Bilinear and quadratic functions in complex space (32.40-32.47)

if $\mathrm{b}(\mathbf{x}, \mathbf{y})$ is a bilinear function in an $n$-dimensional complex arithmetic space, then the function $\mathrm{h}(\mathbf{x}, \mathbf{y}) = \mathrm{b}(\mathbf{x}, \overline{\mathbf{y}})$ is Hermitian bilinear

if b (x, y) is a Hermitian bilinear function in the space $\mathcal{C}_{n}$, then $\mathrm{b}(\mathbf{x}, \mathbf{y}) = \mathrm{h}(\mathbf{x}, \overline{\mathbf{y}})$ is a bilinear function.

$4x_{1}^{2} - 12ix_{1} x_{2} - 9x_{2}^{2}$

$9x_{1}^{2} + 24(i + 1) x_{1} x_{2} + 16x_{2}^{2}$

$x_{1} x_{2}$

$\varepsilon^{2} x_{1}^{2} - \varepsilon x_{1} x_{2} + x_{2}^{2}, \varepsilon = e^{2\pi i / 3}$

$(1 + i) x_{1}^{2} + (2 + 2i) x_{1} x_{3} + ix_{2}^{2} + 3x_{3}^{2}$

$x_{1}^{2} + (2 - 2i) x_{1} x_{2} + 2x_{1} x_{3} + 2ix_{2}^{2} + (2 + 2i) x_{2} x_{3} + (1 + i) x_{3}^{2}$

$- x_{1}^{2} - 4ix_{1} x_{2} - (2 - 2i) x_{1} x_{3} + 4x_{2}^{2} - (4 + 4i) x_{2} x_{3} + 2ix_{3}^{2}$.

$- ix_{1} \bar{y}_{1} \quad (n = 1)$

$- ix_{1} \bar{y}_{1} \quad (n = 2)$

$3x_{1} \bar{y}_{1} + 4ix_{1} \bar{y}_{2} - 5x_{2} \bar{y}_{1} + ix_{2} \bar{y}_{2} \quad (n = 2)$

$- 3ix_{1} \bar{y}_{1} + 2x_{1} \bar{y}_{2} + 2x_{2} \bar{y}_{1} + (1 - i) x_{2} \bar{y}_{2} \quad (n = 2)$

$(1 + i) x_{1} \bar{y}_{2} + (1 + i) x_{2} \bar{y}_{1} - 5x_{2} \bar{y}_{2} \quad (n = 2)$

$(1 + i) x_{1} \bar{y}_{2} + (1 - i) x_{2} \bar{y}_{1} - 5x_{2} \bar{y}_{2} \quad (n = 2)$

$x_{1} \bar{y}_{1} - 3x_{2} \bar{y}_{2} + (2 + i) x_{3} \bar{y}_{3} - ix_{1} \bar{y}_{2} + (4 + i) x_{3} \bar{y}_{1} \quad (n = 3)$

$2x_{1} \bar{y}_{1} - 6x_{2} \bar{y}_{2} + (1 + 3\sqrt{2}) x_{3} \bar{y}_{3} + 3x_{1} \bar{y}_{2} + 3x_{2} \bar{y}_{1} + (2 - 5i) x_{1} \bar{y}_{3} + (2 + 5i) x_{3} \bar{y}_{1} + 4ix_{2} \bar{y}_{3} - 4ix_{3} \bar{y}_{2} \quad (n = 3)$

$\sum_{i = 1}^{n} x_{i} \bar{y}_{i}$

$\mathbf{A}_{47}$. $\mathbf{A}_{47} = \left[\begin{smallmatrix} 5 & -2 \\ -2 & 8 \end{smallmatrix}\right]$.

$\mathbf{A}_{79}$ (при $\varepsilon = e^{2\pii / 3}$). $\mathbf{A}_{79} = \left[\begin{smallmatrix} 0 & \varepsilon \\ \varepsilon^{-1} & 0 \end{smallmatrix}\right]$.

$\mathbf{A}_{280}$. $\mathbf{A}_{280} = \left[\begin{smallmatrix} 0 & 2 & 1 \\ 2 & 8 & 2 \\ 1 & 2 & 0 \end{smallmatrix}\right]$.

$\mathbf{A}_{491}$. $\mathbf{A}_{491} = \left[\begin{smallmatrix} 3 & 1 & 0 & 2 i \\ 1 & 3 & -2 i & 0 \\ 0 & 2 i & 1 & 1 \\ -2 i & 0 & 1 & 1 \end{smallmatrix}\right]$.

$2\left|x_{1}\right|^{2} + ix_{1} \bar{x}_{2} - ix_{2} \bar{x}_{1} + 2\left|x_{2}\right|^{2} \quad (n = 2)$

$\left|x_{1}\right|^{2} + (3 - 4i) x_{1} \bar{x}_{2} + (3 + 4i) x_{2} \bar{x}_{1} + \left|x_{2}\right|^{2} \quad (n = 2)$

$\left|x_{1}\right|^{2} + \varepsilon x_{1} \bar{x}_{2} + \bar{\varepsilon } x_{2} \bar{x}_{1} + \left|x_{2}\right|^{2} \quad \left(\varepsilon = e^{2\pii / 3}\right) \quad (n = 2)$

$3\left|x_{1}\right|^{2} + 3\left|x_{2}\right|^{2} - 5\left|x_{3}\right|^{2} - ix_{1} \bar{x}_{2} + ix_{2} \bar{x}_{1} \quad (n = 3)$

$\left|x_{1}\right|^{2} + \left|x_{2}\right|^{2} + \left|x_{3}\right|^{2} + x_{1} \bar{x}_{2} + x_{2} \bar{x}_{1} + ix_{1} \bar{x}_{3} - ix_{3} \bar{x}_{1} + ix_{2} \bar{x}_{3} - - ix_{3} \bar{x}_{2} \quad (n = 3)$

$12\left|x_{1}\right|^{2} - (1 + i) x_{1} \bar{x}_{2} - (1 - i) x_{2} \bar{x}_{1} + 2x_{1} \bar{x}_{3} + 2x_{3} \bar{x}_{1} + (3 + 3i) x_{1} \bar{x}_{4} + (3 - 3i) x_{4} \bar{x}_{1} + 12\left|x_{2}\right|^{2} + (1 - i) x_{2} \bar{x}_{3} + (1 + i) x_{3} \bar{x}_{2} - - 2x_{2} \bar{x}_{4} - 2x_{4} \bar{x}_{2} + 8\left|x_{3}\right|^{2} - (1 + i) x_{3} \bar{x}_{4} - (1 - i) x_{4} \bar{x}_{3} + 8\left|x_{4}\right|^{2} (n = 4)$.