The Cholesky Factorization, Schur Complements, Correlation Coefficients, Angles between Vectors, and the QR Factorization.
Abstract
An m x m symmetric nonnegative definite matrix Sigma has Cholesky factorization Sigma = u-transpose u. By carrying out the factorization in a particular way for positive definite Sigma, the Schur complements of all the leading principal submatrices of Sigma are produced, as well as their Cholesky factors. It is shown how the same can be done for generalized Schur complements when Sigma is singular. When Sigma is the population covariance matrix of a multivariate random distribution, partial covariances and correlations can be defined in terms of the elements of such Schur complements. It follows that these can be produced efficiently and reliably from the Cholesky factorization. When n x m A is given and Sigma = A-transpose A, the Cholesky factor U may be found directly from the QR factorization A = Q1U, Q1-transpose Q1 = I, and this is preferable in many numerical computations. This QR factorization, or the modified Gram-Schmidt orthogonalization, produces projections of later columns of A onto spaces orthogonal to earlier columns. It is shown how the cosines of the angles between such projected vectors can be found using the elements of U. These cosines produced from A turn out to be the previously mentioned partial correlation coefficients produced from Sigma, when Sigma = A-transpose A. When A is obtained from observations of random variables, these are the sample correlation coefficients. It is shown how such correlation coefficients can be efficiently obtained when observations are added or deleted. This corresponds to altering all of A in a certain simple way, and adding or deleting rows.
Document Details
- Document Type
- Technical Report
- Publication Date
- Feb 01, 1988
- Accession Number
- ADA192398
Entities
People
- C. C. Paige
- I. C. Ipsen
- J.-m. Delosme
Organizations
- Yale University