How do you solve the Eigenvalue problem for your homework assignments?

[temaire]

Homework Statement
[PLAIN]http://img14.imageshack.us/img14/7826/70745131.jpg

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

 

[vishal007win]

2..
is dis correct!

 

[temaire]

2..
is dis correct!

?

 

[Mark44]

Maybe, maybe not.

 

[Mark44]

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?

 

[Mark44]

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?

hope this give answer to your question!

 

[Mark44]

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

 

[Mark44]

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

 

[Mark44]

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

 

[temaire]

All would be 1x

 

[Mark44]

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

 

[temaire]

Also 1x

 

[Mark44]

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?

 

[Mark44]

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.)