Timeline for Why were matrix determinants once such a big deal?
Current License: CC BY-SA 2.5
36 events
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| Oct 17, 2017 at 1:15 | history | protected | Benjamin Steinberg | ||
| Jun 2, 2013 at 10:34 | comment | added | Zsbán Ambrus | I think determinants are still such a big deal. They come up in a lot of seemingly unrelated subjects. | |
| Sep 25, 2012 at 17:46 | answer | added | Jim Dukelow | timeline score: 3 | |
| Sep 25, 2012 at 17:20 | answer | added | Jorge Stolfi | timeline score: 6 | |
| Sep 25, 2012 at 16:38 | answer | added | Sigurd Angenent | timeline score: 14 | |
| Jun 7, 2012 at 2:13 | answer | added | kjetil b halvorsen | timeline score: 3 | |
| Dec 23, 2010 at 16:46 | answer | added | Jose Brox | timeline score: 6 | |
| Dec 22, 2010 at 21:33 | answer | added | BSteinhurst | timeline score: 5 | |
| Dec 22, 2010 at 20:10 | history | edited | arsmath | CC BY-SA 2.5 |
fixed typo in 'parallelepiped'
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| Dec 22, 2010 at 6:57 | answer | added | Vamsi | timeline score: 5 | |
| Sep 22, 2010 at 10:29 | answer | added | Denis Serre | timeline score: 130 | |
| Aug 19, 2010 at 3:00 | answer | added | Matus Telgarsky | timeline score: 8 | |
| Aug 19, 2010 at 2:57 | answer | added | Alexander Woo | timeline score: 6 | |
| Aug 19, 2010 at 1:46 | answer | added | Richard Stanley | timeline score: 66 | |
| Aug 18, 2010 at 22:33 | answer | added | Pace Nielsen | timeline score: 12 | |
| Aug 18, 2010 at 21:36 | answer | added | Andrew | timeline score: 5 | |
| Aug 18, 2010 at 20:59 | comment | added | Per Vognsen | Victor: Agreed that exterior algebra is a wonderful tool. One should elucidate its algebraic, geometric and combinatorial structure rather than try to banish its use. As for computing eigenvalues, the preferred tools in applications are Krylov subspace methods. Though you don't usually see it mentioned in books on numerical analysis, their structure is very much based on notions that fall out of this k[x]-module approach. | |
| Aug 18, 2010 at 20:41 | answer | added | KConrad | timeline score: 21 | |
| Aug 18, 2010 at 20:24 | comment | added | Victor Protsak |
Per, as the title of this paper clearly indicates, Axler is obsessed with eliminating a very useful tool, determinants, from linear algebra. But as any effort along these lines, it has a cost: if you need to compute the eigenvalues of a matrix, say $$\begin{pmatrix}1 & 2\\ 3 & 4\end{pmatrix}, $$ it's little consolation that the minimal polynomial exists by an abstract argument! To me, this doctrinal approach appears just as fruitless as the attempts to base real analysis on "constructible" numbers only (since countably many reals expressible in a finite way "should be sufficient").
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| Aug 18, 2010 at 20:03 | answer | added | Bruce Westbury | timeline score: 9 | |
| Aug 18, 2010 at 19:43 | comment | added | Per Vognsen | Victor: You can develop the theory of eigenvalues without using the characteristic polynomial as the jumping-off point. Axler's Down with Determinants paper (which should have been called Down with Characteristic Polynomials) implicitly takes the k[x]-module approach. For example, every linear map $T$ on a complex vector space $V$ has an eigenvalue because you can factor an annihilator polynomial of T by the fundamental theorem of algebra; an annihilator polynomial is guaranteed to exist by the finite dimensionality of End(v). | |
| Aug 18, 2010 at 19:15 | answer | added | Deane Yang | timeline score: 18 | |
| Aug 18, 2010 at 19:03 | answer | added | Thierry Zell | timeline score: 33 | |
| Aug 18, 2010 at 18:16 | comment | added | Victor Protsak | I agree with the previous commenters above that it's a sociological issue, $\textit{viz}$ elimination of content from standard courses, rather than a substantive one. (One can likewise ask: "why were epsilons and deltas/precise definitions/axioms of Euclidean geometry/etc once thought to be such a big deal?"). Even within linear algebra, determinants play a decisive role: for example, the theory of eigenvalues rests on the fact that $\lambda$ is an eigenvalue of $A$ if and only if $\det(A-\lambda I)=0.$ | |
| Aug 18, 2010 at 18:10 | answer | added | Helge | timeline score: 7 | |
| Aug 18, 2010 at 17:59 | history | edited | Jiahao Chen | CC BY-SA 2.5 |
Generalized and turned into CW; deleted 4 characters in body; Post Made Community Wiki
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| Aug 18, 2010 at 17:52 | comment | added | Mariano Suárez-Álvarez | @Spiro: why should they be treated as signed volumes? I have never ever thought of them as signed volumes, because I never deal with volumes... | |
| Aug 18, 2010 at 17:42 | comment | added | J. M. isn't a mathematician | The theoretical analysis of a number of convergence acceleration methods (e.g. Wynn epsilon or the Levin transforms) rests on being able to express them as ratios of determinants. | |
| Aug 18, 2010 at 17:33 | comment | added | Tom Goodwillie | Not so convenient for making machine computations, either, I guess. I should have mentioned the characteristic polynomial, with the resulting insight into eigenvalues and eigenvectors, as another "cool" thing about determinants. I think the basic answer is that they have gone out of style for numerical purposes because they're not the best way; but for theoretical purposes they will never go out of style. | |
| Aug 18, 2010 at 17:31 | comment | added | Spiro Karigiannis | I think determinants have fallen out of favour in first or second year undergraduate courses because they are difficult to teach. There are very few textbooks at this level that treat determinants in the way they should be treated, which is in terms of signed volumes. The applications of determinants to modern differential geometry are very plentiful. We should always tell that to students, since most of them do not enjoy determinants... | |
| Aug 18, 2010 at 17:31 | comment | added | darij grinberg | (which is the same up to isomorphism). These $n$-th exterior powers have the advantages of being basis-independent and sometimes easier to use. | |
| Aug 18, 2010 at 17:31 | comment | added | darij grinberg | (a) Theoretically there is a lot you can do with determinants. The classical approach to invariant theory goes through Cappelli's formula and other determinant relations. Determinant ideals play a big role in the theory of modules over commutative rings. Resultants and discriminants (the oldest and main method of solving systems of nonlinear polynomial equations with arbitrary precision - at least in theory) are defined as particular determinants. (b) They have been replaced by more abstract notions. For instance, the determinant of a linear map has been replaced by the $n$-th exterior power | |
| Aug 18, 2010 at 17:25 | comment | added | Tom Goodwillie | They're not all that convenient for making mundane linear computations by hand, are they, except in the 2x2 case? Yet they play a big part on the theoretical side, for example in understanding spaces of matrices in topology or algebraic geometry, ... What they are "really" about is alternating (antisymmetrized) multilinear algebra. I have the impression that up to a certain point they played a big role in teaching linear algebra, until somebody had the bright idea that they weren't that helpful at that level. | |
| Aug 18, 2010 at 17:25 | comment | added | Mariano Suárez-Álvarez | Determinants were computed before there was any linear algebra to speak of... | |
| Aug 18, 2010 at 17:25 | answer | added | J. M. isn't a mathematician | timeline score: 7 | |
| Aug 18, 2010 at 17:09 | history | asked | Jiahao Chen | CC BY-SA 2.5 |