Decomposition of a sum of matrix products with itself [closed]

matrices matrix-decomposition

Given tall matrix $B \in \mathbf{R}^{n \times r}$, where $n \gg r$, and vector $c \in \mathbf{R}^n$, let

$$ A A^T := B B^T + c c^T $$

I would like to get any possible matrix $A$ efficiently. If possible, smaller than a square matrix. Is there a solution? Would it help if $BB^T$ is a diagonal matrix?


$$A = \begin{bmatrix} B & c \end{bmatrix}$$

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