Transcription of Jordan Normal Form - Texas A&M University
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Jordan Normal Form - Texas A&M University
Chapter 8 Jordan Normal Minimal PolynomialsRecallpA(x)=det(xI A) is called the characteristic polynomial of Mn. Then there exists a unique monic polyno-mialqA(x)of minimum degree for whichqA(A)= (x)is any polyno-mial such thatp(A)=0,thenqA(x)dividesp(x). there is a polynomialpA(x)forwhichpA(A) = 0, there is one ofminimal degree, which we can assume is monic. by the Euclidean algorithmpA(x)=qA(x)h(x)+r(x)where degr(x)<degqA(x). We knowpA(A)=qA(A)h(A)+r(A).Hencer(A) = 0, and by the minimality assumptionr(x) 0. ThusqAdividedpA(x) and also any polynomial for whichp(A) = 0.
222 CHAPTER 8. JORDAN NORMAL FORM Corollary 8.1.1. If A,B ∈Mn are similar, then they have the same min- imal polynomial. Proof. B = S−1AS qA(B)=qA(S−1AS)=S−1qA(A)S = qA(A)=0. If there is a minimal polynomial for B of smaller degree, say qB(x), then qB(A) = 0 by the same argument.This contradicts the minimality of qA(x). Now that we have a minimum polynomial for …
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