Eigenvalues of the Frenet formulas and angular velocity

[ForMyThunder]

So there's a circular helix parametrized by \vec x(t)=(a\cos(\alpha t), a\sin(\alpha t), bt) and you have the matrix K given in the Frenet-Serret formulas. In the book I'm reading it says that -\alpha^2 is the nonzero eigenvalue of K^2. Can someone explain how they know this is? I understand that you can compute the eigenvalues of the matrix to verify this but how can you say this without computation?

 

[lavinia]

So there's a circular helix parametrized by \vec x(t)=(a\cos(\alpha t), a\sin(\alpha t), bt) and you have the matrix K given in the Frenet-Serret formulas. In the book I'm reading it says that -\alpha^2 is the nonzero eigenvalue of K^2. Can someone explain how they know this is? I understand that you can compute the eigenvalues of the matrix to verify this but how can you say this without computation?

Can you show me why the eigen value is -alpha^2?

I get +-alpha^2/(a.alpha^2 + b^2)