Re: [math-fun] [seqfan] Re: EXACT matrix factorizations
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, . . . -----
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To be precise, I should have called it "Frobenius Normal Form" or "rational canonical form": http://en.wikipedia.org/wiki/Frobenius_normal_form . I thought of it as the same thing since the proof that I knew came from the structure theorem for finitely generated modules over a PID, as Adam mentioned. Victor On Thu, May 23, 2013 at 6:30 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
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, . . . -----
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Adam P. Goucher -
Victor Miller