When does a matrix have $LDL^T$ decomposition?

Question

I have limited experience with matrix analysis, yet I've come across several related questions in practical applications.

In all questions, permutations are allowed, i.e., it's ok to have $A=PLDL^TP^{-1}$.

1. Suppose that $A$ is a $n\times n$ real symmetric matrix. What is the necessity and sufficiency for $A$ to have $LDL^T$ decomposition (say $L$ is a lower triangular matrix and $D$ is a diagonal matrix)?

2. What if $A$ is complex symmetric (not Hermitian)?

3. What if $A$ is Hermitian?

4. Can arbitrary invertible complex symmetric matrix $A$ have $LDL^T$ decomposition?

As for $LU$ decomposition with permutations, it seems that $A$ has $LU$ decomposition $\Leftrightarrow$ $A$ is invertible.

Anything could be helpful! Thank you in advance.

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