Exercise 3.4.1 (Definite and indefinite forms)

Proof. Using (2.11), we obtain

(a)

Here $d=-4<0,a=1>0$, thus $x^2+y^2$ is positive definite.

(b)

$d=-4<0,a=-1<0$, thus $-x^2-y^2$ is negative definite.

(c)

$d=8>0$, thus $x^2-2y^2$ is indefinite.

(d)

$d=9^2-4\cdot 8\cdot 10=-239,a=10>0$, thus $10x^2-9\mathit{xy}+8y^2$ is positive definite.

(e)

$d=5$, thus $x^2-3\mathit{xy}+y^2$ is indefinite.

(f)

$d=26^2-4\cdot 17\cdot 10=-4,a=17>0$, thus $17x^2-26\mathit{xy}+10y^2$ is positive definite.