Issues-with-covaraince-matrix.md

@@ -128,4 +128,10 @@ The vertical dashed lines in the plots refer to a threshold of maximum rank for
 The vertical lines of GW190412 and GW190814 events are drawn at k=370 and k=330, respectively.
 
 ### Envelop of the spread of difference norms
-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 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" >
+
+
+