1sir actually my question is that can a square matrix be called nilpotent even if Ak=0 but k is NOT the order of the matrix
i mean if the order is 3 and A2=0 , THEN can we say that A is nilpotent ?
Which of the following definition is correct?
A square matrix A is called nilpotent if there exists a positive integer m such that Am= 0 , where m is the least positive integer such that Am = 0 , then m is called the index of matrix A.
A square matrix A is called nilpotent of order m provided it satisfies the relation Ak=0 and Ak-1≠0 , where k is positive integer and 0 is null matrix and k is the order of the nilpotent matrix A .
I got the above two from different books .
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9 Answers
62Both are teh same definitions...
arent they ?
Think a bit more ..
1but sir, in the second one it has been specified that k should be the order of the square matrix while in the first one nothing is specified?
62It is ..
see there you have m..
if m is the smallest integer, it means that for no integer less than m shoudl Am be zero....
ie am-1≠0
Even though I have nto proved the equivalence of both these definitions.. (I hope this much understanding does help you!)
162that will never happen...
if A2 = 0 then the order by the definition of order becomes 2..
it is the smallest such number
(In your first definition, index is the same as the order of the matrix in the 2nd)
1i don't know sir actually i got a matrix like that
row1= 0 0 0
Row2 = 0 0 0
row3= 0 1 0
A2=0 but 2 is not the order of the matrix
62The order of this matrix is 2..
Where did you find that the order is given to be 3!!
(Some times order of a 3x3 matrix is also used for the dimension of a matrix.. ) I guess you are confused with that!
1i guess you are right !!
then how do we decide the order of a matrix??