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In the expression for , what is and ? Deepak 16:54, 31 March 2006 (UTC)
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
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).
Need definition of this term -- does it refer to the inner product of xi and xj?
It seems to me that, because of the conjugate symmetry property of inner products, the Gramian matrix is not symmetric but Hermitian. MorpheusCO ( talk) 18:19, 21 April 2009 (UTC)
It would be nice to know why the Gramian is pos. def. —Preceding unsigned comment added by 188.60.12.40 ( talk) 16:59, 12 December 2010 (UTC)
Google does not find any Dave Gramian!? — Preceding unsigned comment added by 217.75.195.210 ( talk) 13:21, 11 October 2012 (UTC)
This is a mess. If the vectors really are centered random variables, that is, they are vectors of the Hilbert space of all square integrable functions on a probability space (with zero mean), then the Gramian is exactly the covariance matrix (just by definition), with no scaling. Apparently it was meant that the vectors are a sample from a multidimensional probability distribution. Boris Tsirelson ( talk) 06:14, 10 May 2015 (UTC)
the corresponding French wikipedia page is https://fr.wikipedia.org/wiki/D%C3%A9terminant_de_Gram . Where do I connect it here ? — Preceding unsigned comment added by 194.199.26.79 ( talk) 14:20, 24 April 2023 (UTC)
This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||||||||||||||||
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In the expression for , what is and ? Deepak 16:54, 31 March 2006 (UTC)
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
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).
Need definition of this term -- does it refer to the inner product of xi and xj?
It seems to me that, because of the conjugate symmetry property of inner products, the Gramian matrix is not symmetric but Hermitian. MorpheusCO ( talk) 18:19, 21 April 2009 (UTC)
It would be nice to know why the Gramian is pos. def. —Preceding unsigned comment added by 188.60.12.40 ( talk) 16:59, 12 December 2010 (UTC)
Google does not find any Dave Gramian!? — Preceding unsigned comment added by 217.75.195.210 ( talk) 13:21, 11 October 2012 (UTC)
This is a mess. If the vectors really are centered random variables, that is, they are vectors of the Hilbert space of all square integrable functions on a probability space (with zero mean), then the Gramian is exactly the covariance matrix (just by definition), with no scaling. Apparently it was meant that the vectors are a sample from a multidimensional probability distribution. Boris Tsirelson ( talk) 06:14, 10 May 2015 (UTC)
the corresponding French wikipedia page is https://fr.wikipedia.org/wiki/D%C3%A9terminant_de_Gram . Where do I connect it here ? — Preceding unsigned comment added by 194.199.26.79 ( talk) 14:20, 24 April 2023 (UTC)