Maximum Likelihood Estimation of Multivariate Normal distribution parameters

In the multivariate statistical analysis, the Multivariate Normal Distribution has a core position (which is easy to be compared with the one-dimensional statistical analysis). Today, we use its distribution density function and maximum likelihood estimation (ML) the simple derivation process and results are recorded here for me to lay the foundation for moving towards SEM. First, the density function:

For samples y ~ from Multivariate Normal Distribution population ~ Nm (μ, V ),

Obviously, it is easy to write the joint distribution density of the N samples:

According to the regular routines of ML, take the logarithm (note to write convenience now orders = V-1 ):

Now we need to introduce several marks based on the derivation:

The specific derivation process is far from being used. It is troublesome and incredible. Due to the fact that my basic matrix knowledge is not strong, I have not been able to read the detailed process. The final result is: