Can vectors be inverted? I wish to enquire if it is possible to solve the below for $C$.
$$B^{-1}(x-\mu) = xc $$
Here obviously $B$ is an invertible matrix and both $c$ and $\mu$ are column vectors.
Would the solution be $$x^{-1}B^{-1}(x-\mu) = c $$ is it possible to invert vectors ?
How about if it was the other way
$$B^{-1}(x-\mu) = cx $$
Is there any other way to do this ?
Thanks in advance. 

Vectors, in general, can't be inverted under matrix multiplication, as only square matricies can inverses. However, in the situation you've described, it's possible to compute $c$ anyway, assuming the equation is satisfied for some $c$. If we multiply both sides by $X^T$, the result is $x^T B^{-1} (x-\mu) = x^T x c = |x|^2 c$. Assuming x is not the zero vector (in which case any $c$ will still have $xc=0$ so any choice of $c$ should work), we just get $c= \frac{1}{|x|^2} x^T B^{-1} (x-\mu)$.
I must caution that the expression above for $c$ is defined even when there is no solution of the original equation, which will be almost all of the time for randomly generated vectors and matrices. Hence, if you are going to use it, you should check that this works by plugging what you get for $c$ back into the original expression and see if it works. Also, the choice of $x^T \over |x|^2$ is not unique; any row vector $v$ such that $vx=1$ will work equally well in the above expression.
