Introduction E9F5FC

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Consider three (bilinear or sesquilinear) forms

{$x=\varepsilon_1 x_1 + \dots + \varepsilon_n x_n \text{ where } \varepsilon_i \in \mathbb{R}, \mathbb{C} \text{ or } \mathbb{H}$}

{$y = \eta_1 y_1 + \dots + \eta_n y_n \text{ where } \eta_i \in \mathbb{R}, \mathbb{C} \text{ or } \mathbb{H}$}

{$\phi (x,y) = \sum \varepsilon_i \phi_{ij} \eta_j \text{ or } \phi (x,y) = \sum \overline{\varepsilon_i} \phi_{ij} \eta_j$}

{$ \phi(x,y) = \pm \overline{\varepsilon_1} \eta_1 \pm \overline{\varepsilon_2} \eta_2 \pm \cdots \pm \overline{\varepsilon_n} \eta_n $}

{$ \phi(x,y) = \varepsilon_1 \eta_1 + \varepsilon_2 \eta_2 + \cdots + \varepsilon_n \eta_n $}

{$ \phi(x,y) = (\varepsilon_1 \eta_{m+1} - \varepsilon_{m+1} \eta_{1}) + (\varepsilon_2 \eta_{m+2} - \varepsilon_{m+2} \eta_{2}) + \cdots + (\varepsilon_m \eta_{2m} - \varepsilon_{2m} \eta_{m}) $}

Idea to work on:

{$\overline{\varepsilon}\eta = (\varepsilon_1 - \varepsilon_{m+1}i)(\eta_1 + \eta_{m+1}i)$}

{$=\varepsilon_1 \eta_1 + \varepsilon_{m+1} \eta_{m+1} + (\varepsilon_1 \eta_{m+1} - \varepsilon_{m+1} \eta_1)i$}

complex = real + i * quaternion

unitary = orthogonal + i * symplectic

Readings

• Gunnar Traustason. Diagonalisation of real quadratic forms