Consider a complex square matrix $A\in\mathcal{C}^{n\times n}$. Now let us discuss two kinds of the factorization of $A$, say, eigendecomposition and Schur decomposition because both of them are often used to factorize square matrices.
My concerns are as follows:
Which one of two above-mentioned decomposition techniques is preferable to the other?
Which case we should use eigendecomposition instead of Schur decomposition? Also, could you please give me a detailed explanation?
Given that $A$ has extra properties: Hermitian positive definite matrix. In this case, will Cholesky be a better choice than two above-mentioned techniques?
1-2: It depends, and it's hard to say that one is inherently preferable. It's also good to consider Jordan form decomposition, which generalizes Eigendecomposition.
Eigendecomposition, when possible to find, makes computations easier, especially where polynomial and power series are involved.
Schur decomposition makes more sense when conjugate-transposes (adjoints) are involved
Schur decomposition is easier to find numerically
Schur decomposition, unlike Eigendecomposition, works for any square matrix
3: Cholesky decomposition is used for different reasons. It's akin to $LU$ decomposition, but is the better option when possible.