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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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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