How Do You Solve for the Smallest Eigenvalue in a Quadratic Equation?

[amninder15]

Can someone help me with this question?

I know we have to set up the auxiliary equation and then solve for λ but for some reason I am not getting the right answer.

My equation is:

m^2 + 4m + (5λ + 3) = 0

then we get -2 ± sqrt(5λ-1)i

Now can somebody explain what I have to do after this?

 

[Mark44]

Can someone help me with this question?

I know we have to set up the auxiliary equation and then solve for λ but for some reason I am not getting the right answer.

My equation is:

m^2 + 4m + (5λ + 3) = 0

then we get -2 ± sqrt(5λ-1)i

Now can somebody explain what I have to do after this?

You are assuming that 1 - 5λ < 0 or IOW, that λ > 1/5. Are you given any information about λ?

I would break this into three cases:
λ < 1/5, implying two real values of m.
λ = 1/5, implying one real and repeated value of m; namely m = -2.
λ > 1/5, implying two complex values of m.

Since the question asks for the smallest eigenvalue, I take it that they are referring to the roots of the characteristic equation, or m. Also, since complex numbers aren't ordered (you can't compare them with < or >), I guess that limits things to the first two cases above.