Matrix A s.t. A^3=0 but A^2 doesn't

  • Thread starter NullSpace0
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  • #1
NullSpace0
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Homework Statement


Find a matrix, A, such that A3=0, but A2≠0.


Homework Equations





The Attempt at a Solution


I don't actually think this is possible. My only other thought is to take A2*A where this product equals zero, and then decompose A. But I don't know how to do that such that it is the "square root" of A.
 

Answers and Replies

  • #2
Dustinsfl
2,285
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Homework Statement


Find a matrix, A, such that A3=0, but A2≠0.


Homework Equations





The Attempt at a Solution


I don't actually think this is possible. My only other thought is to take A2*A where this product equals zero, and then decompose A. But I don't know how to do that such that it is the "square root" of A.

Try a 3x3 matrix with 1s above the diagonal.
 
  • #3
NullSpace0
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How would I have known to look for that?

From what I know about matrix multiplication, I know that the zeroes below the diagonal help when I think of multiplying by taking linear combinations of the columns, but how would I reason this out in general?
 
  • #4
Dustinsfl
2,285
5
How would I have known to look for that?

From what I know about matrix multiplication, I know that the zeroes below the diagonal help when I think of multiplying by taking linear combinations of the columns, but how would I reason this out in general?

Let A be an n xn matrix for which A^k = 0 for some k > n. Prove that A^n = 0.

This theorem tells me to work on nxn matrices since I know I know I can construct one that is A^n = 0. So your case we needed A^3. Hence I picked a 3x3
 
  • #5
Ray Vickson
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How would I have known to look for that?

From what I know about matrix multiplication, I know that the zeroes below the diagonal help when I think of multiplying by taking linear combinations of the columns, but how would I reason this out in general?

Re "How would I have known to look for that?" I'll bet your book or notes have some relevant material. Alternatively, you could Google 'nilpotent matrix', but I guess you would need to know that is what such A are called.

RGV
 

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