where is the gram form/matrix, denotes the volume of a subset of , and is the unit ball of centered at .
Let , where is the Gram bilinear form, then .
If is injective and is an isometry, then .
Let be an endomorphism of and its matrix. Let be the polar decomposition of , is a symmetric positive matrix and an orthogonal matrix. Let the be the columns of () and let be the Gram matrix the . Then . (N.B.: this is one way to prove the existence of the polar decomposition, take the square root of the Gram matrix, multiply by its inverse and it easily follows that what is obtained is an orthogonal matrix).
They are utilized in statistics in Principal components analysis. One wants to determine the general trend in terms of few characteristics ( of them) in a large sample ( individuals). Each represents the results of the individuals in the sample for the characteristic. Each one of the 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.
|Date of creation||2013-03-22 17:56:55|
|Last modified on||2013-03-22 17:56:55|
|Last modified by||lalberti (18937)|
|Synonym||Gram bilinear form|