I'm going to partition a symmetric matrix (covariance matrix) into columns/rows that are independent. Analysis of eigenvalues was what I had in mind but I have difficulty mapping out eigenvalues/vectors to the respective elements in the matrix. Is this what I need to do? Are there any other techniques that I can use?
If a vector y has covariance matix C, you can find a lower triangular matrix L, so that y=Lx, where the elements of x are independent with variance 1, and where L*L^t=C. If C is positive definite, x has the same dimension as y, and you can find L by Cholesky decomposition of C. If C is only positive semidefinite (i.e. C is singular), you need to modify the Cholesky decomposition algorithm slightly in order to return a valid result L. I have attached an Excel workbook with a macro Cholesky_D that returns a matrix L of same dimension as C. If the columns of L containing only zeroes are deleted, then y=Lx, where the dimension of x is the number of remaning columns in L, which is equal to the rank of C.
First find out the rank of the matrix, which is nothing but the dimension of the vector space concerned. After that if you find that n is the dimension, select any n rows to form a square matrix, and that would give the solution you need.
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