# Gram matrix

For a vector space  $V$ of dimension  $n$ over a field $k$, endowed with an inner product $<.,.>:V\times V\to k$, and for any given sequence  of elements  $x_{1},\ldots,x_{l}\in V$, consider the following inclusion map  $\iota$ associated to the $(x_{i})_{i=1,\ldots,l}$:

 $\begin{array}[]{cccc}\iota:&k^{l}&\to&k^{n}=V\\ &(\lambda_{1},\ldots,\lambda_{l})&\mapsto&\sum_{i=1}^{l}\lambda_{i}x_{i}\end{array}$

The Gram bilinear form of the $(x_{i})_{i=1,\ldots,l}$ is the function

 $x,y\in k^{l}\times k^{l}\mapsto<\iota(x),\iota(y)>$

The Gram matrix of the $(x_{i})_{i=1,\ldots,l}$ is the matrix associated to the Gram bilinear form in the canonical basis of $k^{l}$. The Gram form (resp. matrix) is a symmetric bilinear form  (resp. matrix).

Gram forms/matrices are usually considered with $k=\mathbb{R}$ and $<.,.>$ the usual scalar products  on vector spaces over $\mathbb{R}$. In that context they have a strong geometrical meaning:

They are utilized in statistics in Principal components analysis. One wants to determine the general trend in terms of few characteristics ($l$ of them) in a large sample ($n$ individuals). Each $x_{i}(\in{\mathbb{R}}^{n})$ represents the results of the $n$ individuals in the sample for the $i^{\mathrm{th}}$ characteristic. Each one of the $l$ dimensions represents a characteristic and one wants to know what are the predominant characteristics and if they bear some kind of linear relations  between them. This is achieved by diagonalizing the Gram matrix (often called dispersion matrix or covariance matrix in that context). The higher the eigenvalue     , the more important the eigenvector    associated to it.

Title Gram matrix GramMatrix 2013-03-22 17:56:55 2013-03-22 17:56:55 lalberti (18937) lalberti (18937) 5 lalberti (18937) Definition msc 15A63 Gram matrices Gram bilinear form