Is the Scalar Product of Stress Tensors in Cartesian Components Correct?

[The Alchemist]

Homework Statement

stress tensor in cartesian components.
$$\sigma$$ is the stress tensor.
$$e_i$$ are the basis vectors

Homework Equations

$$\sigma \cdot \sigma$$

The Attempt at a Solution

I tried to write out the components with a cartesian basis:
$$\sigma=\sigma_{ij} (e_i \otimes e_j)$$
But then I'm stuck on
$$\sigma \cdot \sigma = \sigma_{ij} (e_i \otimes e_j) \cdot \sigma_{ji} (e_j \otimes e_i)$$

How can that be a scalar, since it is the scalar product...

I have no idea if this is the right approach, should I explicit use the unit vectors e_i to emphasize the cartesian components?

Thanks in advance.

 

[The Alchemist]

Okay, I made my way through this.

$$\sigma_{ij} (e_i \otimes e_j) \cdot \sigma_{kl} (e_k \otimes e_l) = \sigma_{ij} \delta_{ik} \delta_{jl} \sigma_{kl} = \sigma_{ij}^2$$
This is indeed a scalar, since there is no tensor space to span.
The key was to create the kronecker deltas.

Thanks anyway.