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2.2 Inner products

Inner products allow for the introduction of intuitive geometrical concepts,such as the length of a vector and the angle or distance between two vectors. A major purpose of inner products is to determine whether vectors are orthogonal to each other.

Dot Product

It is a special type of inner product.The scalar product/dot product of two vectors in $\mathbb{R}^n$ is given by

$x^T y= \sum_{i=1}^n x_iy_i $

Example
from numpy import inf
from numpy import array
a = array([1, 2, 3])
b = array([2, 3, 4])
print(a)
print(b)
print("dot product")
print(a.dot(b))

O/P
[1 2 3]
[2 3 4] 
dot product 
20



General Inner products
Let $V$ be a vector space and  $\Omega : V \times V \to \mathbb{R}$ be a bilinear mapping that takes two vectors and maps them onto real number. Then

$\Omega$ is called symmetric if $\Omega(x,y)=\Omega(y,x)$ for all $x,y \in V$, i.e., the order of the arguments does not matter.
$\Omega$ is called positive definite if
$\forall x \in V / \{0\}$: $\Omega(x,x) > 0, \Omega(0,0)=0 $

Definition:Let $V$ be a vector space and  $\Omega : V \times V \to \mathbb{R}$ be a bilinear mapping that takes two vectors and maps them onto real number. Then

  • A positive definite, symmetric bilinear mapping $\Omega : V \times V \to \mathbb{R}$ is called an inner product on $V$.We typically write $<x,y>$ instead of  $\Omega(x,y)$
  • The pair $(V,<\cdot,\cdot>)$ is called an inner product space or (real) vector space with inner product.If we use the dot product defined as $x^Tx$, we call $(V,<\cdot,\cdot>)$ a Euclidean vector space
Inner Product that is not dot product

Consider $V=\mathbb{R}^2$. If we define
$<x,y>:=x_1y_1-(x_1y_2+x_2y_1)+2x_2y_2$
then $<\cdot,\cdot>$ is an inner product but different from the dot product.
$<x,y>:=x_1y_1-(x_1y_2+x_2y_1)+2x_2y_2$
$\qquad= y_1x_1 - (y_1x_2 + y_2x_1) + 2y_2x_2 = <y,x>$

Thus it is symmetric

It holds that
$<x,x> = x_1^2- (2x_1x_2) + 2x_2^2$
$= (x_1 - x_2)^2 + x_2^2$

This is a sum of positive terms for $x \ne 0$. Moreover, this expression shows that if $<x,x> = 0$ then $x_2 = 0$ and then $x_1 = 0$, i.e., $x = 0$. Hence, $<.,.>$ is positive definite.


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