For an axis-aligned ellipsoid
$a_{11}x_1^2+a_{22}x_2^2+a_{33}x_3^2+a_{44}x_4^2=1$
it is easy to see that $x_1$ is in the range $-1/\sqrt{a_{11}}$ to $1/\sqrt{a_{11}}$. However, in the general case of a rotated ellipsoid,
$\sum_{i,j} a_{ij}x_ix_j=1$ (eq. 1)
where $a_{ij}$ forms a positive definite matrix, how do I calculate the limits for $x_1$?
In other words, I want to project the ellipsoid (which in my case has ~100 000 dimensions) onto its first dimension.
What I have come up with so far is solving eq. 1 wrt. $x_1$
$x_1 = \frac1{a_{11}}\left[-\sum_{j\neq1}x_ja_{1j}\pm\sqrt{a_{11}(\sum_{i\neq1, j\neq1}x_ix_ja_{ij})-((\sum_{j\neq1}x_ja_{1j})^2) }\right]$
and taking its first derivative with respect to the other $x_k$
$\frac{dx_1}{dx_k}=\frac1{a_{11}}\left[-a_{1k}\pm\frac{a_{11}x_k(\sum_{j\neq1}x_ja_{kj})-A_{1k}}{\sqrt{a_{11}(\sum_{i\neq1, j\neq1}x_ix_ja_{ij})-((\sum_{j\neq1}x_ja_{1j})^2) }}\right]$
to exploit the necessary condition $\frac{dx_1}{dx_k}=0$ for extrema. However, the resulting system of equations does not look like I could find a general solution easily, especially since all the sums still contain $x_k$ :-(.
Is there possibly a much smarter way to solve this problem? Or am I stuck with plugging the formula for $x_1$ and its derivative into a numerical solver to find minimum and maximum?