The SVD of the matrix $A$ is $A = U \Sigma V^T$, where $A\in R^{m\times n}$ is symmetric positive definite or semi positive definite matrix and $U$ and $V$ are square orthogonal matrices.
Does $A$ has to be positive or semi positive? If $A\in R^{n\times n}$ is symmetric but indefinite can we still have a SVD? Also if $A\in R^{m\times n}$ but $U$ and $V$ are not orthogonal will $A$ and $UAV$ still have same singular values?
Basicaly I want to understand if we can have SVD for any matrix and if the case when $A$ is symmetric positive definite with orthogonal eigenvectos just a special case of SVD?
SVD decomposition exists for any real matrices. No other additional requirement is needed. $\Sigma$ share the same size as $A$ and $U$ and $V$ are orthogonal matrices.
As for your another question when $U$ and $V$ need not be orthogonal, not true in general, for example if you let $U=0$ then the singular value is just $0$.
When a matrix is symmetric, we know that it has an eigenvalue decomposition, that is you can pick $U$ and $V$ to be the same orthogonal matrix.