# Linear Algebra/Orthogonality

## Cauchy-Schwarz inequality

The Cauchy-Schwarz inequality states that the magnitude of the inner product of two vectors is less than or equal to the product of the vector norms, or: ${\displaystyle |\langle x,y\rangle |\leq \|x\|\|y\|}$ .

## Definition

For any vectors ${\displaystyle x}$  and ${\displaystyle y}$  in an inner product space ${\displaystyle V}$ , we say ${\displaystyle x}$  is orthogonal to ${\displaystyle y}$ , and denote it by ${\displaystyle x\bot y}$ , if ${\displaystyle \langle x,y\rangle =0}$ .

## Applications

### How to orthogonalize a basis

Suppose to be on a vector space V with a scalar product (not necessarily positive-definite),
Problem: Construct an orthonormal basis of V starting by a random basis { v1, ... }.
Solution: Gram-Schmidt for non isotropic vectors, otherwise choose v_i + v_j and reiterate.