Talk:Gramian matrix

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Help with this template Comments: matrix/Comments|action=edit}} edit – matrix/Comments|action=history}} history – matrix/Comments|action=watch}} watch – matrix|action=purge}} refresh Added template, rated article. Nbarth (email) (talk) 02:27, 18 January 2008 (UTC)

[edit] Limits

In the expression for G_ij, what is t_D and t_f ? Deepak 16:54, 31 March 2006 (UTC)

t₀ and t_f are initial and final time; the functions whose Gramian is being computed are defined on the interval [t₀,t_f]. I've clarified this in the article.

Nbarth (email) (talk) 02:17, 18 January 2008 (UTC)

[edit] Too much physics-oriented

The Gramian matrix can be calculated and is important in any inner product space. The integral of the multiplication of two functions which is shown in the article is just one case of an inner product.

Here is a good page on the subject: http://www.jyi.org/volumes/volume2/issue1/articles/barth.html

Yes, I agree. Gram matrices show up also in machine learning, where a number of methods depend on a set of input vectors in (for some finite n) only through the Gram matrix of this set. One may construct the Gram matrix using the standard inner product (dot product) on , or -- very usefully -- an arbitrary inner product, which corresponds to mapping the input vectors nonlinearly into some usually higher-dimensional space and taking the dot product there (known as the kernel trick). Either way, integrals are not involved. Eclecticos 05:04, 24 September 2006 (UTC)

Also, the usual name in the machine learning literature, and AFAIK in the linear algebra literature too, is "Gram matrix." I have never run across the variant "Gramian matrix" before, but perhaps it is used in physics? Eclecticos 05:04, 24 September 2006 (UTC)

I am aquainted with "Gramian" from mathematics. The main point is that the gramian matrix of some base (not necessarily orthonormal) of an euclidean (= inner product) space contains all the information on the geometry (the inner products) of that space.

In addition, checking for linear dependencies is only a specific case of determining the volume of the parallelopiped spanned by some vectors, which can be done easily by the gramian matrix. It is different from the determinant, since it applies to non-rectangular matrices as well.

For example - to calculate the area of a parallelogram given within a 3-d space by determinant is hard, because we need to find an orthonormal base for the plane in which the parallelogram lies and transform the vectors to that base, but using the gramian matrix it is very simple (see the external link).

Thank you for the kind words on the JYI article, which I wrote (many years ago).

I've significantly revised the page, stating it more generally (and making other revisions).

Nbarth (email) (talk) 02:21, 18 January 2008 (UTC)

[edit] What does (xi|xj) mean?

Need definition of this term -- does it refer to the inner product of xi and xj?

Yes—I've updated the article accordingly to clarify.

Nbarth (email) (talk) 02:18, 18 January 2008 (UTC)