what is signature of q?

Question

For $α ∈ R$, let $q(x_1, x_2) =x_1^2 + 2αx_1x_2 + \frac{1}{2}x_2^2$ for $(x_1,x_2) \in R^2$

b) find all values of $\alpha$ for which the signature of q is 1.

my attempts : as i know that

signature (s) =total number of positive entries - total no of negative entries

=$ p -(r-p) = 2p-r$ where r is the rank of matrix

$q = \begin{bmatrix} 1& 1\\1&\frac{1}{2}\end{bmatrix}$

now by elementray operation

$q = \begin{bmatrix} 1& 1\\0&0\end{bmatrix}$

here $s = 2.1 -1$ where$ r= 1,p =1 $

therefore signature(s) of $ q =1 $for $\alpha =1$

is my answer is correct or not ?????

Answer 1

HINT

Let consider the associated matrix

$$\begin{bmatrix} 1& \alpha\\\alpha&\frac12\end{bmatrix}$$

by Sylvester criterion note that

• $\det(1)=1$
• $\det A=\frac12-\alpha^2\implies \det A>0 \iff-\frac{\sqrt 2}2<\alpha<\frac{\sqrt 2}2$