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If A is nxn nilpotent matrix, this char(A) = x^n

  • #1
1,456
44

Homework Statement


If ##A## is an ##n \times n## nilpotent matrix, then the characteristic polynomial of ##A## is ##x^n##

Homework Equations




The Attempt at a Solution


Suppose that ##A## has an eigenvalue with corresponding eigenvector such that ##A v = \lambda v##. Then ##A^k v = \lambda^k v = 0##, and since ##v \ne \vec{0}##, ##\lambda^k = 0 \implies \lambda = 0##. Since ##0## is the only eigenvalue, and since the characteristic polynomial must be of degree n, the characteristic polynomial must be ##x^n##.
 

Answers and Replies

  • #2
33,900
9,615
You should introduce k. Apart from that: right.
 
  • #3
12,502
8,909
Yes.
 

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