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)