Let A,B in M_n(CC) be matrices and f in C[X] such that Af(B)=B. Prove that if f(B) is not invertible, f(0)=0. Prove that if f(B) is not invertible, f(0)=0.

Mauricio Mathis 2022-07-22 Answered
Let A , B M n ( C ) be matrices and f C [ X ] such that A f ( B ) = B. Prove that if f ( B ) is not invertible, f ( 0 ) = 0
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Mireya Hoffman
Answered 2022-07-23 Author has 14 answers
Let f ( x ) = k = 0 n a k x k . Suppose that f ( B ) is not invertible, i.e. there exists a nonzero vector v, such that f ( B ) v = 0. Then from A f ( B ) = B it follows that B v = 0, and therefore
0 = f ( B ) v = a 0 v + k = 1 n a k B k v = a 0 v a 0 = 0.

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