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(Blank Abstract)

De nuestro entendimiento de eigenvalores y eigenvectores hemos descubierto ciertas cosas sobre nuestro operador, la matriz A . Sabemos que los eigenvectores de A generan el espacio n y sabemos como expresar cualquier vector x en términos de v 1 v 2 v n , entonces tenemos el operador A calculado. Si tenemos A actuando en x , después esto es igual a A actuando en la combinación de los eigenvectores.

Todavía tenemos dos preguntas pendientes:

  • ¿Cuándo los eigenvectores v 1 v 2 v n de A generan el espacio n (asumiendo que v 1 v 2 v n linealmente independientes)?
  • ¿Cómo expresamos un vector dado x en términos de v 1 v 2 v n ?

1 respuesta a la pregunta #1

¿Cuándo los eigenvectores v 1 v 2 v n de A generan el espacio n ?
Si A tiene n diferentes eigenvalores i i j λ i λ j donde i y j son enteros, entonces A tiene n eigenvectores linealmente independientes. v 1 v 2 v n que generan el espacio n .
La demostración de esta proposición no es muy difícil, pero no es interesante para incluirla aquí. Si desea investigar esta idea, léase Strang G.,“Algebra Lineal y sus aplicaciones”para la demostración.
Además, n diferentes eigenvalores significa que A λ I c n λ n c n 1 λ n 1 c 1 λ c 0 0 tiene n raíces diferentes.

Respuesta a la pregunta #2

¿Cómo expresamos un vector dado x en términos de v 1 v 2 v n ?
Queremos encontrar α 1 α 2 α n tal que
x α 1 v 1 α 2 v 2 α n v n
Para poder encontrar el conjunto de variables, empezaremos poniendo los vectores v 1 v 2 v n como culumnas en una matriz V de n×n. V   v 1 v 2 v n   Ahora la se convierte en x   v 1 v 2 v n   α 1 α n ó x V α Lo que nos da una forma sencilla de resolver para la variable de nuestra pregunta α : α V -1 x Notese que V es invertible ya que tiene n columnas linealmnete independientes.

Comentarios adicionales

Recordemos el conocimiento de funciones y sus bases y examinemos el papel de V . x V α x 1 x n V α 1 α n donde α es solo x expresada en una base diferente: x x 1 1 0 0 x 2 0 1 0 x n 0 0 1 x α 1 v 1 α 2 v 2 α n v n V transforma x de la base canónica a la base v 1 v 2 v n

DiagonalizaciÓN de matrices y salidas

También podemos usar los vectores v 1 v 2 v n para representar la salida b , del sistema: b A x A α 1 v 1 α 2 v 2 α n v n A x α 1 λ 1 v 1 α 2 λ 2 v 2 α n λ n v n b A x   v 1 v 2 v n   λ 1 α 1 λ 1 α n A x V Λ α A x V Λ V -1 x donde Λ es la matriz con eigenvalores en la diagonal: Λ λ 1 0 0 0 λ 2 0 0 0 λ n Finalmente, podemos cancelar las x y quedarnos con una ecuación final para A : A V Λ V -1

1 interpretaciÓN

Para nuestra interpretación, recordemos nuestra formulas: α V -1 x b i α i λ i v i podemos interpretar el funcionamiento de x con A como: x 1 x n α 1 α n λ 1 α 1 λ 1 α n b 1 b n Donde los tres pasos (las flechas) en la ilustración anterior representan las siguientes tres operaciones:

  • Transformar x usando V -1 , nos da α
  • Multiplicar por Λ
  • Transformada Inversa usando V , lo que nos da b
¡Este es el paradigma que usaremos para los sistemas LTI!

Ilustración simple del sistema LTI.

Questions & Answers

What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
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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.
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
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Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
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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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Señales y sistemas. OpenStax CNX. Sep 28, 2006 Download for free at http://cnx.org/content/col10373/1.2
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