Orthogonality and orthonormality

Dependencies:

  1. Inner product space

Let $S = \{v_1, v_2, \ldots, v_n\}$ be a set of vectors from an inner product space $V$.

$S$ is orthogonal iff $(\forall i, v_i \neq 0) \wedge (\forall i \neq j, \langle v_i, v_j \rangle = 0)$.

$S$ is orthonormal iff $S$ is orthogonal and $\forall i, \|v_i\|^2 = 1$.

Dependency for:

  1. Standard normal random vector on vector space
  2. Symmetric operator on V has a basis of orthonormal eigenvectors
  3. Coordinatization over orthogonal vectors
  4. A set of mutually orthogonal vectors is linearly independent
  5. Gram-Schmidt Process
  6. Orthonormal basis change matrix
  7. Orthogonal matrix

Info:

Transitive dependencies:

  1. Group
  2. Ring
  3. Field
  4. Vector Space
  5. Inner product space