I have this equation:
$$H=\frac{t\cdot n^T}{n^T\cdot x}$$
with $t$, $n$ and $x$ being $3\times 1$ column vectors and $H$ a $3\times 3$ matrix, and where $\cdot$ is matrix multiplication. Notice that on the RHS the numerator results in a $3\times 3$ matrix and the denominator in a scalar.
How can I isolate the vector $n$?
UPDATE:
This is the part of a larger equation that if I can solve I can solve the original. I've been messing around but I'm stuck and I don't know how to solve for vector n. I have tried to use sympy to isolate n but there I have some other errors which I have ask here (slightly different form of the equation). This is not homework.
If $n = n_0$ satisfies the equation $H = \dfrac{tn^T}{n^Tx}$, then so does $n = Cn_0$ for any constant $C \neq 0$.
So, we can only determine $n$ up to a constant factor.
To get one solution for $n$, we can do the following:
$$H = \dfrac{tn^T}{n^Tx}$$
$$t^TH = \dfrac{t^Ttn^T}{n^Tx}$$
$$t^TH = \dfrac{t^Tt}{n^Tx}n^T$$
$$H^Tt = \dfrac{t^Tt}{n^Tx}n$$
$$\dfrac{n^Tx}{t^Tt}H^Tt = n$$
So $n = H^Tt$ up to a scalar constant factor (provided that $t^Tt = \|t\|^2 \neq 0$).
It's not hard to check that $n = CH^Tt$ satisfies the equation $H = \dfrac{tn^T}{n^Tx}$ for any constant $C \neq 0$.