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  • 1a. M = N = r : One solution with no error, ε .
  • 1b. M = N > r : b s p a n { A } : Many solutions with ε = 0 .
  • 1c. M = N > r : b n o t s p a n { A } : Many solutions with the same minimum error.
  • 2a. M > N = r : b s p a n { A } : One solution ε = 0 .
  • 2b. M > N = r : b n o t s p a n { A } : One solution with minimum error.
  • 2c. M > N > r : b s p a n { A } : Many solutions with ε = 0 .
  • 2d. M > N > r : b n o t s p a n { A } : Many solutions with the same minimum error.
  • 3a. N > M = r : Many solutions with ε = 0 .
  • 3b. N > M > r : b s p a n { A } : Many solutions with ε = 0
  • 3c. N > M > r : b n o t s p a n { A } : 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, b is necessarily in the span of 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.

a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 x 1 x 2 x 3 = b 1 b 2 b 3

If the matrix has rank 3, then the b vector will necessarily be in the space spanned by the columns of A which puts it in case 1a. This can be solved for x by inverting A or using some more robust method. If the matrix has rank 1 or 2, the 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 A is 4 x 3, then we have more equations than unknowns or the overspecified or overdetermined case.

a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 a 41 a 42 a 43 x 1 x 2 x 3 = b 1 b 2 b 3 b 4

If this matrix has the maximum rank of 3, then we have case 2a or 2b depending on whether b is in the span of A or not. In either case, a unique solution x exists which can be found by [link] or [link] . For case 2a, we have a single exact solution with no equation error, ϵ = 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 A is 3 x 4, then we have more unknowns than equations or the underspecified case.

a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 x 1 x 2 x 3 x 4 = b 1 b 2 b 3

If this matrix has the maximum rank of 3, then we have case 3a and b must be in the span of A . For this case, many exact solutions x exist, all having zero equation error and a single one can be found with minimum solution norm | | 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 ε be minimized and, if there is ambiguity (several solutions with the same minimum error), the l 2 norm squared of x also be minimized. A useful alternative tominimizing the norm of x is to require certain entries in x to be zero (sparse) or fixed to some non-zero value (equality constraints).

Questions & Answers

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.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
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
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
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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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