# SVD of symmetric but indefinite matrix

The SVD of the matrix AA is A=UΣVTA = U \Sigma V^T, where A∈Rm×nA\in R^{m\times n} is symmetric positive definite or semi positive definite matrix and UU and VV are square orthogonal matrices.

Does AA has to be positive or semi positive? If A∈Rn×nA\in R^{n\times n} is symmetric but indefinite can we still have a SVD?
Also if A∈Rm×nA\in R^{m\times n} but UU and VV are not orthogonal will AA and UAVUAV still have same singular values?

Basicaly I want to understand if we can have SVD for any matrix and if the case when AA is symmetric positive definite with orthogonal eigenvectos just a special case of SVD?

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If AA is m×nm \times n and m≠nm \ne n, it can’t be symmetric.
– Robert Israel
2 days ago

In that question it is stated that A∈RnxnA\in R^{nxn}
– Biljana
2 days ago

In the first line it says m×nm \times n.
– Robert Israel
2 days ago

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