I have the somewhat ugly expression:
$((V_iV_j\lambda_2 + X_iX_j\lambda_1)^2 + \lambda_1\lambda_2(V_iX_j - X_iV_j)^2)^{\frac{1}{2}}$
Every single term here is a scalar over the reals.
The goal is to try to isolate that left term in the sum i.e I need to get either:
$((V_iV_j\lambda_2 + X_iX_j\lambda_1)^2)^{0.5} + T$
Where $T$ is some term that makes the expression equal to the one I gave. This is the nicest possible result. If that is not possible then at least something of the form:
$((V_iV_j\lambda_2 + X_iX_j\lambda_1)^2)^{0.5} \cdot T$
Once again $T$ being some expression that makes the claim true.
I have tried and experimented with completing the square and adding 0, multiplying by 1 and similar tricks but I can't find a good way to change the expression into either of the above forms.
I hope this helps \begin{align} ((V_iV_j\lambda_2 + X_iX_j\lambda_1)^2 + \lambda_1\lambda_2(V_iX_j - X_iV_j)^2)^{\frac{1}{2}} = \Big((V_iV_j\lambda_2 + X_iX_j\lambda_1)^2\Big)^{1/2}\left(1 + \frac{\lambda_1\lambda_2(V_iX_j - X_iV_j)^2}{(V_iV_j\lambda_2 + X_iX_j\lambda_1)^2}\right)^{1/2} \end{align} valid when the first term is non zero.