On covariance and covariance matrix formula It is well known that the covariance (x1-e) is not one of the covariance (x1-1)

Because the covariance matrix is an estimate, that is, the population variance is estimated by the sample variance. In order to meet the unbiasedness, we use n-1. We can check any probability and statistics textbook for specific derivation. Unbiasedness means that the expected value is equal to the real value

What is the module of a matrix
How to define the module of a matrix?

If the mark you see is ||||||||||||||||||||||||||||||
If the sign you see is | a |, then this determinant det (a) is often used to represent a, and sometimes it is also used to represent the matrix obtained by moduling all the elements of A

Covariance matrix?
What is it?

For example, X is a variable, μ is the mean, and the covariance matrix is equal to E [(x - μ) (x - μ) ^ t]. The physical meaning is like this, for example, x = (x1, X2,..., Xi), then the number of columns n in row m of the covariance matrix is the covariance of XM and xn, if M = n, If the elements of X are independent, then only the diagonal of the covariance matrix has a value, because if x is independent, the covariance of XM and xn is 0 for m ≠ n. in addition, the covariance matrix is symmetric
Covariance matrix is usually used in multivariable distribution (such as multivariate Gaussian distribution). Covariance matrix is also used to analyze the properties of uncertain stationary signal and define the distance of uncertain vector (Mahalanobis norm) in engineering

On covariance and covariance matrix formula It is well known that the covariance (x1-e) is not one of the covariance (x1-1)