3
$\begingroup$

I am interested in solving the following quadratic equation: $$x^{\top} A x = \sqrt{x^{\top} B x}$$

Here, $x \in \mathbb{R^q}$ is an unknown vector, and A and B are two q$\times$q-dimensional symmetric positive definite real matrices.

I am also interested, in particular, in the specific case where $A=f(B)$, with $f$ a linear function. Many thanks!

Improve this question
$\endgroup$
2
  • 5
    $\begingroup$ If I am not mistaken, you can always find solutions by taking an arbitrary vector $v$ and rescaling it suitably. $\endgroup$
    – Federico Poloni
    Commented Oct 23, 2020 at 13:25
  • 2
    $\begingroup$ Diagonalizing $A$ and $B$ simultaneously, on obtains an equation $(\sum a_i y_i^2)^2 = \sum(b_i y_i^2)$ for real numbers $a_i, b_i$, and a vector $y=(y_1,\ldots,y_q)=Lx$, where $L$ is a real $q \times q$ matrix. Substituting $z_i=y_i^2$ one obtains a considerably simpler equation. $\endgroup$
    – Martin Seysen
    Commented Oct 23, 2020 at 16:30

0

Reset to default

You must log in to answer this question.