Talk:Determinant

Right handed coordinante

the following sentence is not clear. "The determinant of a set of vectors is positive if the vectors form a right-handed coordinate system, and negative if left-handed." what does "right-handed coordinate system" means? the "coordinate system" article does not mention it. amit man

Possible to do/see also items

linear algebra/analytic geometry

linear independence/collinearity, Gram determinant, tensor, positive definite matrix (Sylvester's criterion), defining a plane, Line-line intersection, Cayley–Hamilton_theorem, cross product, Matrix representation of conic sections, adjugate matrix, similar matrix have same det (Similarity invariance), Cauchy–Binet formula, Trilinear_coordinates, Trace diagram, Pfaffian

types of matrices

special linear group, special orthogonal group, special unitary group, indefinite special orthogonal group, modular group, unimodular matrix,

number theory/algebra

Pell's equation/continued fraction?, discriminant, Minkowski's theorem/lattice, Partition_(number_theory), resultant, field norm, Dirichlet's_unit_theorem, discriminant of an algebraic number field

geometry, analysis

conformal map?, Gauss curvature, orientability, Integration by substitution, Wronskian, invariant theory, Monge–Ampère equation, Brascamp–Lieb_inequality, Liouville's formula, absolute value of cx numbers and quaternions (see 3-sphere), distance geometry (Cayley–Menger determinant), Delaunay_triangulation

open questions

Jacobian conjecture, Hadamard's maximal determinant problem

algorithms

polar decomposition, QR decomposition, Dodgson_condensation, Matrix_determinant_lemma, eigendecomposition a few papers: Monte carlo for sparse matrices, approximation of det of large matrices, The Permutation Algorithm for Non-Sparse Matrix Determinant in Symbolic Computation, DETERMINANT APPROXIMATIONS

examples

reflection matrix, Rotation matrix, Vandermonde matrix, Circulant matrix, Hessian matrix (Blob_detection#The_determinant_of_the_Hessian), block matrix, Gram determinant, Elementary_matrix, Orr–Sommerfeld_equation, det of Cartan matrix

generalizations

Hyperdeterminant, Quasideterminant, Continuant (mathematics), Immanant of a matrix, permanent, Pseudo-determinant, det's of infinite matrices / regularized det / functional determinant (see also operator theory), Fredholm determinant, superdeterminant

other

Determinantal point process, Kirchhoff's theorem,

books

[1]

No determinants over a non-commutative ring

A recent edit injected the following text into this article

For matrices over non-commutative rings, the equivalence remains true provided the scalar multiplication is consistently applied (say, on the left); this can be verified by a similar process.

This surprised me a lot, notably the claim that Lang would write such nonsense. He doesn't, chapter XIII clear states that throughout R will be a commutative ring (my emphasis), and there is absolutely no mention of non-commutativity of scalars or the distinction of left and right multiplication in section 4 of that chapter. The sentence "the equivalence remains true" does not even make sense until a determinant is defined in this setting, which would require a specification of the order of products in (for instance) the Leibniz formula (and which would make the equivalence fail). Would it be too much to ask checking whether a reference actually supports a claim before inserting the claim and the reference into an article? To avoid all doubt, I'll explain why over a non-commutative ring there is no such thing as an n-linear form on matrices taking the value 1 on the identity matrix, for n ≥ 2. Here the linearity of the columns is taken in the same sense, say left-linear: multiplying any column on the left by a scalar should multiply the result of the function on the left by the same scalar. Now the a, b be two non-commuting scalars. Then

$ab = ab \left|\begin{matrix}1&0\\0&1\end{matrix} \right| = a \left|\begin{matrix}1&0\\0&b\end{matrix} \right| = \left|\begin{matrix}a&0\\0&b\end{matrix} \right| = b \left|\begin{matrix}a&0\\0&1\end{matrix} \right| = ba \left|\begin{matrix}1&0\\0&1\end{matrix} \right|= ba,$

contradicting the assumption on a, b. The best one can do in a non-commutative setting is define a function on 2×2 matrices that is (say) left-linear in the first column and right-linear in the second column (the function $\tbinom{a~c}{b~d}\mapsto ad-bc$ is an example), but even then it would not be alternating, nor linear of any kind in the rows. There are fruitful attempts to define some substitute for determinants in a non-commutative setting (which for instance allow special linear groups to be defined for the quaternions; interesting detail: it is of real codimension 1 in the set of all matrices, not 4 as one would expect by analogy to the complex special linear groups) but it is not by way of an expression taking values in the ring, as in the commutative caseMarc van Leeuwen (talk) 10:04, 12 September 2011 (UTC)

New mnemonic

What do people think of this alternative mnemonic for a 3 × 3 (only) determinant? Its referanced and perhaps clearer in signs and permutations than the current one (no offence to the author of that diagram...).

Visual mnemonic for a 3 × 3 (only - no other order) determinant, to help obtain the correct sign and permutation of expanded terms. (REF TO ADD: Linear Algebra, S. Lipschutz, M. Lipson, Schuam's Outline Series, McGraw Hill (USA), 2009, ISBN 978-0-07-154352-1)

--Maschen (talk) 00:01, 4 December 2011 (UTC)

If you're going to stick in arows you should at least put the terms in the order encountered on the arrows. The order the terms are written seems better than the order in the diagrams. I think it is a bad mnemonic if they ignore it and use some other system of their own. Dmcq (talk) 00:34, 4 December 2011 (UTC)

Thanks for feedback. I made the change (in the same place as above). Any chance now?--Maschen (talk) 08:28, 4 December 2011 (UTC)

Facts can be verifiable but still trivia so having a reference isn't sufficient for inclusion. Mnemonics in general have, at best, marginal encyclopedic value (see WP:NOTTEXTBOOK), so I'd like to see at least two independent sources to show that someone beside the person who made it up thinks it's useful. If it's in common use then it shouldn't be too hard to find more than one source. Btw, the diagram shown only works for 3x3, the 2x2 rule is slightly different and may need a separate image.--RDBury (talk) 12:37, 4 December 2011 (UTC)

Your'e correct about the 2 × 2 determinant, I seemed to have looked at the paths incorrectly - the caption has been repaired (but the 2 × 2 det is trivial anyway - no real need for an image).

I included the referance since thats where I found it, and its in a renowned series (Schuam's) so in that case I fail to see how it can be hard to find. Also the referance is not for persuasion or to "add value for inclusion" etc.

This was just a suggestion by the way - i'm not forcefully trying to include it. If anyone wants/allows it - take it and use it. If not - leave it and forget it...--Maschen (talk) 13:25, 4 December 2011 (UTC)

This looks like a different visual presentation of Sarrus' rule to me. It corresponds roughly to what I do mentally for a 3×3 determinant, so it's fine with me, but it does not seem to be much of a big deal. Another mnemonic that works for n=3 (but not for any other n (except n=0 :-)) is that the sign is negative iff exactly one main diagonal entry is chosen. Marc van Leeuwen (talk) 13:34, 4 December 2011 (UTC)

I find the diagram that's already in the article more intuitively clear. -- Elphion (talk) 15:31, 4 December 2011 (UTC)

Same here and in addition I wonder if even that might give the erroneous impression that 4th order determinants follow the same pattern, it'd be better if they got to this via the adjugates first I think. However when push comes to shove it is whether there is a lot of support for it out there or not. In this though I think Schaum certainly gives very good evidence I'd like to see two instances as the circular path doesn't strike me as something that good as a mnemonic. Dmcq (talk) 17:54, 4 December 2011 (UTC)

The article already warns the reader that the Rule of Sarrus does not generalize, though perhaps we should say it louder. I disagree about introducing adjugates first; the typical reader is interested primarily in 2 and 3 dimensions, and those should be covered first with their simpler rules. The general rule will just scare people away. (I'm always struck by how our natural tendency to generalize right out of the gate manages to make us generally unintelligible!) -- Elphion (talk) 19:49, 4 December 2011 (UTC)

I found another source for the diagram, see Thomas Muir's classic text. Also there are probably those who prefer this to expanding the matrix as in Sarrus. It might be better to add the image to Rule of Sarrus rather than this article though.--RDBury (talk) 16:47, 5 December 2011 (UTC)