Let A M3 If A notequalto and A2 notequalto I but A3 I find

Let A M_3 If A notequalto and A^2 notequalto I, but A^3 = I, find the Jordan canonical form of A.

Solution

Note that A3=I

A3=I, means that

A3I=0

A3I=0, which means that the minimal

polynomial qA(t)of A divides

t31=(t1)(t)(t2)

where =e2i/ is a root of unity. Note that the maximal degree of any factor of the minimal polynomial is 1, which means that the maximum length Jordan block for any given eigen value is 1