Question

设A,B都是实数域R上的n×n矩阵,证明:AB,BA的特征多项式相等

Answer 1

就是要证明|λE-AB| = |λE-BA|.

考虑分块矩阵P =

E 0

-A E

与分块矩阵Q =

λE B

λA λE

可算得PQ =

λE B

0 λE-AB

有λ^n·|λE-AB| = |λE|·|λE-AB| = |PQ| = |P|·|Q|.

另一方面QP =

λE-BA B

0 λE

有λ^n·|λE-BA| = |λE-BA|·|λE| = |QP| = |Q|·|P|.

于是|λE-AB| = |λE-BA|.