Linear Algebra -- Projection matrix question

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  • #1
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Homework Statement


Let A be an n×n matrix which has the property that A^2 =A.
(i) Write down the most general polynomial in A


Homework Equations




The Attempt at a Solution


My biggest problem is that I don't even understand what the question is asking

Is it just sum (alphaA^n)=0

but A^n=A

sum(alpha A)=0 ?

I know its not the equation to find the eigenvalues as that follows, and I'm fine with that

Av=pv where p are the eigenvalues, and v the corresponding eigenvectors
A^2v=Apv
Av=p(Av)
pv=p^2v
v(p)(p-1)=0 and hence p= 0 or 1

But I just don't understand the first bit

Many thanks in advance :)
 

Answers and Replies

  • #2
Simon Bridge
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The absolute most general polynomial, degree N, in some arbitrary A would be ##P_N(A)=\sum_{n=0}^N a_nA^n## wouldn't it?
What is special about this polynomial?
 
  • #3
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The absolute most general polynomial, degree N, in some arbitrary A would be ##P_N(A)=\sum_{n=0}^N a_nA^n## wouldn't it?
What is special about this polynomial?
Well since A is a projection matrix, surely that would imply than ##A^n=A## and hence ##P_N(A)=\sum_{n=0}^N a_nA^n## -> ##P_N(A)=\sum_{n=0}^N a_nA##
 
  • #4
Simon Bridge
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i.e. the most general polynomial is 1st order. Well done.
 
  • #5
LCKurtz
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Wouldn't you want to include ##A^0 = I##?
 
  • #6
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Wouldn't you want to include ##A^0 = I##?
Yes, yes I would ;) Thank you
 
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