Curious how to retrieve original vector $x$ from an $x^Tx$ operation I am doing some analysis on a signal $x(t)$ which is a vector of length $m$. After performing the operation $x^Tx$ of the vector on itself, I get a matrix $Y$ of size $m\times m$. Now, how can I retrieve the original $x(t)$?
Should this be the elements in the diagonal of $Y$? I did some basic plotting and while the diagonal looks similar to original signal vector, they are not equal. Looks straightforward but I am missing something, probably need to normalize it somehow.
In summary,

*

*Is it always possible to retrieve the original signal without information loss?


*If it is possible, how is it done?

The diagonal holds the squares of the original vector.  With it, and the signs of the other entries, you can find $x$ or $-x$ but can't decide between the two.
