3

In computer vision, there are lots of formula came in being with geometry form. And this equation is very about Fundamental matrix in epipole geometry. So, can some one proof this equation? $${\displaystyle (M\mathbf {a} )\times (M\mathbf {b} )=(\det M)\left(M^{-1}\right)^{\mathrm {T} }(\mathbf {a} \times \mathbf {b} )=\operatorname {cof} M(\mathbf {a} \times \mathbf {b} )}$$

Ben Grossmann
  • 225,327
ZhaoAlpha
  • 31
  • See this post. By the way, I've rolled this post back to include your original version of the equation. The only thing missing was the pair of $$ delimiters. – Ben Grossmann Sep 10 '20 at 07:29
  • @Ben Grossmann, feel free to do that :)! – ZhaoAlpha Sep 10 '20 at 07:34
  • There are some other proofs given here that you might find helpful. I like Orman's proof, but I suspect that it's presented a bit too succinctly for those not used to these kinds of linear algebra proofs. – Ben Grossmann Sep 10 '20 at 07:38
  • If there is something specific that you would like clarified (i.e. you're stuck on a step from one of the proofs or you would like a proof via a specific alternative approach), then please say so. Otherwise, I'd like to close this question as a duplicate. – Ben Grossmann Sep 10 '20 at 07:40
  • I know $|M^T|$ = |M|, but why ⟨Ma×Mb,Mx⟩=⟨MT(Ma×Mb),x⟩ in Omran Kouba proof given by your link above – ZhaoAlpha Sep 10 '20 at 07:53
  • In general, we have $\langle x, My \rangle = \langle M^Tx, y\rangle$. To put this strictly in matrix-multiplication notation, we have $$ x^T(My) = (x^TM)y = (M^Tx)^T y $$ – Ben Grossmann Sep 10 '20 at 07:57
  • can u give me some inspiration about det(u,v,x)=⟨w,x⟩,in other words, how can i derive dot product from triple scalar product ? thank u! – ZhaoAlpha Sep 10 '20 at 08:02
  • I really think that you should try to look at my own proof, which is the first proof that I linked. It presents a proof that roughly follows the same steps as Orman's, but is framed entirely in terms of matrices. – Ben Grossmann Sep 10 '20 at 08:11
  • Also I don't understand your latest question. The vector triple product is written with a dot product. So what exactly do you mean by "derive dot product" from the scalar triple product? – Ben Grossmann Sep 10 '20 at 08:15
  • ok, I mean how can I compute this formulation <a$\times$b,c >=<a,b,c>.By the way, I have found its proof. And thank u for your answers! – ZhaoAlpha Sep 10 '20 at 09:51
  • And , u can close this question,whenever u want .And thank u for your patiently answers. – ZhaoAlpha Sep 10 '20 at 09:53

0 Answers0