# 0.2 General solutions of simultaneous equations  (Page 2/4)

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• 1a. $M=N=r$ : One solution with no error, $\epsilon$ .
• 1b. $M=N>r$ : $\mathbf{b}\phantom{\rule{0.222222em}{0ex}}\in span\left\{\mathbf{A}\right\}$ : Many solutions with $\epsilon =\mathbf{0}$ .
• 1c. $M=N>r$ : $\mathbf{b}\phantom{\rule{0.222222em}{0ex}}not\in span\left\{\mathbf{A}\right\}$ : Many solutions with the same minimum error.
• 2a. $M>N=r$ : $\mathbf{b}\phantom{\rule{0.222222em}{0ex}}\in span\left\{\mathbf{A}\right\}$ : One solution $\epsilon =\mathbf{0}$ .
• 2b. $M>N=r$ : $\mathbf{b}\phantom{\rule{0.222222em}{0ex}}not\in span\left\{\mathbf{A}\right\}$ : One solution with minimum error.
• 2c. $M>N>r$ : $\mathbf{b}\phantom{\rule{0.222222em}{0ex}}\in span\left\{\mathbf{A}\right\}$ : Many solutions with $\epsilon =\mathbf{0}$ .
• 2d. $M>N>r$ : $\mathbf{b}\phantom{\rule{0.222222em}{0ex}}not\in span\left\{\mathbf{A}\right\}$ : Many solutions with the same minimum error.
• 3a. $N>M=r$ : Many solutions with $\epsilon =\mathbf{0}$ .
• 3b. $N>M>r$ : $\mathbf{b}\phantom{\rule{0.222222em}{0ex}}\in span\left\{\mathbf{A}\right\}$ : Many solutions with $\epsilon =\mathbf{0}$
• 3c. $N>M>r$ : $\mathbf{b}\phantom{\rule{0.222222em}{0ex}}not\in span\left\{\mathbf{A}\right\}$ : Many solutions with the same minimum error.

Figure 1. Ten Cases for the Pseudoinverse.

Here we have:

• case 1 has the same number of equations as unknowns ( A is square, $M=N$ ),
• case 2 has more equations than unknowns, therefore, is over specified ( A is taller than wide, $M>N$ ),
• case 3 has fewer equations than unknowns, therefore, is underspecified ( A is wider than tall $N>M$ ).

This is a setting for frames and sparse representations.

In case 1a and 3a, $\mathbf{b}$ is necessarily in the span of $\mathbf{A}$ . In addition to these classifications, the possible orthogonality of thecolumns or rows of the matrices gives special characteristics.

## Examples

Case 1: Here we see a 3 x 3 square matrix which is an example of case 1 in Figure 1 and 2.

$\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right]=\left[\begin{array}{c}{b}_{1}\\ {b}_{2}\\ {b}_{3}\end{array}\right]$

If the matrix has rank 3, then the $\mathbf{b}$ vector will necessarily be in the space spanned by the columns of $\mathbf{A}$ which puts it in case 1a. This can be solved for $\mathbf{x}$ by inverting $\mathbf{A}$ or using some more robust method. If the matrix has rank 1 or 2, the $\mathbf{b}$ may or may not lie in the spanned subspace, so the classification will be 1b or 1c and minimization of ${||x||}_{2}^{2}$ yields a unique solution.

Case 2: If $\mathbf{A}$ is 4 x 3, then we have more equations than unknowns or the overspecified or overdetermined case.

$\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\\ {a}_{41}& {a}_{42}& {a}_{43}\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right]=\left[\begin{array}{c}{b}_{1}\\ {b}_{2}\\ {b}_{3}\\ {b}_{4}\end{array}\right]$

If this matrix has the maximum rank of 3, then we have case 2a or 2b depending on whether $\mathbf{b}$ is in the span of $\mathbf{A}$ or not. In either case, a unique solution $\mathbf{x}$ exists which can be found by [link] or [link] . For case 2a, we have a single exact solution with no equation error, $ϵ=\mathbf{0}$ just as case 1a. For case 2b, we have a single optimal approximate solution with the least possible equation error. If the matrix hasrank 1 or 2, the classification will be 2c or 2d and minimization of ${||x||}_{2}^{2}$ yelds a unique solution.

Case 3: If $\mathbf{A}$ is 3 x 4, then we have more unknowns than equations or the underspecified case.

$\left[\begin{array}{cccc}{a}_{11}& {a}_{12}& {a}_{13}& {a}_{14}\\ {a}_{21}& {a}_{22}& {a}_{23}& {a}_{24}\\ {a}_{31}& {a}_{32}& {a}_{33}& {a}_{34}\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\\ {x}_{4}\end{array}\right]=\left[\begin{array}{c}{b}_{1}\\ {b}_{2}\\ {b}_{3}\end{array}\right]$

If this matrix has the maximum rank of 3, then we have case 3a and $\mathbf{b}$ must be in the span of $\mathbf{A}$ . For this case, many exact solutions $\mathbf{x}$ exist, all having zero equation error and a single one can be found with minimum solution norm $||\mathbf{x}||$ using [link] or [link] . If the matrix has rank 1 or 2, the classification will be 3b or 3c.

## Solutions

There are several assumptions or side conditions that could be used in order to define a useful unique solution of [link] . The side conditions used to define the Moore-Penrose pseudo-inverse are that the ${l}_{2}$ norm squared of the equation error $\epsilon$ be minimized and, if there is ambiguity (several solutions with the same minimum error), the ${l}_{2}$ norm squared of $\mathbf{x}$ also be minimized. A useful alternative tominimizing the norm of $\mathbf{x}$ is to require certain entries in $\mathbf{x}$ to be zero (sparse) or fixed to some non-zero value (equality constraints).

where we get a research paper on Nano chemistry....?
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
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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
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