Jordan Normal Form follows from the Structure Theorem for finitely-generated modules over a PID, by considering C^n as a C[x]-module and applying the Fundamental Theorem of Algebra. Hence, algebraic completeness is indeed a sufficient condition for Jordan normal form to exist. You can embed the field in its algebraic closure to derive a Jordan Normal Form for a matrix, which will itself be a matrix over that field as long as the eigenvalues all inhabit that field. Consequently, for a matrix to be expressible in JNF, it is necessary and sufficient that the eigenvalues lie in that field. Sincerely, Adam P. Goucher
----- Original Message ----- From: Dan Asimov Sent: 05/23/13 07:07 AM To: math-fun Subject: Re: [math-fun] [seqfan] Re: EXACT matrix factorizations
What are the conditions for a ring of matrices over a field to have Jordan form? I know the theorem over the complexes and imagine that algebraic completeness of the field is at least sufficient, if not necessary.
--Dan
On 2013-05-22, at 2:12 PM, Victor Miller wrote: ----- . . . . . . I look at all possible Jordan canonical forms for an n by n matrix, . . . -----
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun