Editing Hermitian matrix (section)

===Complex Hermitian forms vector space over {{math|ℝ}}===

The Hermitian complex {{mvar|n}}-by-{{mvar|n}} matrices do not form a [[vector space]] over the [[complex number]]s, {{math|'''ℂ'''}}, since the identity matrix {{math|''I''<sub>''n''</sub>}} is Hermitian, but {{math|''i'' ''I''<sub>''n''</sub>}} is not. However the complex Hermitian matrices ''do'' form a vector space over the [[real numbers]] {{math|'''ℝ'''}}. In the {{math|2''n''<sup>2</sup>}}-[[dimension of a vector space|dimensional]] vector space of complex {{math|''n'' × ''n''}} matrices over {{math|'''ℝ'''}}, the complex Hermitian matrices form a subspace of dimension {{math|''n''<sup>2</sup>}}. If {{math|''E''<sub>''jk''</sub>}} denotes the {{mvar|n}}-by-{{mvar|n}} matrix with a {{math|1}} in the {{math|''j'',''k''}} position and zeros elsewhere, a basis (orthonormal with respect to the Frobenius inner product) can be described as follows:
<math display=block>E_{jj} \text{ for } 1 \leq j \leq n \quad (n \text{ matrices}) </math>

together with the set of matrices of the form
<math display=block>\frac{1}{\sqrt{2}}\left(E_{jk} + E_{kj}\right) \text{ for } 1 \leq j < k \leq n \quad \left( \frac{n^2-n} 2 \text{ matrices} \right) </math>

and the matrices
<math display=block>\frac{i}{\sqrt{2}}\left(E_{jk} - E_{kj}\right) \text{ for } 1 \leq j < k \leq n \quad \left( \frac{n^2-n} 2 \text{ matrices} \right) </math>

where <math>i</math> denotes the [[imaginary unit]], <math>i = \sqrt{-1}~.</math>

An example is that the four [[Pauli matrices]] form a complete basis for the vector space of all complex 2-by-2 Hermitian matrices over {{math|'''ℝ'''}}.