I have a real matrix $A \in \mathbb{R}^{n\times n}$ such that:

• $A$ is symmetric
• All the off-diagonal terms are known and positive
• Has rank $k<n$

Unfortunately I don't know the values of the diagonal terms $A_{i,i}$, but I know that they are positive; what can be said in this case about SVD decomposition? Is there a way to calculate an approximate SVD having diagonal terms of the matrix to be decomposed undetermined?