Issues-with-covaraince-matrix.md

@@ -35,7 +35,7 @@ Now, we discuss that the covariance matrix is ill-conditioned (i.e., nearly sing
 The above plot shows the array of the eigenvalues. This plot indicates that the covariance matrix is ill-conditioned since several eigenvalues are close to zero. Also, the eigenvalues are oscillating around and after the index number 400 (the eigenvalues below ~1e-11), which occurs due to the numerical inaccuracies. This can affect the inverse calculation since the inverse is proportional to the determinant of that matrix. Therefore, we impose a threshold on eigenvalues to calculate the inverse in the `pinv` method. In other words, a reduced singular matrix can be used in the SVD based inverse computation, which can capture the maximum feature of the covariance matrix and also excludes the numerical instabilities. The parameter `rcond` in `numpy.linalg.pinv` function controls this criterion such that singular values less than or equal to `rcond * largest_singular_value` are set to zero.
 
 # Choice of the rank of the covariance matrix
-
+## GW190814
 First, we demonstrate an example from the GW190814 event to find an appropriate cut-off on the low-rank approximation of the covariance matrix, and its effect on $`\beta`$. We choose top 5 PE samples and plot $`\beta`$ as a function of the rank of the covariance matrix. The spectrum of the eigenvalues is also plotted. 
 
 <img src="uploads/c93fa55e8c322397cdd710513d862054/beta_lambda_all.png"  width="440" >