Linear Algebra/Orthogonality

< Linear algebra

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: $$| \langle x,y \rangle | \le  \| x \| \| y \| $$.

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

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.