Eigenvalue Problem

oomjos

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Nov 6, 2014
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6
Hello.

I have had some trouble solving the following problem.

| -2 β |
| 1 -3 |



The question asks to determine all values for β which A has distinct real eigenvalues.

This is what I have done so far.
|AI| = 0

| -2-λ β ....| = 0
| 1 -3-λ |


=>(-2-λ)(-3-λ) - β = 0

Now my thinking was that If β =0 => λ = -3, -2

And if β =6 => λ2+5λ=0
=> λ(λ+5)=0
=> λ=0,-5

That is all I can think of for now. But this does not seem right.
I am sure that I have not covered all the possible values for β.

If someone can point out to me how to get all the possible values for β it would be much appreciated.

Thanks
 
Managed to figure something out

So I did the following.

λ2 +5λ +6 - β = 0

Then I applied the quadratic formula.

gif.latex



Then I only looked at the discriminant [D = 25-2(6-β) ]

So for real and distinct eigenvalues
gif.latex

 
Hello.

I have had some trouble solving the following problem.

| -2 β |
| 1 -3 |



The question asks to determine all values for β which A has distinct real eigenvalues.

This is what I have done so far.
|AI| = 0

| -2-λ β ....| = 0
| 1 -3-λ |


=>(-2-λ)(-3-λ) - β = 0
Yes, this is correct so far but it might be better to multiply it out and get \(\displaystyle \lambda^2+ 5\lambda+ 6- \beta\).

Now my thinking was that If β =0 => λ = -3, -2

And if β =6 => λ2+5λ=0
=> λ(λ+5)=0
=> λ=0,-5

That is all I can think of for now. But this does not seem right.
I am sure that I have not covered all the possible values for β.
I don't understand why you are taking specific values for β. If you are working with matrices, eigenvalues, and differential equations, surely you must know the quadratic formula
roots of \(\displaystyle ax^2+ bx+ c= 0\) are \(\displaystyle x= \frac{-b\pm\sqrt{b^2- 4ac}}{2a}\) and, in particular, those roots will be distinct real roots if and only if the discriminant, \(\displaystyle b^2- 4ac\) is positive.

If someone can point out to me how to get all the possible values for β it would be much appreciated.

Thanks
 
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Solved

Ah yes I did not think of that.

Thank you for your reply.

I should be able to find all the values for β now.
 
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