Let $Q(x_1,x_2, \ldots ,x_n)$ be a positive definite real quadratic form in the variables $x_1, \ldots ,x_n$. It is not hard to see that the function $f(x_1, x_2, \ldots ,x_n)=\frac{Q(x_1,x_2, \ldots ,x_n)}{x_n^2}$ attains a minimum value on ${{\mathbb R}^{n-1}} \times {{\mathbb R}^*}$, an that this value (call it $\lambda$) is positive.
Then $R(x_1,x_2, \ldots, x_n)=Q(x_1,x_2, \ldots ,x_n)-{\lambda}x_n^2$ is a nonnegative real quadratic form. Is it true that $R$ can always be written as a sum of $n-1$ squares (of linear forms) ?