existance of a solution to quadratic form equation

Question

Let $\lambda$ is an unknown scalar and;

$Q=Q_1 - \lambda*Q_2$ where $Q_1, Q_2$ are $NxN$ positive symetric matrices,

$B=B_1 - \lambda*B_2$, where $B_1, B_2$ are $Nx1$ vectors,

$m=m_1 - \lambda*m_2$, where $m_1, m_2$ are scalars,

The problem is, i want to find the maximum value of $\lambda$ which a solution to the following equation exist:

$x'*Q*x + x'*B +m=0$

where $x$ is a $Nx1$ solution vector.

Thank you

Ayhan

Answer 1

If $m_2<0$, then there is no maximum value of $\lambda$. Indeed, there is $u>0$ s.t. $\lambda>u$ implies that $m>0$ and $Q$ is symmetric $<0$. Let $f(x)=x^TQx+x^TB+m$; then $f(0)>0$ and, for sufficiently great $||x||$, $signum(f(x))=signum(x^TQx)=-1$.