Having large matrices, $W$ (the unknown) and $M$ (known), is it possible to solve for $W$ in this equation $$W \cdot W^{T} = M,$$ where $M$ can have negative entries.
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You might be looking for the Cholesky decomposition. The referenced article also contains an example for $M$ having negative entries. Note the constraints on $M$ for this decomposition to exist.
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$\begingroup$ Even in Cholesky decomposition, the equation is solved by assuming $W$ to be made up of rows containing $a_{11},a_{12}\ldots a_{1n}$ and then equating the corresponding entries of $WW^T$with $M$. Isn't this the brute force algorithm? $\endgroup$– nitishchCommented Mar 5, 2014 at 15:07