For which values, $a$ satsfies that $(x,y) = x_1y_1 + x_2y_2 + x_3y_3 + ax_2y_3 + ax_3y_2$ is an inner product in $R_3$

Question

For which values, $a$ satsfies that $(x,y) = x_1y_1 + x_2y_2 + x_3y_3 + ax_2y_3 + ax_3y_2$ is an inner product in $R_3$

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I looked at the properties of inner product, and it seems that only the the property that satisfies: $$ (x,x) \geq 0 $$

May help.

But i got:

$$ (x,x) = x_1^2 + x_2^2 + x_3^2 +ax_2x_3 + ax_3x_2 \geq 0 $$

Getting:

$$ a \geq \frac{-x_1^2 - x_2^2 - x_3^2}{2x_2x_3} $$

But the answers say:

$$ |a| < 1 $$

How did they get this and why?

Thanks.

Answer 1

Hint

$$(x,y)=x^T\begin{pmatrix}1&0&0\\0&1&a\\0&a&1\end{pmatrix}y$$

Answer 2

Hint

$$ x_1^2 + x_2^2 + x_3^2 +2ax_2x_3 =x_1^2+(x_2+ax_3)^2+(1-a^2)x_3^2$$

Alternately, the above is a quadratic form. Write the correspoding symmetric matrix, and check when the eigenvalues are positive.