Question

Find rank, index, signature and nature of the Quadratic form by reducing it into

Canonical form by orthogonal transformation 2x2 + 2y2 + 322 + 2xy - 4y2 - 42x.

Answer 1

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

The matrix of the quadratic form:

$A=\begin{pmatrix} 2 & 1& -2\\ 1 & 2&-2\\ -2 & -2&3 \end{pmatrix}$

$\begin{vmatrix} 2-\lambda & 1& -2\\ 1 & 2-\lambda&-2\\ -2 & -2&3-\lambda \end{vmatrix}=0$

$\lambda_1=1,\lambda_2=3-2\sqrt 2,\lambda_3=3+2\sqrt 2$

$x_1=\begin{pmatrix} -1 \\ 1\\ 0 \end{pmatrix}$ , $x_2=\begin{pmatrix} \sqrt 2/2 \\ \sqrt 2/2\\ 1 \end{pmatrix}$ , $x_3=\begin{pmatrix} -\sqrt 2/2 \\ -\sqrt 2/2\\ 1 \end{pmatrix}$

modal matrix:

$\begin{pmatrix} -1 & \sqrt2/2&- \sqrt2/2\\ 1 & \sqrt2/2&- \sqrt2/2\\ 0&1&1 \end{pmatrix}$

$|x_1|=\sqrt 2, |x_2|=\sqrt 2,|x_3|=\sqrt 2$

normalized matrix:

$N=\begin{pmatrix} -1/\sqrt2 & 1/2&- 1/2\\ 1/\sqrt2 & 1/2&- 1/2\\ 0&1&1 \end{pmatrix}$

$N^T=\begin{pmatrix} -1/\sqrt2 & 1/\sqrt2&0\\ 1/2 & 1/2&1\\ -1/2&-1/2&1 \end{pmatrix}$

diagonal matrix $D=N^TAN$ :

$D=\begin{pmatrix} -1/\sqrt2 & 1/\sqrt2&0\\ 1/2 & 1/2&1\\ -1/2&-1/2&1 \end{pmatrix}\begin{pmatrix} 2 & 1& -2\\ 1 & 2&-2\\ -2 & -2&3 \end{pmatrix}\begin{pmatrix} -1/\sqrt2 & 1/2&- 1/2\\ 1/\sqrt2 & 1/2&- 1/2\\ 0&1&1 \end{pmatrix}=$

$=\begin{pmatrix} -1/\sqrt2 & 1/\sqrt2&0\\ -1/2 & -1/2&1\\ -7/2&-7/2&5 \end{pmatrix}\begin{pmatrix} -1/\sqrt2 & 1/2&- 1/2\\ 1/\sqrt2 & 1/2&- 1/2\\ 0&1&1 \end{pmatrix}=\begin{pmatrix} 1 & 0&0\\ 0 & 1/2&0\\ 0&0&17/2 \end{pmatrix}$

canonical form:

$Q=y_1^2+y_2^2/2+17y_3^3/2$

The rank of the quadratic form is equal to the number of non zero Eigen values of the matrix of quadratic form

rank = 3

The index of the quadratic form is equal to the number of positive Eigen values of the matrix of quadratic form

index = 3

The signature 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

signature = 3

Nature of the quadratic form is Positive definite: all the Eigen values of the matrix of quadratic form are positive