norm — Jemoka Knowledge Base

The norm is the “length” of a vector, defined generally using the inner product as:

\begin{equation} \|v\| = \sqrt{\langle v,v \rangle} \end{equation}

additional information properties of the norm nonnegativity: \norm{v} \geq 0 zero: \|v\| = 0 IFF v=0 first-degree homogeneity: \|\lambda v\| = |\lambda|\|v\| triangle inequality: \norm{x+y} \leq \norm{x} + \norm{y} inner product is a norm Inner product is a norm: By definition of an inner product, \langle v,v \rangle = 0 only when v=0 See algebra: \begin{align} |\lambda v|^{2} &= \langle \lambda v, \lambda v \rangle \ &= \lambda \langle v, \lambda v \rangle \ &= \lambda \bar{\lambda} \langle v,v \rangle \ &= |\lambda |^{2} |v|^{2} \end{align} motivating the norm using actual numbers In linear algebra, the norm of a vector in a real vector space is defined as follows:

\begin{equation} \| x\| = \sqrt{{{x_1}^{2} + \dots + {x_n}^{2}}} \end{equation}

Note that, given the definition of dot product, \| x \|^{2} = x \cdot x. The norm in complex vector space requires taking the absolute value (for a+bi, |a+bi| = \sqrt{{a^{2}+b^{2}}}) of each slot. That is, for Euclidean Inner Product spaces:

\begin{equation} \|z\| = \sqrt{|z_1|^{2} + \dots |z_{n}|^{2}} \end{equation}

otherwise, simply squaring the complex number (giving us a^{2}-b^{2}) may very well yield negative numbers, which means we’d have an imaginary norm! Euclidean norm \begin{equation} \norm{u}_{2} = \left(u^{T}u\right)^{\frac{1}{2}} \end{equation}