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