Need Help On Eigenvalue and Eigenvector

[shermaine]

Hi Guys,

I have got some enquires for eigenvalue and eigenvector.

Consider the 1st matrix:

A = [ 1 2 3]
[ 0 5 6]
[ 0 6 5]

The characteristic polynomial is

det(A-λI) = [ 1-λ 2 3]
[ 0 5-λ 6]
[ 0 6 5-λ]

= (1-λ) [ (5-λ)^2 - 36]

The eigenvaules are D(λ) = 0 ---> λ1 = 1, λ2=-1 and λ3 =11
May i know how do we get the λ1 , λ2 and λ3? Can someone advise me?
Dont seem quadratic is working for this??


Matrix (2)

A = [3 5 3]
[0 4 6]
[0 0 1]

Is the characteristic polynomial: det(A - λI) = [ 3-λ 5 3]
[ 0 4-λ 6]
[ 0 0 1-λ]

= (3-λ)[(4-λ)(1-λ)- 0]?


Does the characteristic polynomial of matrix 2 also the same as Matrix (3)

A = [ 3 0 0]
[ 4 4 0]
[ 5 6 1]

Please advise. Thanks

 

[shermaine]

how to get the attached circled?

 

[Hayato]

Yes, they are the same.

Just solve the det(A - λI).

After you find the Eigenvalues, you need to find the Eigenvectors, by the method

(A - λi*I)*Vi = 0, where i varies from 1 to n; If you have 3 eigenvalues, then n = 3...

In the picture, you have a 3x3 matrix, so V is formed by 3 components (a,b,c) or (u,v,w)...