From: rusin@math.niu.edu (Dave Rusin) Newsgroups: sci.math.research Subject: Re: Continuity of the Jordan Decomposition Map Date: 27 Feb 1998 23:09:27 GMT In article <6ctpud\$5t@senator-bedfellow.MIT.EDU>, Lones Smith wrote: >Consider the Jordan decomposition of an nXn matrix M = Q D Q^{-1}. > >Question: If each matrix is so indexed as M_t = Q_t D_t Q_t^{-1}, >and if M_t is continuously (in R^{n^2}) deformed as t changes, >then does there exist a continuous selection t --> (Q_t, P_t) >thru Q_0=Q, D_0=D? >(If not, could I have chosen Q and D so that this is true.) I assume this is a typo: you want t --> (Q_t, D_t) to be a continuous map into the subset X x Y of (R^{n^2})^2 where X=GL(n,R) is the set of invertible matrices and Y is the set of those in Jordan canonical form. If that's your goal, you should think for a minute about what continuous change for the D_t will look like. If for example n=2, Y is the set of diagonal matrices together with the set of matrices [[a,1],[0,a]], that is, a plane through the origin and a line parallel to it, in R^4. So a continuous motion will have to stay either within the set of diagonalizable matrices, or within the set of nondiagonalizable ones. On the other hand, as M is continously deformed, it can easily change from diagonalizable to not (e.g. vary from the identity to [[1,epsilon],[0,1]] ). So no, one can't always accomplish what you ask, even if you allow yourself the luxury of adjusting your presentation of M in Jordan canonical form Q D Q^{-1}. dave