Issues-with-covaraince-matrix.md

@@ -17,9 +17,9 @@ To resolve this issue, we propose to use the Moore-Penrose pseudo-inverse method
 <img src="uploads/4692d03ff868fcc538ab06803102a7c3/pinv.png"  width="440" ><img src="uploads/8d64ddc718f1fe06d54b62ca6028d123/pinv_off_diag.png"  width="440" >
 In the above figures, the quantity 'pinv' refers to the `numpy.linalg.pinv` function. The figures indicate that the inner product using the right-hand inverse matrix is consistent with the left-hand inverse matrix, and also the off-diagonal elements are nearly equal to zero. Therefore, `pinv` more robust.
 
-Now, we discuss that the covariance matrix is ill-conditioned (i.e., nearly singular). Let us focus on the eignevalues of the covaraince matrix.
+Now, we discuss that the covariance matrix is ill-conditioned (i.e., nearly singular). Let us focus on the eigenvalues of the covariance matrix.
 <img src="uploads/b4604ec7877c560d6f7aa9232ffd4c08/eigenvalues1.png"  width="440" >
 
-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. The parameter `rcond` `numpy.linalg.pinv` function controls this criteria such that singular values less than or equal to `rcond * largest_singular_value` are set to zero.
+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.