathbf{W}\mathbf{A}\right]}^{-\mathbf{1}}{\mathbf{A}}^{\mathbf{T}}{\mathbf{W}}^{\mathbf{T}}\mathbf{W}\mathbf{b}$

To derive the necessary conditions for minimizing the Euclidian norm ${||x||}_{2}$ when there are few equations and many solutions to [link] , we define a Lagrangian

$\mathcal{L}\left(\mathbf{x},\mu \right)={||\mathbf{W}\mathbf{x}||}_{2}^{2}+{\mu }^{\mathbf{T}}\left(\mathbf{Ax}-\mathbf{b}\right)$

take the derivatives in respect to both $\mathbf{x}$ and $\mu$ and set them to zero.

${\nabla }_{\mathbf{x}}\mathcal{L}=\mathbf{2}{\mathbf{W}}^{\mathbf{T}}\mathbf{W}\mathbf{x}+{\mathbf{A}}^{\mathbf{T}}\mu =\mathbf{0}$

and

${\nabla }_{\mu }\mathcal{L}=\mathbf{Ax}-\mathbf{b}=\mathbf{0}$

Solve these two equation simultaneously for $\mathbf{x}$ eliminating $\mu$ gives the pseudoinverse in [link] and [link] result.

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

Because the weighting matrices $\mathbf{W}$ are diagonal and real, multiplication and inversion is simple. These equations are used in the Iteratively Reweighted LeastSquares (IRLS) algorithm described in the next chapter.

## Regularization

To deal with measurement error and data noise, a process called “regularization" is sometimes used [link] , [link] , [link] .

## Least squares approximation with constraints

The solution of the overdetermined simultaneous equations is generally a least squared error approximation problem. A particularly interesting anduseful variation on this problem adds inequality and/or equality constraints. This formulation has proven very powerful in solving theconstrained least squares approximation part of FIR filter design [link] . The equality constraints can be taken into account by using Lagrange multipliers and the inequality constraints can use theKuhn-Tucker conditions [link] , [link] , [link] . The iterative reweighted least squares (IRLS) algorithm described in the next chaptercan be modified to give results which are an optimal constrained least p-power solution [link] , [link] , [link] .

## Conclusions

There is remarkable structure and subtlety in the apparently simple problem of solving simultaneous equations and considerable insight can begained from these finite dimensional problems. These notes have emphasized the ${l}_{2}$ norm but some other such as ${l}_{\infty }$ and ${l}_{1}$ are also interesting. The use of sparsity [link] is particularly interesting as applied in Compressive Sensing [link] , [link] and in the sparse FFT [link] . There are also interesting and important applications ininfinite dimensions. One of particular interest is in signal analysis using wavelet basis functions [link] . The use of weighted error and weighted norm pseudoinverses provide a base for iterative reweighted leastsquares (IRLS) algorithms.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!