Timeline for Should the formula for the inverse of a 2x2 matrix be obvious?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Aug 14, 2020 at 19:14 | comment | added | LSpice | @YemonChoi, done. | |
| Aug 14, 2020 at 18:46 | comment | added | Yemon Choi | @LSpice I've deleted my well-meant but perhaps ill-judged comment, so perhaps you can delete your response to me | |
| Aug 14, 2020 at 18:15 | comment | added | LSpice | Perhaps @FedericoPoloni's point was that you claimed that "remember the formula in special cases" could be made a rigorous proof, and that, while it certainly can, it requires first observing that the proposed formula for the inverse is anti-multiplicative—surely an unappealing recipe for intuition? | |
| Aug 14, 2020 at 16:48 | comment | added | Daniel Litt | @FedericoPoloni: write a generic matrix as a product of generic elements of N, A, K, invert each and multiply the results together. Then observe that the formula you get agrees with the standard one. I agree this certainly isn't a good computational tool! | |
| Aug 14, 2020 at 15:45 | comment | added | Federico Poloni | How does this method yield the formula for the inverse? If I understand correctly, what you suggest is (1) compute this Iwasawa decomposition of $A$ (2) invert each term one by one, to get a different decomposition of $A^{-1}$ (because the factors are in the reverse order). But at this point why not using any other classical matrix decomposition, for instance the SVD? And how does it help me computing the inverse of $\begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}$, assuming that I cannot compute its Iwasawa decomposition in my head? | |
| Aug 14, 2020 at 15:38 | history | edited | Daniel Litt | CC BY-SA 4.0 |
typos, brief warning
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| Feb 23, 2012 at 3:55 | comment | added | Frank Thorne | Since KConrad entered the discussion, I'll mention that he wrote up a great treatment of the Iwasawa decomposition: math.uconn.edu/~kconrad/blurbs/grouptheory/SL(2,R).pdf | |
| Feb 22, 2012 at 18:19 | comment | added | Daniel Litt | I also wanted to note that Frank's method of using a few special cases where the geometry was obvious (e.g. unipotents) to remember the general formula actually amounts to a geometric proof of the general formula, if one does enough geometric special cases. | |
| Feb 22, 2012 at 18:14 | comment | added | Daniel Litt | @KConrad: In practice I do actually just recall the formula from memory; but just dredging it up from memory isn't by favorite way to remember it. In my ideal world, perhaps, we would have much less burned into our brains in high school; rather we would develop understanding and intuition (like this and other answers purport to give). On the other hand, I guess, sometimes you just gotta invert some $2\times 2$ matrices, and thinking about the upper half-plane is probably not the easiest way to do that ;-). | |
| Feb 22, 2012 at 15:19 | comment | added | KConrad | Daniel, you really remember the inversion for 2 x 2 matrices by this method? I remember it the same way I remember the quadratic formula: I burned it into my brain back in high school. What you describe seems more like a way to understand the formula than to remember it. | |
| Feb 21, 2012 at 15:07 | vote | accept | Frank Thorne | ||
| Feb 21, 2012 at 15:07 | comment | added | Frank Thorne | This is a spectacular answer. +1, sir. | |
| Feb 21, 2012 at 2:58 | history | answered | Daniel Litt | CC BY-SA 3.0 |