Orthogonality

Meaning

In mathematics, orthogonality represents a general description of objects being perpendicular some sense. In the case of vectors, it is quite intuitive to check as the dot products of two perpendicular (orthogonal) vectors is zero

• Two vectors $\mathbf{a},\mathbf{b}\in {\mathbb{R}}^n$ are orthogonal if

  $$(\mathbf{a},\mathbf{b}):={\mathbf{a}}^{\top }\mathbf{b}=0.$$

• Two functions $f,g$ are orthogonal if

  $$(f,g):={\int }_a^{b}f(x)g(x)dx=0,$$

  Example: $\sin$, $\cos$ functions

• Two (mean-free) random variables $X,Y$ are orthogonal if they are uncorrelated, i.e.,

  $$(X,Y):=\mathbb{E}[XY]=0.$$