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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:

1) Which one of two above-mentioned decomposition techniques is preferable to the other?

2) Which case we should use eigendecomposition instead of Schur decomposition? Also, could you please give me a detailed explanation?

3) Given that $A$ has extra properties: Hermitian positive definite matrix. In this case, will Cholesky be a better choice than two above-mentioned techniques?

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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.

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