For a real positive definite matrix $A$, we can find the Cholesky decomposition $LL^T = A$. If we relax the constraints that L has to be lower triangular, we should be able to find a whole host of decompositions $FF^T=A$. Is there anything known about how to find the general solution to this?

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    $\begingroup$ If you have $G$ such that $GG^T=I$ then $(FG)(FG)^T=FGG^TF^T=FF^T = A$. Not sure if it covers all possibilities though. The other famous option is the matrix composed of eigenvectors times square roots of eigenvalues. $\endgroup$ – Yulia V Oct 7 '14 at 9:28

I think that the following is correct:

We use the QR decomposition: $F^T=QR$ where $Q$ is orthogonal and $R$ is upper triangular. Then $A=FF^T=R^TQ^TQR=R^TR$. Then it remains to solve the equation in $L$: $A=LL^T$ where $L$ is lower triangular.

Conversely, We apply to $A$ the Cholesky's algorithm except that we do not necessarily put on each element of the diagonal of $L$ the positive square root. It is easy to see that we have $2^n$ possible values for the diagonal $(L_{i,i})_{i,i}$ and consequently, it is not difficult to see that we obtain $2^n$ distinct values for the lower triangular matrix $L$. In the following way we obtain all the required solutions of our problem: $F=QR$ where $Q$ is any orthogonal matrix and $R$ is one of the $2^n$ solutions of the previous equation.

Note that it is linked to the Yulia V's comment.


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