Define:
All vectors in a orthogonal basis are othogonal to each other.
Define:
Orthonormal: orthogonal + normalized
An orthonormal matrix contains normalized vectors which are orthogonal to each other; here we denote a orthonormal matrix \(Q\).
(i)
\(Q^TQ = I\). Moreover, if \(Q\) is square, \(Q^{-1} = Q^T\)
(ii)
An Orthonormal matrix preserves properties such as norm and dot product of vectors it multiplies with.
\(\|Q\boldsymbol{x}\| = \|\boldsymbol{x}\|\) and \((Q\boldsymbol{x})^T(Q\boldsymbol{y}) = \boldsymbol{x^Ty}\)
A systematic way to create orthogonal basis from any basis. (actually orthonormal)
Note: Orthonormal basis is often(?) called only orthogonal basis.
Algorithm GS (Produce an orthonormal basis). Given a ordinary basis \(\{\boldsymbol{v_1},...,\boldsymbol{v_n}\}\), convert it to an orthonormal basis.
I. [Initailize] Let \(\boldsymbol{e_1} = \boldsymbol{v_1}, \boldsymbol{q_1} = \boldsymbol{e_1}/\|\boldsymbol{e}\|\)
II. [Convertion] Set \(\boldsymbol{e_k} = \sum_{1\leq i\leq k-1}(\boldsymbol{q_i^T\boldsymbol{v_k}})\boldsymbol{q_i}\) cont.