Inner Product Space - Part I
July 13, 2018
Hello! Inner products are the generalized form of dot products and is useful in dimesnsion reduction in principal component analysis, which is a major field of Machine learning & Data science.
Basics of Inner Products
Inner product, V is the function on cross product.
Let, V = (C ^ n) : an inner product, where C is the complex vector space
on V is a function < . , . > : V * V -> C
which satisfies the following condition.
- <x,x> must be non-negative real number , where x belong to V : Positive definiteness of Inner product
- <x+y,z> = <x,z> + <y,z> for x,y,z in V : Linearity with respect to first argument/variable (Bilinear)
- <Mx,y> = M<x,y> , true for all M in C : Linearity with respect to first argument/variable (Bilinear)
- <y,x> is the complex conjugate of the image of <x,y> : Conjugate symmetry
A vector space, V together with an inner product < . , . > is called an inner product space (V , <.,.>).
To learn more about Inner products, Stay Tuned.
Hope this helps! Keep tuned for more blogs from ML series.
Rajiv Jha :)