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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.

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martinako
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  • Welcome to MathSE! You are more likely to get a good answer to your question if you follow a few guidelines. In particular, what have you tried so far, and just where are you stuck? This is not a homework-answering site: we want to see that you have put some work into the problem. – Frentos Jan 18 '16 at 23:33
  • Ok, thanks. I have updated the question – martinako Jan 18 '16 at 23:40

1 Answers1

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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 $C \neq 0$.

JimmyK4542
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  • That's great! thanks. I only need the direction of n so that's perfect. I think I need to brush up my linear algebra! – martinako Jan 19 '16 at 00:29