Norms and angles of vectors

In an inner product space, the norm (AKA length) of a vector is defined as \({\left\Vert v\right\Vert\equiv\sqrt{\left\langle v,v\right\rangle }}\), leading to common relations such as the Cauchy-Schwarz inequality \({\left|\left\langle v,w\right\rangle \right|\leq\left\Vert v\right\Vert \left\Vert w\right\Vert}\) and the triangle inequality (see below).

Alternatively, we can define a norm on a vector space as a non-negative real function that only vanishes for the zero vector and satisfies \({\left\Vert av\right\Vert =\left|a\right|\left\Vert v\right\Vert}\) as well as the triangle inequality \({\left\Vert v+w\right\Vert \leq\left\Vert v\right\Vert +\left\Vert w\right\Vert}\). This makes the space a normed vector space. Existence of a norm does not in general imply the existence of an inner product, but if a norm satisfies the parallelogram identity

\(\displaystyle \left\Vert v+w\right\Vert ^{2}+\left\Vert v-w\right\Vert {}^{2}=2\left(\left\Vert v\right\Vert {}^{2}+\left\Vert w\right\Vert {}^{2}\right), \)

an inner product can be obtained using the polarization identity, which for a real vector space is

\begin{aligned} \left\langle v,w\right\rangle &=\frac{1}{4}\left(\left\Vert v+w\right\Vert {}^{2}-\left\Vert v-w\right\Vert {}^{2}\right)\\&=\frac{1}{2}\left(\left\Vert v\right\Vert {}^{2}+\left\Vert w\right\Vert {}^{2}-\left\Vert v-w\right\Vert {}^{2}\right)\\&=\frac{1}{2}\left(\left\Vert v+w\right\Vert {}^{2}-\left\Vert v\right\Vert {}^{2}-\left\Vert w\right\Vert {}^{2}\right),\end{aligned}

and for a complex vector space is

\begin{aligned}\left\langle v,w\right\rangle & =\frac{1}{4}\left(\left\Vert v+w\right\Vert {}^{2}-\left\Vert v-w\right\Vert {}^{2}+i\left\Vert v-iw\right\Vert {}^{2}-i\left\Vert v+iw\right\Vert {}^{2}\right).\end{aligned}

In a real inner product space, we can define the angle between vectors by \({\cos\theta\equiv\left\langle v,w\right\rangle /\left(\left\Vert v\right\Vert \left\Vert w\right\Vert \right)}\). In a complex vector space \({V}\), taking the real part of this cosine defines the Euclidean angle \({\cos\theta_{E}\equiv\mathrm{Re}\left(\left\langle v,w\right\rangle \right)/\left(\left\Vert v\right\Vert \left\Vert w\right\Vert \right)}\), which is the angle between the vectors using the real inner product defined by the orthonormal basis in the decomplexification \({V_{\mathbb{R}}}\). Alternatively, taking the modulus of this cosine defines the Hermitian angle \({\cos\theta_{H}\equiv\left|\left\langle v,w\right\rangle \right|/\left(\left\Vert v\right\Vert \left\Vert w\right\Vert \right)}\), which is the ratio of the orthogonal projection of \({v}\) onto \({w}\) over the norm of \({v}\) (or the reverse).

Δ Note that a Euclidean angle of \({\pi/2}\) does not ensure a vanishing inner product, and that the orthogonal projection used in the interpretation of the Hermitian angle uses the complex inner product, so that parallel vectors in \({V}\) may be orthogonal using the corresponding real inner product in \({V_{\mathbb{R}}}\).

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