If I had a set of matrices a,b, and c, and I knew a*b=c, how would I find matrix a?
assuming;
matrix a= a 3x4 matrix, values unknown
matrix b= a 4x1 matrix, values = M1 M2 M3 M4
matrix c= a 3x1 matrix, values = 0 0 0
$\displaystyle \displaystyle \mathbf{A}\mathbf{B} = \mathbf{C}$
$\displaystyle \displaystyle \mathbf{A}\mathbf{B}\mathbf{B}^T = \mathbf{C}\mathbf{B}^{T}$
$\displaystyle \displaystyle \mathbf{A}\mathbf{B}\mathbf{B}^T(\mathbf{B}\mathbf {B}^T)^{-1} = \mathbf{C}\mathbf{B}^T(\mathbf{B}\mathbf{B}^T)^{-1}$
$\displaystyle \displaystyle \mathbf{A}\mathbf{I} = \mathbf{C}\mathbf{B}^T(\mathbf{B}\mathbf{B}^T)^{-1}$
$\displaystyle \displaystyle \mathbf{A} = \mathbf{C}\mathbf{B}^T(\mathbf{B}\mathbf{B}^T)^{-1}$.
Are you guaranteed that $\displaystyle BB^{T}$ is invertible? In fact, given that you're generating a whole matrix out of one column vector, I can't help but wonder if you aren't guaranteed that it doesn't have an inverse! Is there another way to do this, do you think, Prove It?