Have a matrix dot its transpose, what is the original matrix?

I have a matrix that equals the dot product of a matrix A with its transpose. How do I get the matrix A?

Ex.: $$AA^T$$ = [a given matrix]

• In general there will be multiple solutions. That is so say, one cannot uniquely recover $A$ from the given data. Even in the case of $1\times1$ matrices. – kimchi lover Feb 18 at 1:09
• You can't. If you have symmetric $B$ also positive definite, it can be written as $B = C C^T.$ However, given any orthogonal $P,$ meaning $PP^T = P^T P = I,$ we also have $B = (CP)(CP)^T$ – Will Jagy Feb 18 at 1:09

In general, you can't. For example, $$[1\ 8]\begin{pmatrix}1 \\ 8\end{pmatrix} = [4\ 7]\begin{pmatrix}4 \\ 7\end{pmatrix} = [65].$$