Why do we calculate eigenvalues instead of using Sylvester’s criterion(wiki)?
They both give us the same results in the end, but Sylvester’s criterion is a bit easier for hand calculation.
Hello @denkalenichenko and welcome to the DeepLearning community!
Quite a tough one! 
but I do think it’s because the Eigenvalues will give us more values that will help us determine the behavior of the matrix and find its eigenvectors, and determine its stability. while Sylvester’s criterion (I am not much familiar with it!) will only help us know whether a given matrix is positive definite or not.
Happy Learning
Isaak