Make a Change of Variable, X = Py, That Transforms $$Q ( x ) = 5 x _ { 1 } ^ { 2 } + 8 x _ { 2 } ^ { 2 } + 4 x _ { 1 } x _ { 2 }$$

Make a change of variable, x = Py, that transforms the given quadratic form into a quadratic form with no cross-product
term. Give P and the new quadratic form.
- $$Q ( x ) = 5 x _ { 1 } ^ { 2 } + 8 x _ { 2 } ^ { 2 } + 4 x _ { 1 } x _ { 2 }$$

A)
$$\mathrm { P } = \left[ \begin{array} { r r } 1 & - 2 \\2 & 1\end{array} \right] ; 9 \mathrm { y } _ { 1 } + 4 \mathrm { y } _ { 2 }$$
B)
$$\mathrm { P } = \left[ \begin{array} { c c } 1 / \sqrt { 5 } & - 2 / \sqrt { 5 } \\2 / \sqrt { 5 } & 1 / \sqrt { 5 }\end{array} \right] ; 9 \mathrm { y } _ { 1 } ^ { 2 } + 4 \mathrm { y } _ { 2 } ^ { 2 }$$
C)
$$P = \left[ \begin{array} { r r } 1 & - 2 \\2 & 1\end{array} \right] ; 9 y _ { 1 } ^ { 2 } + 4 y _ { 2 } ^ { 2 }$$

D)

$$P = \left[ \begin{array} { r r } - 1 / \sqrt { 5 } & - 2 / \sqrt { 5 } \\2 / \sqrt { 5 } & 1 / \sqrt { 5 }\end{array} \right] ; - 9 y _ { 1 } ^ { 2 } - 4 y _ { 2 } ^ { 2 }$$