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Eigenvalu problem

  • Thread starter temaire
  • Start date
  • #1
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Homework Statement
[PLAIN]http://img14.imageshack.us/img14/7826/70745131.jpg [Broken]

The attempt at a solution
How do I go about solving this problem?
 
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Answers and Replies

  • #2
2..
is dis correct!
 
  • #3
279
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  • #4
33,483
5,173
Maybe, maybe not.
 
  • #5
33,483
5,173
What does it mean to say that a number lambda is an eigenvalue of a matrix?
 
  • #6
it means that no. is the eigen value of new matrix formed by the expression..
A^2010 -2A+3I
 
  • #7
if for an eigen vector,eigen value is 1...dis means for that e-vector, matrix is behaving like an identity matrix...so the same eigen vector this expression will have eigen value :2
 
  • #8
279
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How did you show it was 2?
 
  • #9
33,483
5,173
it means that no. is the eigen value of new matrix formed by the expression..
A^2010 -2A+3I
You didn't answer my question. Here's a slightly different question. What does it mean to say that 3 is an eigenvalue of a matrix A?
 
  • #10
matrix can act in two ways ...rotating a vector or changing it length...
this value...(e-value) is a factor by which matrix changes the length of vector...
 
  • #11
You didn't answer my question. Here's a slightly different question. What does it mean to say that 3 is an eigenvalue of a matrix A?
hope this give answer to your question!
 
  • #12
33,483
5,173
I'm looking for an equation that involves A and 3.
 
  • #13
279
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I'm looking for an equation that involves A and 3.
Is it:

(A - 3I)x = 0
 
  • #14
33,483
5,173
Yes, or equivalently, Ax = 3x.

With your problem, you know that 1 is an eigenvalue of A, so Ax = 1x. You are trying to find an eigenvalue of A2010 - 2A + 3I.

(A2010 - 2A + 3I)x = ?x
 
  • #15
33,483
5,173
Think about the other problem you posted. If Ax = 1x, what are A2x, A3x, A4x, ...?
 
  • #16
279
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All would be 1x
 
  • #17
33,483
5,173
OK, so what would (A2010 - 2A + 3I)x be?
 
  • #18
279
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Also 1x
 
  • #19
33,483
5,173
  • #20
279
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  • #21
279
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So would it be enough to say that since A=1, A^2010 - 2A + 3I = 1 -2 + 3 = 2?
 
  • #22
33,483
5,173
So would it be enough to say that since A=1, A^2010 - 2A + 3I = 1 -2 + 3 = 2?
No, A is a matrix, so it can't be equal to any number, and A^2010 - 2A + 3I [itex]\neq[/itex] 2

However, you know that Ax = 1x, so (A2010 - 2A + 3I)x = ___x? (Fill in the blank.)
 

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