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Generalidades de Eigenvectores y Eigenvalores

Module by: Michael Haag, Justin Romberg. E-mail the authorsTranslated By: Fara Meza, Erika Jackson

Based on: Eigen-stuff in a Nutshell by Michael Haag, Justin Romberg

Summary: Este modulo nos da un pequeño repaso de la importancia de los eigenvectores y eigenvalores en el análisis y entedimiento de los sistemas LTI.

La Matriz y sus Eigenvectores

La razón por la cual estamos recalcando la importancia de los eigenvectores es por que la acción de una matriz AA en uno de sus eigenvectores vv es

  1. Extremadamente fácil (y rápido) de calcular
    Av=λv A v λ v
    (1)
    solo multiplicar vv por λ λ.
  2. fácil de interpretar: AA solo escala vv, manteniendo su dirección constante y solo altera la longitud del vector.
Si solo cada vector fuera un eigenvector de AA....

Usando el Espacio Generado por los Eigenvectores

Claro que no todos los vectores pero para ciertas matrices (incluidas aquellas con eigenvalores λλ's), cuyos eigenvectores generan el subespacio Cn n , lo que significa que para cada xCn x n , podemos encontrar α 1 α 2 α n C α 1 α 2 α n tal que:

x= α 1 v 1 + α 2 v 2 ++ α n v n x α 1 v 1 α 2 v 2 α n v n
(2)
Dada la ecuación 2, podemos reescribir Ax=b A x b . Esta ecuación esta modelada en nuestro sistema LTI ilustrado posteriormente:

Figura 1: Sistema LTI.
Figura 1 (eigv_sys.png)

x=i α i v i x i α i v i b=i α i λ i v i b i α i λ i v i El sistema LTI representado anteriormente representa nuestra ecuación 1. La siguiente es una ilustración de los paso para ir de xx a bb. x(α=V-1x)(ΛV-1x)(VΛV-1x=b) x α V -1 x Λ V -1 x V Λ V -1 x b Donde los tres pasos (las flechas) de la ilustración anterior representan las siguientes tres operaciones:

  1. Transformar x x usando V-1 V -1 - nos da αα
  2. Acción de AA en una nueva base- una multiplicación por Λ Λ
  3. Regresar a la antigua base- transformada inversa usando la multiplicación por V V, lo que nos da bb

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