Please help with this matrix question Solve for Eigenvales, and Eigenvectors

[sidinsky]

M = (a c)
(c b)

Sorry for the double sets of brackets, its all in one. I'll also show as far as i got below:

[a-λ c] => (a-λ)(b-λ) - c^2 = λ^2 + (-a-b)λ + (ab-c^2) =0
[c b-λ] =>

then using the quadratic formula: λ = [-(-a-b) +/- Sqrt{(-a-b)^2 - 4(1)(ab-c^2)}]/ 2

then after some algebra I got stuck: λ = [(a+b) +/- Sqrt{a^2 + 2ab + b^2 -4ac^2}]/2

λ = [(a+b) +/- Sqrt{a^2 - 2ab + b^2 + c^2}]/2 ==>> THIS IS WHERE I GOT STUCK. PLEASE HELP

 

[tiny-tim]

welcome to pf!

hi sidinsky! welcome to pf!

(have a square-root: √ and a ± and try using the X² button just above the Reply box )

λ = [(a+b) +/- Sqrt{a^2 - 2ab + b^2 + c^2}]/2

looks ok …

that gives you the two eigenvalues,

so now use the standard techniques to find an eigenvector for each ​

 

[sidinsky]

is there any way this expression can be simplified further? I am sort of having trouble with the algebra :S

 

[tiny-tim]

is there any way this expression can be simplified further?

i don't think so

how far have you got?​

 

[sidinsky]

well I managed to clean up the expression inside the sqrt a bit to: [(a-b)² + c ² ]/2

a² -2ab + b² + c² = (a-b)² + c²

λ₁ = (a+b) + sqrt{(a-b)² +c ²}$/2$

and λ₂ = (a+b) - sqrt{(a-b)² + c²}$/2$

now these expressions for Lambda are too difficult for me to use to solve for eigenvectors