How do I find the resulting 4x4 matrix for the square of \eta?

[quixi]

Homework Statement

From a given $\left| \psi \right>$ I have calculated an expression for $\eta$ which results in a 4x4 matrix (as required) however I now need to find $\eta^{2}$ but don't know how to do this with still keeping a resulting 4x4 matrix.

Homework Equations

$$\eta = \alpha \left| 0000 \right> + \beta \left| 0001 \right> + ... + \theta \left| 1111 \right>$$ etc.

The Attempt at a Solution

I'm pretty sure I can't just do:

$$\eta^{2} = \alpha^{2} \left| 0000 \right> + \beta^{2} \left| 0001 \right> + ... + \theta^{2} \left| 1111 \right>$$ etc.

But if I do $\eta^{2}$ I'll end up with:

$$\eta^{2} = \alpha^{2} \left| 00000000 \right> + \beta^{2} \left| 00010001 \right> + ... + \theta^{2} \left| 11111111 \right>$$ etc.

which isn't any good, I need a 4x4 matrix not a 8x8 matrix.

Perhaps:

$$\eta^{2} = \eta \eta*$$

but I still don't think that would do.

Hmm.

 

[betel]

The eta you gave is no 4x4 matrix. It is vector in a 2^4 dimensional space.
If you are give $$\psi$$ then $$\eta$$ is defined as $$\eta = |\psi><\psi|$$. Then you can square the matrix which will result in having the scalar product between your basis vectors.