Talk:Trace (linear algebra)

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I do not think that the article title "Trace (matrix)" is any clearer than "Trace of a matrix". In fact, quite the opposite.

Wikipedia articles are about concepts, not about words. This is not about the word "trace" in the context of matrix theory (in which case it should properly be called "Trace (matrix theory)" or "Trace (linear algebra)"), but it is about the concept "Trace of a matrix", which is a perfectly self-explanatory headline of the article. AxelBoldt 23:10 Jan 31, 2003 (UTC)

Trace is empty. let's move it there -- Tarquin 23:26 Jan 31, 2003 (UTC)

I would like to see 2 things: Brief paragraph on the importace of the trace, and an intoative proof of the main feature (is it main?) tr(AB) = tr(BA). thanks

I agree that this article should be renamed. Mainly because the concept of the trace is more general that that of a matrix. In other words there is a class of linear transformations for which the trace makes sense which are not matrices. move the article to Trace -Lethe | Talk

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[edit] Trace of matrix products

I would like to suggest a change regarding the trace of the product of several (>2) matrices. Specifically the part beginning "If A, B, and C are square...". The phrase "all permutations" is, strictly speaking, not correct. The last three terms in the chained equality, while looking like general permutations, are transposes of the first three with the transpose marks omitted due to symmetry. My motivation for the change is that if more than three matrices are involved only the cyclic permutation may be used to reorder the matrices. Transposing with symmetry assumed will make more (apparent) permutations possible, but not all posible permutations. My quick and dirty suggestion is to delete the entire paragraph along with the chained equality. It could be fixed but it is just a special case involving the previously discussed cyclic permutation and transpose along with the symmetry condition. Doggit 17:19, 10 August 2006 (UTC)

[edit] Direct Link

I set the link from spur to this article. --Kajaktiger 16:47, 4 November 2006 (UTC)

[edit] Derivatives

Don't revert back to d Tr = I. This is not true. There is a vectorisation operator present if you must present the information in this form. The peterson & pederson seem to have dropped this when they copy pasted from brookes.

Just goes to show what sort of crap you can find on the internet. —The preceding unsigned comment was added by 150.203.45.188 (talk) 09:35, 5 December 2006 (UTC).

[edit] Inner product

I think that following <A,B> = tr\left(A\,^*B\right), it should be mentioned that * B is the conjugate transpose of B. 131.130.90.152 10:36, 22 January 2007 (UTC)

[edit] Trace of a 1×1 matrix

Trace can be used in a non-obvious way where one considers a scalar the trace of a 1×1 matrix. For example, if X is a n×1 column vector and A is a n×n square matrix, then XTAX is a 1×1 matrix which is often regarded as a scalar, but it can be beneficial to use the trace operator instead (to use the cyclic property). This technique is used in the Estimation of covariance matrices article. Can anyone provide a simpler example to add to the main article? Roman V. Odaisky 17:23, 4 October 2007 (UTC)

[edit] tr(AB)n

I’m confused by the formula 0 \leq \mathrm{tr}(AB)^n \leq \mathrm{tr}(A)^n \mathrm{tr}(B)^n. Shouldn’t it have read tr((AB)n) instead of the middle term? Or perhaps the formula should be split into two inequalities:

\mathrm{tr} AB \leq \mathrm{tr} A \cdot \mathrm{tr} B
\mathrm{tr} A^n \leq (\mathrm{tr} A)^n (if n is natural then this follows from the previous inequality)

Roman V. Odaisky 17:33, 4 October 2007 (UTC)

[edit] Various edits

Hi---just wanted to leave a note about the various edits I've been making to the article. If anyone has objections or comments, please respond here---thanks. Spireguy (talk) 18:28, 4 March 2008 (UTC)

[edit] Intuition/example

I know the determinate of a transformation matrix can be thought of as the change in volume it causes. For example, if det(A) = 1, then A doesn't change the (n-dimensional) volume of an object it is applied to. Could someone provide a similar intuition for trace? What is the meaning of the sum of the eigenvalues? —Ben FrantzDale (talk) 11:59, 27 March 2008 (UTC)

This is addressed in the article under "derivatives", although perhaps it is not clear as stated. The trace measures the infinitesimal change in volume, since it is the derivative of the determinant. Do you think this should be emphasized or re-stated in a more down-to-earth manner? -- Spireguy (talk) 16:41, 27 March 2008 (UTC)

OK, I went ahead and put in an example. Comments? -- Spireguy (talk) 16:49, 27 March 2008 (UTC)