Editing Hodge star operator (section)

== Formal definition for ''k''-vectors ==
Let {{math|''V''}} be an [[Dimension (vector space)|{{math|''n''}}-dimensional]] [[orientation (mathematics)|oriented]] [[vector space]] with a nondegenerate symmetric bilinear form <math>\langle \cdot,\cdot \rangle</math>, referred to here as a scalar product. (In more general contexts such as pseudo-Riemannian manifolds and [[Minkowski space]], the bilinear form may not be positive-definite.) This induces a [[Exterior algebra#Inner product|scalar product]] on [[Multivector|{{math|''k''}}-vectors]] {{nowrap|<math display="inline">\alpha, \beta\in \bigwedge^{\!k}V</math>,}} for <math>0 \le k \le n</math>, by defining it on simple {{math|''k''}}-vectors <math>\alpha = \alpha_1 \wedge \cdots \wedge \alpha_k</math> and <math>\beta = \beta_1 \wedge \cdots \wedge \beta_k</math> to equal the [[Gram determinant]]<ref name="HF">[[Harley Flanders]] (1963) ''Differential Forms with Applications to the Physical Sciences'', [[Academic Press]]</ref>{{rp|14}}
: <math> \langle \alpha, \beta \rangle = \det \left( \left\langle \alpha_i, \beta_j \right\rangle  _{i,j=1}^k\right)</math>
extended to <math display="inline">\bigwedge^{\!k}V</math> through linearity.

The unit {{math|''n''}}-vector <math>\omega\in{\textstyle\bigwedge}^{\!n}V</math> is defined in terms of an oriented [[orthonormal basis]] <math>\{e_1,\ldots,e_n\}</math> of {{math|''V''}} as:
: <math>\omega := e_1\wedge\cdots\wedge e_n.</math>
(Note: In the general pseudo-Riemannian case, orthonormality means
<math>
\langle e_i,e_j\rangle
\in\{\delta_{ij},-\delta_{ij}\}</math> for all pairs of basis vectors.)
The '''Hodge star operator''' is a linear operator on the [[exterior algebra]] of {{math|''V''}}, mapping {{math|''k''}}-vectors to ({{math|''n'' – ''k''}})-vectors, for <math>0 \le k \le n</math>. It has the following property, which defines it completely:<ref name="HF" />{{rp|15}}
: <math>\alpha \wedge ({\star} \beta) = \langle \alpha,\beta \rangle \,\omega </math>  for all {{math|''k''}}-vectors <math>\alpha,\beta\in {\textstyle\bigwedge}^{\!k}V .</math>

Dually, in the space <math>{\textstyle\bigwedge}^{\!n}V^*</math> of {{math|''n''}}-forms (alternating {{math|''n''}}-multilinear functions on <math>V^n</math>), the dual to <math>\omega</math> is the [[volume form]] <math>\det</math>, the function whose value on <math>v_1\wedge\cdots\wedge v_n</math> is the [[determinant]] of the <math>n\times n</math> matrix assembled from the column vectors of <math>v_j</math> in <math>e_i</math>-coordinates. Applying <math>\det</math> to the above equation, we obtain the dual definition:
: <math>\det(\alpha \wedge {\star} \beta) = \langle \alpha,\beta \rangle </math> for all {{math|''k''}}-vectors <math>\alpha,\beta\in {\textstyle\bigwedge}^{\!k}V .</math>

Equivalently, taking <math>\alpha = \alpha_1 \wedge \cdots \wedge \alpha_k</math>, <math>\beta = \beta_1 \wedge \cdots \wedge \beta_k</math>, and <math>{\star}\beta = \beta_1^\star \wedge \cdots \wedge \beta_{n-k}^\star</math>:
: <math>
  \det\left(\alpha_1\wedge \cdots \wedge\alpha_k\wedge\beta_1^\star\wedge \cdots \wedge\beta_{n-k}^\star\right)
  \ = \ \det\left(\langle\alpha_i, \beta_j\rangle\right).
</math>

This means that, writing an orthonormal basis of {{math|''k''}}-vectors as <math>e_I \ = \ e_{i_1}\wedge\cdots\wedge e_{i_k}</math> over all subsets <math>I = \{i_1<\cdots<i_k\}</math> of <math>[n]=\{1,\ldots,n\}</math>, the Hodge dual is the ({{math|''n – k''}})-vector corresponding to the complementary set <math>\bar{I} = [n] \smallsetminus I = \left\{\bar i_1 < \cdots < \bar i_{n-k}\right\}</math>:
: <math>{\star} e_I = s\cdot t\cdot  e_\bar{I} ,</math>
where <math>s\in\{1,-1\}</math> is the [[Parity of a permutation|sign]] of the permutation <math>i_1 \cdots i_k \bar i_1 \cdots \bar i_{n-k}</math>
and <math>t\in\{1,-1\}</math> is the product
<math>\langle e_{i_1},e_{i_1}\rangle\cdots \langle e_{i_k},e_{i_k}\rangle</math>. In the Riemannian case, <math>t=1</math>.

Since Hodge star takes an orthonormal basis to an orthonormal basis, it is an [[isometry]] on the exterior algebra <math display="inline">\bigwedge V</math>.