Casting a function as quadratic

Question

I would like to represent the following function

$$f(\mathbf{x}) = \sqrt{\|\mathbf{y}\|^2 - \langle\,\mathbf{x},\mathbf{y}\rangle ^2}$$

as quadratic, meaning the structure should be:

$$ \mathbf{x}^TA\mathbf{x} + \mathbf{b}^T\mathbf{x} + c $$

Notice that in this function $\mathbf{y}$ is a parameter, not a variable. The part I'm "stuck" with is $2x_ix_jy_iy_j$ under the sqrt. Is there a way for this to be done?

Thanks!

Answer 1

This cannot be done. Take the one dimensional case and let $y=1$. You cannot write $\sqrt {1-x^{2}}$ as a quadratic in $x$.