How can I find a matrix from the multiply of this matrix and it's transpose

Question

I have a matrix $B$ which it's dimension is $nm$ (with $n>m$). During an iterative process I'll change it to get a desired state of matrix $B$, but in each step, I should check a constraint that $\text{Trace}(B^TB)$ must be equal to an integer number $M$. Here $B^T$ is the transpose of $B$.

To meet the constraint, in each step, after getting new state of $B$, I do blow semi code to keep the trace of $B^TB$ equal to the given constant $M$.

$$x = B^TB$$
$$\hat{x} = \frac{M}{\text{Trace}(x)}x$$

actually $\hat{x}$ meets the constraint and now the Trace$(B^TB)$ is equal to $M$. But how can I find the matrix $B$ from $\hat{x}$ or how can I calculate $B$ from multiply of this matrix and it's transpose.

Answer 1

You calculate $x$ as above, call $a = \sqrt{\frac{M}{\text{Trace}(x)}}$. The new $B$ is $aB$.

Answer 2

In general, you cannot hope to find $B$ by knowing $B^t B$. For instance, if $B$ is any rotation matrix, then $B^t B = I$. So if you're told that $B^t B = I$, then you'd have infinitely many possibilities for $B$.