Retrieving unknown $A$ from $B = AA^T$

Question

Suppose I have a matrix $B = AA^T$, such that A is positive definite but generally not symmetric.

Is there a way to retrieve the matrix $A$ from $B$ when $A$ is unknown?

Answer 1

As already has been stated, there is no unique solution to the problem. We can however characterize the solution space:

Let $A^T = QR$, where $Q$ is an orthogonal matrix, and $R$ is an upper triangular matrix in row echelon form (this is called QR-decomposition).

Then $B= AA^T = R^TQ^TQR = R^TR$. Since $R$ is in row echelon form, there is a unique solution for $A$ (see e.g. the Cholesky-decomposition).

On the other hand, as $Q$ annihilates itself in $AA^T$, we can pick it freely, i.e. $Q$ is any orthogonal matrix of fitting dimension.

Answer 2

Since you ask about “positive definite but generally not symmetric” matrices, I’ll assume that we’re talking here about square matrices.

Let $Q$ be an orthogonal matrix of the same order as $A$. Then $$(AQ)(AQ)^T = A(QQ^T)A^T=AA^T,$$ so outside of a few special cases such as $B=0$, if you have any solution to $B=AA^T$, then there is an infinite number of them.