Have a matrix A that's 4x3 and a matrix B that is known. They relate by the following equation: $$ A^T A = B $$ Trying to find A, given B, such that $A^TA$ is as close to B as possible.
I've tried using a least square error, but I get stuck trying to solve $\delta A^T / \delta A$ which I'm not sure how to do.
I've also thought about using the SVD but not sure how to transform that into the above once it's found.
I'm assuming that there will be multiple solutions to this problem, but any help on solving would be great--even if there are a set of solutions.
$\endgroup$ 11 Answer
$\begingroup$Since $B$ is square and symmetric ($B=A^TA$) it has an SVD of the form
$$B=U\Sigma U^T$$
We can augment now an arbitrary vector $a\in\mathbb{C}^3$ such that
$$B=[U \quad a]\left[\matrix{ \Sigma & 0 \\ 0 & 0}\right]\left[\matrix{U^T\\a^T}\right]$$
Thus,
$$A=\left[\matrix{ \Sigma^{1/2} & 0 \\ 0 & 0}\right]\left[\matrix{U^T\\a^T}\right]=\left[\matrix{\Sigma^{1/2}U^T \\ 0}\right]$$
