Say $g$ is a ($d\times d$) matrix which is given as, $g = g_0 + xg_2 + x^2 g_4 .. +x^{d/2 -1}g_{d-2}+ x^{d/2}(g_d + h_d(log (x)))$ where $d$ is an even number and each $g_i$ is a matrix (same dimension as $g$) and $h_d$ is another matrix.
- For such a set of arbitrary matrices, how can one power-series expand $\sqrt {det(g)}$ in $x$?
One related formula (which might help derive the above) that I am aware of is this,
$\sqrt{I + M } = I + \frac{Tr M}{2} + \frac{1}{8} ((Tr[M])^2-2Tr[M^2]) + \frac{1}{48} ( 8Tr[M^3] - 6 Tr[M^2]Tr[M] + (Tr M)^3) + O(M^4) $
- Can someone help derive the above?