Answer to Question #219720 in Linear Algebra for Kailash

Question

Find the nature of the matrix of QF is x2
+y2
+z2-2xy-2yz-2zx.

Answer 1

Solution;

The quadratic equation can be written as;

x²+y²+z²-xy-xy-yz-yz-xz-xz

The matrix of the quadratic form is ;

$A=\begin{bmatrix} 1 & -1&-1\\ -1 & 1&-1\\ -1&-1&1 \end{bmatrix}$

The characteristic equation of the given matrix is |A-$\lambda$I|=0

Find the eigenvalues of the characteristic equation;

$-(\lambda-2)^2(\lambda+1)=0$

The roots are

-1,2 and 2 which are the eigenvalues.

Two are positive and one is the negative.

The given quadratic form is not positive definite or negative definite (it is indefinite).