Question

reduce the quadratic form 6x^2+3y^2+3z^2-4xy-2yz+4xz to the sum of squares form and then find its index and signature.​

Answer 1

Step-by-step explanation:

The index of the quadratic form is equal to the number of positive Eigen values of the matrix of quadratic form. Signature: The index of the quadratic form is equal to the difference between the number of positive Eigen values and the number of negative Eigen values of the matrix of quadratic form.

Answer 2

Given:

$6x^2+3y^2+3z^2-4xy-2yz+4xz$

To find:

sum of squares$(x)$

index$(y)$

signature$(z)$

Step-by-step explanation:

      $6x^2+3y^2+3z^2-4xy-2yz+4xz=0$

$\Longrightarrow(4x^2-4xy+y^2)+2(x^2+2xz+z^2)+y^2+(y^2-2yz+z^2)=0$

$\Longrightarrow{(2x-y)}^2+2{(x+z)}^2+y^2+{(y-z)}^2=0$

Sum of square term $= 0$

$\Longrightarrow$ Each term $=0$

$\Longrightarrow y^2=0$

$\Longrightarrow y=0$

$(2x-y)=0$

$y=2x$                          $(y=0)$

$x=0$

$(y-z)=0$

$\Longrightarrow y=z=0$

$(x+z)=0\leftarrow x\;=\;y\;=\;z\;=\;0$

Hence the answers are  given,

$[x=y=z=0]$