When p(A)=0 iff p(B)=0 for any polynomial,why same minimal polynomial?by td21 Tags: minimal, polynomial 

#1
Jun1311, 02:00 AM

P: 74

For two matrices A and B, when p(A)=0 iff p(B)=0 for any polynomial, what will happen? i read that A and B have the same minimal polynomial, why?




#2
Jun1311, 07:57 AM

P: 4

and [tex]m_B (B) = 0\Rightarrow m_B (B) = 0 \Rightarrow m_A / m_B ^{(2)}[/tex] [tex]\overset {(1), (2)}{\Rightarrow} m_A = k \cdot m_B[/tex] with k a constant. But [tex]m_A, m_B[/tex] are both monic polynomials so, [tex]k=1[/tex] and finally [tex]m_A = m_B.[/tex] 



#3
Jun1311, 02:13 PM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,904

Pretty much the same thing but in slightly differentwords:
Suppose P_{A}(x), of degree n, is the minimal polynomial for A. Then P_{A}(A)= 0 so P_{A}(B)= 0. If This is not the minimal polynomial for B, there exist a polynomial P_{B}, of degree m< n, such that P_{B}(A)= 0. But then P_{B}(A)= 0 contradicting the fact that the mininal polynomial of A has degree n> m. 


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