Two different Covariance Matrices?

Question

I am a little bit confused!

Assume we have observed the Data X = [x1,..,xn] and they are vectors in R^d (with zero mean)

X^T denotes the transposed of X

Sometimes i see that the covariance matrix is in the form of 1/n * X*X^T (e.g. Principal Component Analysis) and sometimes is see it in the form 1/n * X^T*X (e.g. Kernel-Covariance matrix with kernel k(x,y) = x^T*y)

So why are 2 different ways or am i mixing up some things? Thank you for your help.

Answer 1

Well, the results differ in their dimension. One is a nxn-matrix, the other is a dxd-matrix. I don't know the application for nxn-result, but when I used the covariance matrix to denote the variation of a vector in R^d (with measurements X = [x1,..,xn]) the result has to be a dxd-matrix, whose eigenvectors and -values indicate the main axes and extends of an "variance ellipsoid" (which must be given in dxd)

PS: Only half an answer, I know

Addendum: Kernels are used for creating inner products of pairwise features, thus reducing the dimension to 1 to find patterns more easily. Have a look at http://en.wikipedia.org/wiki/Kernel_principal_component_analysis#Introduction_of_the_Kernel_to_PCA to get an impression, what the kernel covariance matrix is used for