From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: Orthogonal transformation Date: 12 Dec 1999 17:29:48 -0500 Newsgroups: sci.math,sci.stat.math Keywords: comparison of various matrix factorizations In article <3852B948.2A6653D1@ucla.edu>, Prabha Siddarth wrote: :Is there an orthogonal transformation that transforms a symmetric :positive definite matrix to an identity matrix? : :Thanks for any help, : You probably merged several true facts into one confusion (real mathices seem to be the context): (1) Every positive definite matrix A can be written as A = C * C' (C' being the transpose of C), where C is invertible (can be stipulated to be upper triangular) This requires finding square roots, so some theories settle for (2) Every positive definite matrix A can be written as A = L * D * L' where D is diagonal with positive entries and L is invertible (can be stipulated to be upper triangular with unit diagonal) Here L and D can have entries from the same field where entries of A come from (e.g. the rationals). Facts (1) and (2) are related to "congruence of matrices": two matrices A, B are called congruent if A = P * B * P' where P is invertible. The statements say that every positive definite matrix is congruent to I, or over the same field, at least to a diagonal p.d. matrix. (It's a part of a more general statement about "inertia of symmetric matrices".) So far, orthogonality has not been mentioned. With orthogonal matrices, (3) every p.d. matrix A can be written as A = U * D * U' with U orthogonal and D diagonal with positive diagonal entries. The field where the entries of D and U come from must contain roots of characterictic polynomials of symmetric matrices. (BTW, is it the same as the real part of the field of algebraic numbers, if the entries of A are restricted to be rational?) And you cannot do any better in general (forcing D to be I), as other respondents wrote. Cheers, ZVK(Slavek).