Issues-with-covaraince-matrix.md

@@ -131,7 +131,10 @@ The vertical lines of GW190412 and GW190814 events are drawn at k=370 and k=330,
 We can see from the above plots that the difference of norms of $`\Sigma(k)\Sigma^{-1}(k)`$ and $`I(k)`$ is flaring-up after a certain rank. We can set a number for the flare-up by looking at the envelope of
 the spread of the difference of norms. The spread is basically the standard deviation of this quantity.
 
-<img src="uploads/caa72d6cc1dcdf4ad4b26bc6babc5bf7/envelop_gw190412.png"  width="440" >
+<img src="uploads/e40fad036f1eb14b88dc87c04afbef4a/envelop_gw190412_1_.png"  width="440" >
 
+The quantity $`\mu(k)`$ is the mean of the difference of norms and $`\sigma(k)`$ is the standard deviation.
+The 10 largest likelihood samples are chosen to compute the spread. We can find the k_max for which the distance between these two curves along the y-axis is greater than a threshold, approximately O(1e-7).
+In our analysis, we set this threshold to be $`2.4 \times 10^{-7}`$.