# Eigenvalu problem

temaire
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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vishal007win
2..
is dis correct!

temaire
2..
is dis correct!

????

Mentor
Maybe, maybe not.

Mentor
What does it mean to say that a number lambda is an eigenvalue of a matrix?

vishal007win
it means that no. is the eigen value of new matrix formed by the expression..
A^2010 -2A+3I

vishal007win
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

temaire
How did you show it was 2?

Mentor
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?

vishal007win
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...

vishal007win
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?

Mentor
I'm looking for an equation that involves A and 3.

temaire
I'm looking for an equation that involves A and 3.

Is it:

(A - 3I)x = 0

Mentor
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

Mentor
Think about the other problem you posted. If Ax = 1x, what are A2x, A3x, A4x, ...?

temaire
All would be 1x

Mentor
OK, so what would (A2010 - 2A + 3I)x be?

temaire
Also 1x

Mentor
Also 1x
No it isn't.

temaire
No it isn't.

Whoops lol, I meant 2.

temaire
So would it be enough to say that since A=1, A^2010 - 2A + 3I = 1 -2 + 3 = 2?

Mentor
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 $\neq$ 2

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