A square matrix A with ker(A^2)= ker (A^3)

[yeland404]

Homework Statement

a square matrix A with ker(A^2)= ker (A^3), is ker(A^3)= ker (A^4),verify...

Homework Equations

The Attempt at a Solution

 

[micromass]

What did you try already??

 

[yeland404]

What did you try already??

it means that A^2*vector x= 0 and A*x=0 has same result,then I really confused how to do the next step

 

[micromass]

it means that A^2*vector x= 0 and A*x=0 has same result,then I really confused how to do the next step

No, it means that

$$A^2x=0~\Leftrightarrow~A^3x=0$$

You need to prove that

$$A^3x=0~\Leftrightarrow~A^4x=0$$

 

[yeland404]

No, it means that

$$A^2x=0~\Leftrightarrow~A^3x=0$$

You need to prove that

$$A^3x=0~\Leftrightarrow~A^4x=0$$

emm..., to the difinition it says that ker(A) is T(x)=A(x)=0

 

[micromass]

emm..., to the difinition it says that ker(A) is T(x)=A(x)=0

Yes, but you're not working with ker(A) here, but with ker(A²).

 

[yeland404]

Yes, but you're not working with ker(A) here, but with ker(A²).

should define a matrix A and write A^2 as dot product of A & A,and also to A^3...it seems to be so complex, or use the block to divide the matrix into some small matrix?

 

[micromass]

should define a matrix A and write A^2 as dot product of A & A,and also to A^3...it seems to be so complex, or use the block to divide the matrix into some small matrix?

Do you understand my Post 4??

 

[yeland404]

Do you understand my Post 4??

so times A on both side of the equation?

 

[micromass]

so times A on both side of the equation?

No, you need to prove two things:

If $A^3x=0$, then $A^4x=0$. This can indeed be accomplished by multiplying sides with A.

But you also need to prove that if $A^4x=0$, then $A^3x=0$.

 

[yeland404]

so Ker (A^2)=0 can lead to Ker(A^4)=0,then?

 

[HallsofIvy]

NO ONE has said that Ker(A^2)= 0 so I do not understand why you are asking this question. You seem, frankly, to have no idea what the question is saying. ker(A^2)= ker(A^3) means, as micromass said, "A^2x= 0 if and only if A^3x= 0". You want to use that to prove "A^3x= 0 if and only if A^4x= 0". As micromass said, the first part is easy: if A^3x= 0 then, applying A to both sides, A^4x= A0= 0 which proves that ker(A^3) is a subset of ker(A^4). You still need to prove the other way: if A^4x= 0, then A^3= 0. You cannot just multiply by A^{-1} because you have no reason to think that A is invertible. But notice that ker(A^2)= ker(A^3) has not yet been used so it might help to note that A^4x= A^2(A^2x).