Inner Products

What is an inner product?

An inner product on a vector space V is a pairing: $<v,w>\in \mathbb{R}$ which satisfies the following properties:

• Bilinearity: $<(cu+dv,w)>=c<u,w>+d<v,w>$
• Symmetry: $<w,v>=<v,w>$
• Positivity: $<v,v>>0,v\ne 0<0,0>=0$

What are examples of inner products?

We have the classic dot product, weighted inner product, and the alternative inner product.

What are properties of the dot product?

The dot product is commutative, distributive, allows scalar multiplication, and demonstrates a relationship with regard to angle.

• Commutative: $v\ast w=w\ast v$
• Distributive: $u(v+w)=uv+uw$
• Scalar Multiplication: $cv\ast w=c(v\ast w)$
• Angle Relation: $v\ast w=||v||||w||cos(\theta )$

What is the norm of a vector?

The norm of a vector is a positive value, which corresponds with the vector’s distance from the origin. It is defined as: $||v||=\sqrt{<v,v>}$It is used for positive definite matrices, least squares, data analysis, and signal processing.

What are inner product on function spaces?

For functions, f, g, defined on [a, b], the inner product is: $<f,g>={\int }_a^{b}f(x)g(x)dx$This makes the norm: $||f||=L^2=\sqrt{{\int }_a^{b}f(x{)}^{2}dx}$ We also have a weighted form of this inner product: $<f,g{>}_w={\int }_a^{b}w(x)f(x)g(x)dx$

What is an inner product space?

An inner product space is a vector space V, with an inner product satisfying linearity, symmetry, & positivity. These properties make sense considering V must remain a vector space. See here: Vector Spaces, Basis Vectors, Subspaces

What is Cauchy-Swartz inequality?

$<v,w{>}^2\le ||v|{|}^2||w|{|}^2=<v,w>\le ||v||||w||$ for all v, w in an inner product space. This leads into the triangle inequality which states: $||v+w|{|}^2\le (||v||+||w||{)}^2=||v+w||\le (||v||+||w||)$

What is the triangle inequality?

In an inner product space, $||v+w|{|}^2\le (||v||+||w||{)}^2=||v+w||\le ||v||+||w||$.

What is the Minkowski’s Inequality?

$\sum (|v_i\ast w_i{|}^p{)}^{1/p}\le \sum (|v_i{|}^p{)}^{1/p}+\sum (|w_i{|}^p{)}^{1/p}$

What is distance in normed space?

Distance between two vectors is the norm of their difference, so: $d(v,w)=||v-w||$. This maintains symmetry, positivity, and the triangle inequality.

What is matrix norm?

Given a norm on n-dimensional space, the corresponding matrix norm is defined as: $||A||=max(||Au|||||u||=1)$. $||A||\ge 0$ unless $||A||=0$, which is only true if and only if $A=0$.

What is gram-schmidt?

Gram-schmidt is a process to convert a linearly independent set into an orthonormal basis.

$u_1=v_1;u_2=v_2-\frac{<u_2,u_1>}{<u_1,u_1>}\ast u_1$. and $e_i=\frac{u_i}{||u_i||}$.

What is a p-norm?

The p-norm of a vector is: $||x|{|}_p=(\sum |x_i{|}^p{)}^{\frac{1}{p}}$.

What is a positive definite matrix?

A symmetric matrix A is a positive definite matrix if and only if:

$x^{T}Ax>0,\forall x\ne 0$. This means the matrix all positive eigenvalues.