Answer to Question #219720 in Linear Algebra for Kailash

Question #219720
Find the nature of the matrix of QF is x2
+y2
+z2-2xy-2yz-2zx.
1
Expert's answer
2021-07-23T07:13:42-0400

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}\n 1 & -1&-1\\\\\n -1 & 1&-1\\\\\n-1&-1&1\n\\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).



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