A problem about calculation of linear algebra and Matlab coding for solve it?

Question

I have encountered questions about a problem of a derivation of math. The question is shown as in the image under. Does any body can provide some suggestions or insight into it, thank you very much.

Question has been captured, please take a look on this image to see formula problem.

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I have also a question for matlab to solve it: I have also ask questions in matlab answers

Answer 1

Note that 4.24 can be rearranged to say $[r_1, r_2, T_0] = A_c^{-1}G^{-1}$. By 4.27, you can make an arbitrary choice, and obtain a matrix $G_0$ such that the actual $G = gG_0$, where $g$ is the scaling factor referred to. So

$$r_1 = ge_1^T(G_0A_c)^{-1}\\r_2 = ge_2^T(G_0A_c)^{-1}$$ where $e_1^T = [1, 0, 0]$ and $e_2^T = [0, 1, 0]$.

Therefore $$1 = \|r_1\|_2 = |g|\|e_1^T(G_0A_c)^{-1}\|_2$$ so $$|g| = \frac 1{\|e_1^T(G_0A_c)^{-1}\|_2}$$ I don't see anything that determines the sign, though.

Answer 2

@Paul Sinclair
Great! Thank you very much!
I think the problem solving idea i instructive: if we want to use (4.22), we need to get representation of r1 an r2.
But I think There is a problem in deduction:
$$\begin{array}{l}[{r_1},{r_2},{T_0}] = {A_c}^{ - 1}{G^{ - 1}}\\{(\lambda A)^{ - 1}} = \frac{1}{\lambda } \times {A^{ - 1}}\\{r_1} = ({A_c}^{ - 1}{G^{ - 1}}){e_1} = ({A_c}^{ - 1}{(g{G_0})^{ - 1}}){e_1} = ({A_c}^{ - 1}\frac{1}{g}{G_0}^{ - 1}){e_1} = \frac{1}{g}({A_c}^{ - 1}{G_0}^{ - 1}){e_1} = \frac{1}{g}{({G_0}{A_c})^{ - 1}}{e_1}\\{r_1} = ({A_c}^{ - 1}{G^{ - 1}}){e_2} = ({A_c}^{ - 1}{(g{G_0})^{ - 1}}){e_2} = ({A_c}^{ - 1}\frac{1}{g}{G_0}^{ - 1}){e_2} = \frac{1}{g}({A_c}^{ - 1}{G_0}^{ - 1}){e_2} = \frac{1}{g}{({G_0}{A_c})^{ - 1}}{e_2}\\{e_1} = {[1,0,0]^T},{e_2} = {[0,1,0]^T}\\1 = {\left\| {{r_1}} \right\|_2} = \left| {\frac{1}{g}} \right|{\left\| {{{({G_0}{A_c})}^{ - 1}}{e_1}} \right\|_2}\\\left| g \right| = {\left\| {{{({G_0}{A_c})}^{ - 1}}{e_1}} \right\|_2}\end{array}$$ I also a problem to consult:
Now I need to put math to application.
But I'm confused how to write matlab code (I have also ask questions in matlab answers)algorithm for this solve question.
Because G_0 has symbolic g , matlab just get numbers without g, how to do it in matlab to calculate G with g1~g9?

This images shows the matlab questions:I have no enough reputation to post images