1

\begin{aligned} Q (\vec{x}) & = {| x_{1} |}^{2} - i x_{1} x_{2} + i x_{1} x_{2} - {| x_{2} |}^{2} \\ = x_{1} x_{1} - i x_{1} x_{2} + i x_{1} x_{2} - x_{2} x_{2} \\ = (x_{1} - x_{2}) (i x_{1} + x_{2}) \\ = - (x_{1} - x_{2}) (x_{1} - x_{2}) \\ = - {| x_{1} - x_{2} |}^{2} \end{aligned}

\begin{array}{cc} q_{1} = - 1 & q_{2} = 0 \end{array}

sig (Q) = (0,1,1)

\begin{array}{cc} α_{1} = x_{1} - x_{2} & α_{2} = t \overset{ˇ}{r} e b a x_{2} \\ x_{1} = α_{1} + α_{2} & x_{2} = α_{2} \end{array}

𝒜 = ((\begin{array}{c} 1 \\ 0 \end{array}), (\begin{array}{c} 1 \\ 1 \end{array}))

2

\begin{aligned} Q (\vec{x}) & = 2 {| x_{1} |}^{2} + x_{1} x_{3} + x_{1} x_{3} - i x_{2} x_{3} + i x_{3} x_{2} \\ = \frac{1}{2} {| 2 x_{1} + x_{3} |}^{2} - \frac{1}{2} {| x_{3} |}^{2} - i x_{2} x_{3} - i x_{3} x_{2} \\ = \frac{1}{2} {| 2 x_{1} + x_{3} |}^{2} - \frac{1}{2} {| x_{3} - 2 i x_{2} |}^{2} + 2 {| x_{2} |}^{2} \end{aligned}

\begin{array}{ccc} q_{1} = \frac{1}{2} & q_{2} = - \frac{1}{2} & q_{3} = 2 \end{array}

sig (Q) = (2,1,0)

\begin{array}{ccc} α_{1} = 2 x_{1} + x_{3} & α_{2} = x_{3} - 2 i x_{2} & α_{3} = x_{2} \\ x_{1} = \frac{1}{2} (α_{1} - α_{2} - 2 i α_{3}) & x_{2} = α_{3} & x_{3} = α_{2} + 2 i α_{3} \end{array}

𝒜 = ((\begin{array}{c} \frac{1}{2} \\ 0 \\ 0 \end{array}), (\begin{array}{c} - \frac{1}{2} \\ 0 \\ 1 \end{array}), (\begin{array}{c} - i \\ 1 \\ 2 i \end{array}))