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

by td21
Tags: minimal, polynomial
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td21
#1
Jun13-11, 02:00 AM
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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?
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#2
Jun13-11, 07:57 AM
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Quote Quote by td21 View Post
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?
Let [tex]m_A, m_B[/tex] be the minimal polynimials of A and B. Then [tex]m_A (A) = 0\Rightarrow m_A (B) = 0 \Rightarrow m_B / m_A^{(1)}[/tex]
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]
HallsofIvy
#3
Jun13-11, 02:13 PM
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Pretty much the same thing but in slightly differentwords:

Suppose PA(x), of degree n, is the minimal polynomial for A. Then PA(A)= 0 so PA(B)= 0. If This is not the minimal polynomial for B, there exist a polynomial PB, of degree m< n, such that PB(A)= 0. But then PB(A)= 0 contradicting the fact that the mininal polynomial of A has degree n> m.


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