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Producto Interno

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

Based on: Inner Products by Michael Haag, Justin Romberg

Summary: Este modulo describe el concepto de producto interno, el cual nos da una introducción a los espacios de HIlbert. Ejemplos y propiedades de estos dos conceptos son dicutidos.

Definición: Producto Interno

De seguro ya tiene una idea del producto interno, también conocido como producto punto, en Rn n de alguno de sus cursos de matemáticas o de cómputo. Si no, definiremos el producto interno de la siguiente manera, tenemos dados algunas xRn x n y yRn y n

Definition 1: Producto Interno
El producto interno esta definido matemáticamente de la siguiente manera:
x,y=yTx=( y 0 y 1 y n 1 ) x 0 x 1 x n 1 = i =0n1 x i y i x y y x y 0 y 1 y n 1 x 0 x 1 x n 1 i n 1 0 x i y i
(1)

Producto Interno en 2-D

Si tenemos xR2 x 2 y yR2 y 2 , entonces podemos escribir el producto interno como:

x,y=xycosθ x y x y θ
(2)
donde θθ es el ángulo entre xx y yy.

Figura 1: Gráfica general de vectores y ángulos mencionados en las ecuaciones anteriores.
Figura 1 (inprod_f1.png)

Geométricamente, el producto interno nos dice sobre la fuerza de xx en la dirección de yy.

Ejemplo 1

Por ejemplo, si x=1 x 1 , entonces x,y=ycosθ x y y θ

Figura 2: Gráfica de los dos vectores del ejemplo anterior.
Figura 2 (inprod_f2.png)

Las siguientes características son dadas por el producto interno:

  • x,y x y mide la longitud de la proyección de yy sobre xx.
  • x,y x y es el máximo (dadas x x , y y ) donde xx y yy estan en la misma dirección ( (θ=0)(cosθ=1) θ 0 θ 1 ).
  • x,y x y es cero cuando (cosθ=0)(θ=90°) θ 0 θ 90° , es decir xx y yy son ortogonales.

Reglas del Producto Interno

En general el producto interno en un espacio vectorial complejo es solo una función (tomando dos vectores y regresando un número complejo) que satisface ciertas condiciones:

  • Simetria Conjugada: x,y=(x,y)* x y x y
  • Escalado: αx,y=α(x,y) α x y α x y
  • Aditividad: x+y,z=x,z+y,z x y z x z y z
  • "Positividad": x,x>0  ,   x0    x x 0 x x 0
Definition 2: Ortogonal
Decimos que xx y yy son ortogonales si: x,y=0 x y 0

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