Skip to main content

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.


Comments

Post a Comment

Popular posts from this blog

Mathematics for Machine Learning- CST 284 - KTU Minor Notes - Dr Binu V P

  Introduction About Me Syllabus Course Outcomes and Model Question Paper University Question Papers and Evaluation Scheme -Mathematics for Machine learning CST 284 KTU Overview of Machine Learning What is Machine Learning (video) Learn the Seven Steps in Machine Learning (video) Linear Algebra in Machine Learning Module I- Linear Algebra 1.Geometry of Linear Equations (video-Gilbert Strang) 2.Elimination with Matrices (video-Gilbert Strang) 3.Solving System of equations using Gauss Elimination Method 4.Row Echelon form and Reduced Row Echelon Form -Python Code 5.Solving system of equations Python code 6. Practice problems Gauss Elimination ( contact) 7.Finding Inverse using Gauss Jordan Elimination  (video) 8.Finding Inverse using Gauss Jordan Elimination-Python code Vectors in Machine Learning- Basics 9.Vector spaces and sub spaces 10.Linear Independence 11.Linear Independence, Basis and Dimension (video) 12.Generating set basis and span 13.Rank of a Matrix 14.Linear Mapping...
Post a Comment
Read more

1.1 Solving system of equations using Gauss Elimination Method

Elementary Transformations Key to solving a system of linear equations are elementary transformations that keep the solution set the same, but that transform the equation system into a simpler form: Exchange of two equations (rows in the matrix representing the system of equations) Multiplication of an equation (row) with a constant  Addition of two equations (rows) Add a scalar multiple of one row to the other. Row Echelon Form A matrix is in row-echelon form if All rows that contain only zeros are at the bottom of the matrix; correspondingly,all rows that contain at least one nonzero element are on top of rows that contain only zeros. Looking at nonzero rows only, the first nonzero number from the left pivot (also called the pivot or the leading coefficient) is always strictly to the right of the  pivot of the row above it. The row-echelon form is where the leading (first non-zero) entry of each row has only zeroes below it. These leading entries are called...
Post a Comment
Read more

2.11 Eigen Value and Eigen Vectors

Eigenvalues and Eigenvectors Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Definition: Let $A \in \mathbb{R}^{n \times n}$ be a square matrix. Then $\lambda \in \mathbb{R}$ is an eigen value of $A$ and $ x \in \mathbb{R}^n /{0} $ is the corresponding eigen vector of  $A$ if       $Ax=\lambda x$ We call this as Eigenvalue equation The following statements are equivalent $\lambda$ is the eigen value of $A$ There exist an $x \in \mathbb{R}^n/0$ with $Ax=\lambda x$ or equivalently $(A-\lambda I_n)x=0$ can be solved non trivially i.e., $x \ne 0$ $rank(A-\lambda I) \le n$ $det(A-\lambda I) =0$ collinearity and codirection Two vectors that point in the same direction are called codirected.Two vectors are collinear if they point in the same or opposite directio...
Post a Comment
Read more