Solve Eigenvector Equation: Prove Roots are Scalars

[student85]

Hi, this is actually for my general relativity class, but I thought I would get more help in the math section of the forums, since it involves very little physics, or even not at all.

Homework Statement

Take T_ab and S_ab to be the covariant components of two tensors. Consider the determinant equation for [itex]\lambda[/itex] :

| [tex]\lambda[/tex]T_ab - S_ab |= 0

Prove that the roots of this equation are scalars, making clear what you mean by scalar.

Homework Equations

The Attempt at a Solution

Well If I solve for the determinant I think I should get a quartic equation for the eigenvalues [itex]\lambda[/itex] of the form
[tex]\lambda[/tex]^4 + a₁[tex]\lambda[/tex]^3 + a₂[tex]\lambda[/tex]^2 + a₃[tex]\lambda[/tex] + a₄ = 0
Or not? Will I get an equation involving the components of the tensors T and S??
I just want to make sure I am understanding the question and I'm headed in the right path.
Any suggestions are greatly appreciated.

 

[bdforbes]

Maybe you need to show that the eigenvalues are dependent on T_ab and S_ab in such a way that makes them invariant under transformations to another reference frame. Is that how "scalar" is defined in GR?

There's probably a clever way to answer the question that won't involve writing out the equations in detail.

 

[student85]

I just don't know where to start. Do you suggest getting the determinant of the matrix and equaling that to 0? That will take so long. Is there some theorem or something? Anybody know? :S

 

[bdforbes]

Sorry I don't think I know enough to help. I'm only studying special relativity, so I don't know how to interpret this determinant equation.

 

[student85]

Yeah this is a pretty weird problem. One of my classmates is helping me now :)
Thanks anyway