Mumbai University > Electronics and Telecommunication > Sem 4 > Applied Maths 4

Marks: 8M

Year: May 2015

The quadratic form can be written as

$\begin{bmatrix} a & h & g \\ h & b & b \\ g & b & c \end{bmatrix}$

Comparing given equation with

$a_{11}^2 x_{1}^2 + a_{22}^2 x_{2}^2 + a_{33}^2 x_{3}^2 + 2a_{12} x_{1}x_{2} + 2a_{13}x_{1}x_{3} + 2a_{23}x_{2}x_{3} \\ \therefore A = \begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix}$

Now we write

$A = IAI \\ \therefore \begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} A \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

Now while working with row change into L.H.S. (I) matrix when changing column the change R.H.S.(I) matrix

$\therefore R_{2}x_{3}, R_{3}x_{3} \\ \begin{bmatrix} 6 & -2 & 2 \\ -6 & 9 & -3 \\ 6 & -3 & 9 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix} A \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \\ \therefore c_{2}x_{3}, c_{3}x_{3} \\ \begin{bmatrix} 6 & -6 & 6 \\ -6 & 27 & -9 \\ 6& -9 & 27 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix} A \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \\ \therefore R_{2} + R_{1}, C_{2} + C_{1} \\ \begin{bmatrix} 6 & 0 & 6 \\ 0 & 21 & -3 \\ 6 & -3 & 27 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix} A \begin{bmatrix} 1 & 1 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix} \\ R_{3} - R_{1}, C_{3} - C_{1} \\ \begin{bmatrix} 6 & 0 & 0 \\ 0 & 21 & -3 \\ 0 & -3 & 27 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 3 & 0 \\ -1 & 0 & 3 \end{bmatrix} A \begin{bmatrix} 1 & 1 & -1 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix} \\ \therefore R_{3} \times 7, C_{3} \times 7 \\ \begin{bmatrix} 6 & 0 & 0 \\ 0 & 21 & -3 \\ 0 & -21 & 1029 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 3 & 0 \\ -7 & 0 & 21 \end{bmatrix} A \begin{bmatrix} 1 & 1 & -7 \\ 0 & 3 & 0 \\ 0 & 0 & 21 \end{bmatrix} \\ R_{3} + R_{2}, C_{3} + C_{2} \\ \begin{bmatrix} 6 & 0 & 0 \\ 0 & 21 & 0 \\ 0 & 0 & 1008 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 3 & 0 \\ -6 & 3 & 21 \end{bmatrix} A \begin{bmatrix} 1 & 1 & -6 \\ 0 & 3 & 3 \\ 0 & 0 & 21 \end{bmatrix} \\ R_{1} / \sqrt{6} , C_{1} / \sqrt{6} \\ R_{2} / \sqrt{21}, C_{2} / \sqrt{21} \\ R_{3} / \sqrt{1008}, C_{3} / \sqrt{1008} \\ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} -\frac{1}{\sqrt{6}} & 0 & 0 \\ \frac{1}{\sqrt{21}} & \frac{3}{\sqrt{21}} & 0 \\ \frac{-6}{\sqrt{1008}} & \frac{3}{\sqrt{1008}} & \frac{21}{\sqrt{1008}} \end{bmatrix} A \begin{bmatrix} \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{21}} & \frac{-6}{\sqrt{1008}} \\ 0 & \frac{3}{\sqrt{21}} & \frac{3}{\sqrt{1008}} \\ 0 & 0 & \frac{21}{\sqrt{1008}} \end{bmatrix}$

The linear tranformation

x = 8y

$x = \frac{1}{\sqrt{6}}u + \frac{1}{\sqrt{21}}v - \frac{b}{\sqrt{1008}}w \\ y = \frac{3}{\sqrt{21}}v + \frac{3}{\sqrt{1008}}w \\ z = \frac{21}{\sqrt{1008}}w$

Transformation the given quadratic equation to $u^2 + v^2 + w^2$

The rank = 3 & signature = 3 - 0 = 3

The value class is position definite.